Book of Abstract of ADENA 2020 International Conference on … · 2020. 10. 22. · Book of Abstract of ADENA 2020 International Conference on Advances in Di erential Equations and

Book of Abstract of ADENA 2020International Conference on Advances in

Differential Equations and Numerical Analysis(ADENA2020)

Indian Institute of Technology Guwahati,October 12 - 15, 2020

Contents

1 Invited Speakers Talk 12Boundary Value Problems for Delay Differential Equations . . . . . . 13Runge–Kutta methods and B-series . . . . . . . . . . . . . . . . . . . . 14Invariant Domain Preserving Approximation of Nonlinear Conserva-

tion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16A Well-balanced Positivity-preserving Quasi-Lagrange Moving Mesh

DG Method for the Shallow Water Equations . . . . . . . . . . 17Dynamics of the Matryoshka Cavity Generated due to Impact of High-

speed Train of Microdrops on a Liquid Pool . . . . . . . . . . . 18Nondegenerate Solitons and Their Collisions in the Two Component

Manakov Nonlinear Partial Differential Equations . . . . . . . 19Development and Analysis of an Unconditional Stable Method for Acous-

tic Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . 20Computational Data Modelling: Methods and Applications . . . . . . 21Mathematical Models and Analytical Methods for the Hydroelastic Re-

sponses of a Very Large Floating Structure . . . . . . . . . . . 22Epidemiological Short-term Forecasting with Model Reduction of Para-

metric Compartmental Models . . . . . . . . . . . . . . . . . . 23Lax-Phillips Scattering Theory for Simple Wave Scattering . . . . . . 24PDEs and Optimal Control Problems in Domains with Highly Oscil-

lating Boundaries: Asymptotic Analysis . . . . . . . . . . . . . 25Derivation of Ray Equations of a Polytropic Gas from Fermat’s Principle 26Modeling of Fluid-poroelastic Structure Interaction . . . . . . . . . . . 27Semantic Technologies in a Decision Support System . . . . . . . . . . 28Recent Advances in Numerical Methods for Singular PDEs . . . . . . 29Stability of Time Discretizations for Semi-discrete High Order Schemes

for Time-dependent PDEs . . . . . . . . . . . . . . . . . . . . . 30The Numerical Solution of Time-fractional Initial-boundary Value Prob-

lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Stability of Nature-Inspired Algorithms Using Dynamical System Theory 32Godunov Type Solvers for Hyperbolic Systems Admitting . . . . . . . 33Turnpike Control and Deep Learning . . . . . . . . . . . . . . . . . . . 34Higher Order PDE Based Image Processing:Theory, Computation &

Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2 Minisymposia: Recent Trends in Computational Methods for Sin-gularly Perturbed Differential Equations 36

Revisiting the Slow Manifold of the Lorenz-Krishnamurthy Quintet . . 38Isogeometric Analysis for Singularly Perturbed Problems . . . . . . . . 39Global Accuracy for Singularly Perturbed Reaction-Diffusion Problems

with Non-smooth Data . . . . . . . . . . . . . . . . . . . . . . . 40FEM and SDFEM on Graded Meshes for a Problem with Characteristic

Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Solution of Singularly Perturbed Boundary Layer Problems . . . . . . 42Numerical Methods for Singularly Perturbed Delay Differential Equations 43New Finite Difference Schemes for a Class of Singularly Perturbed

Differential-Difference Equations . . . . . . . . . . . . . . . . . 44Numerical Method for a Weakly Coupled System of Singularly Per-

turbed Parabolic Convection-Diffusion Equations with Non-smoothSource Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Generation of Layer-adapted Meshes using Mesh PDE methods: aComputational Study . . . . . . . . . . . . . . . . . . . . . . . 48

Numerical Treatment of Singularly Perturbed Differential-DifferenceEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Parameter-uniform Convergence Analysis for Numerical Approxima-tion of System of Singularly Perturbed Time-dependent Convection-diffusion Problems . . . . . . . . . . . . . . . . . . . . . . . . . 52

Richardson Extrapolation Technique for 2D Singularly Perturbed De-lay Parabolic Partial Differential Equations . . . . . . . . . . . 53

The ADI type Operator Splitting SDFEM for the Singularly Perturbed2D Parabolic PDE . . . . . . . . . . . . . . . . . . . . . . . . . 54

A Second Order Finite Difference Scheme for a Class of SingularlyPerturbed Mixed Parabolic-elliptic Problem . . . . . . . . . . . 55

Discontinuous Galerkin Method with Interior Penalties for SingularlyPerturbed Convection-Diffusion Problems on EquidistributedMesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A Nonstandard Method for a Coupled System of Singularly PerturbedDelay Differential Equations . . . . . . . . . . . . . . . . . . . . 57

Hybrid Method for Two Parameter Singularly Perturbed Elliptic Bound-ary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . 58

Numerical Approximation of a Two-parameter Singularly PerturbedParabolic Weakly Coupled System with Discontinuous Convec-tion and Source Terms . . . . . . . . . . . . . . . . . . . . . . . 59

The Numerical Solution of Two-Dimensional Singularly Perturbed Convection-Diffusion Problems . . . . . . . . . . . . . . . . . . . . . . . . . 60

Fitted Special Finite Difference Scheme for Delay Differential Equationwith Dual Boundary Layers . . . . . . . . . . . . . . . . . . . . 61

An SDFEM for a Singularly Perturbed Fourth Order Ordinary Differ-ential Equation with Mixed Boundary Conditions . . . . . . . 62

Parameter-Uniform Numerical Method for Singularly Perturbed 2DElliptic Convection-Diffusion Problem with Boundary and In-terior Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Robust Numerical Method for a Partially Singularly Perturbed ParabolicSystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

The Bakhvalov Mesh: a History, Recent Results and Some Open Ques-tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Discrete Approximation for a Two Parameter Singularly PerturbedTwo Point Boundary Value Problem with a Turning Point . . . 66

High-Order Finite Difference Scheme for a System of Singularly Per-turbed Convection-Diffusion Equations . . . . . . . . . . . . . 67

Higher-Order Difference Scheme for Parabolic Singularly PerturbedProblem with Time delay . . . . . . . . . . . . . . . . . . . . . 68

Numerical Investigation for Time Delay Singularly Perturbed ParabolicProblems Involving Space Shifts . . . . . . . . . . . . . . . . . 69

Uniformly Convergent Computational Method for a System of Singu-larly Perturbed Parabolic Reaction-diffusion Initial BoundaryValue Problems Using Moving Mesh Refinement . . . . . . . . 70

Uniformly Convergent Quadratic B-spline Collocation Method for Sin-gularly Perturbed Parabolic Partial Differential Equation withTwo Small Parameters . . . . . . . . . . . . . . . . . . . . . . 71

Superconvergence Error Estimates of the NIPG Method for SingularlyPerturbed 2D Elliptic BVPs . . . . . . . . . . . . . . . . . . . . 72

Superconvergence Study of Galerkin FEM for Singularly PerturbedSystems with Multiple Scales . . . . . . . . . . . . . . . . . . . 73

Uniformly Convergent Numerical Scheme for Singularly Perturbed ParabolicPartial Differential Difference Equations . . . . . . . . . . . . . 74

Investigation of an Efficient Numerical Method for Singularly Per-turbed Time-dependent Convection-diffusion Problems with Non-smooth Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

A Finite Difference Scheme to Solve Convection-Reaction Equationwith a Delay Parameter in Convection and Reaction term . . . 76

Challenges and Advantages of Deep Learning for Solving SingularlyPerturbed Partial Differential Equations . . . . . . . . . . . . . 77

A Higher Order Numerical Method for Singularly Perturbed ParabolicConvection-diffusion Problem with Interior Turning Point . . . 78

A Finite Difference Scheme to Solve Convection-Reaction Equationwith a Delay Parameter in Convection and Reaction term . . . 79

3 Minisymposia: Advances in Computational Multiphase Flows 80Analysis of Particulate Flow Through Screw Conveyors . . . . . . . . 81Avalanche Flows of Grains on Minor Planets . . . . . . . . . . . . . . 82On the Effect of Interfacial Tension and Wettability on Pore-Scale

Two-phase Flow Mechanisms in a Three-Dimensional PorousMedium using Pore-Resolved Volume-of-Fluid Simulations . . . 83

Dynamics of Rayleigh Breakup of Charged Droplets . . . . . . . . . . 84Entropic Lattice Boltzmann Model . . . . . . . . . . . . . . . . . . . . 85Thin Film Drainage Between Bubbles Colliding with Bubbles and Solids 86Principles of Particle Technology and Energy Utilisation in Mineral,

Metallurgical and Energy Related Industries . . . . . . . . . . 87CFD Simulations of Bubble Column Bioreactors: Closure models, Nu-

merics and Validation . . . . . . . . . . . . . . . . . . . . . . . 88Chaotic Orbits of Ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . 89Particle Migration in a liquid-liquid Stratified Flow in a Microchannel 90A Consistent and Well-Balanced Immersed Boundary Method for In-

compressible Multi-Component Flows . . . . . . . . . . . . . . 91Control Mechanisms of Miscible Viscous Fingering Instability . . . . . 92Coalescence Dynamics of Unequal Sized Drops . . . . . . . . . . . . . 93Simulation of Blast Waves and its mitigation using Smoothed Particle

Hydrodynamics (SPH) . . . . . . . . . . . . . . . . . . . . . . . 94Turbulence Collapses at a Threshold Particle Loading in a Dilute Particle-

gas Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Secondary Atomization: Breakup of a Liquid Drop in a High Speed Flow 96Simulation of Droplet Impact at High Dynamics using a Diffuse Inter-

face Phase-Field Methods in OpenFOAM . . . . . . . . . . . . 97Thin Film Drainage Between Colliding Bubbles . . . . . . . . . . . . . 98Velocity and Thermal Slip Effects on MHD Convective Two Phase Flow

in an Asymmetric Convergent Channel . . . . . . . . . . . . . . 99

4 Minisymposia: Advances in Computational Science and ParallelComputing 100

Scalable Asynchronous Solvers for Partial Differential Equations . . . 101Parallel Algorithms for Optimal Vehicle Routing . . . . . . . . . . . . 102Parallel Algorithms for Centrality Computations . . . . . . . . . . . . 103Parallel and Space-time Adaptive Numerical Simulation of 3D Cardiac

Electrophysiology . . . . . . . . . . . . . . . . . . . . . . . . . 104A Scalable, Hybrid Fourier-Compact Finite-Difference Large Eddy Sim-

ulation Code for Wind Energy Applications . . . . . . . . . . 105Mixed-precision Subspace Iteration Algorithm for Large-scale Nonlin-

ear Eigenvalue Problems Towards Quantum-mechanical Mod-eling of Materials. . . . . . . . . . . . . . . . . . . . . . . . . . 106

Intra- and Interconnect Contention-aware Data Movement . . . . . . 107

Turbulent ows in Turbomachines: Role of High Performance Comput-ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Bulk-Synchronization Avoiding Algorithms in Graph Applications andIterative Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Higher Order and Power Efficient Numerical Methods for ModelingContinuum Phenomena on Modern Computer Architectures . 110

Prediction of Data Distributions Using Deep Learning . . . . . . . . . 111Parallel Numerical Solution of Bates PIDE for Pricing Options on a

GPU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5 Minisymposia: Delay, Functional and Dynamic Equations with Ap-plications 113

Qualitative Theory of Dynamic Equations on Time Scale . . . . . . . 114Bifurcation Results for Fractional Laplace Equation . . . . . . . . . . 115Fractional Differential Operators: Different definitions and various ap-

proaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Extended Fractional Derivative Operators and Applications . . . . . . 117Wavelet Preconditioned Method for the Numerical Solution of Stochas-

tic Differential Equations . . . . . . . . . . . . . . . . . . . . . 118Conditional and Unconditional Finite Difference Methods for a Cou-

pled System of Hyperbolic Delay Differential Equations . . . . 119Stability Analysis of a Spatially Coupled Model with Delayed Dispersal 120Existence of Hydra Effect in Unstructured Predator-prey Model . . . 121Stage Structured Prey Predator Model with Maturation and Gestation

Delay for Predator using Holling Type 2 Functional Resonse . 122

6 Minisymposia: Mathematical Aspects of Water Waves and Applica-tions 123

Bragg Scattering of Long Waves by an Array of Floating Flexible Platesin the Presence of Multiple Submerged Breakwaters . . . . . . 124

Localized Structures in Integrable Models . . . . . . . . . . . . . . . . 125Flexural Gravity Wave Blocking in Coupled Plate System . . . . . . . 126Linear Water Wave Interaction With a Composite Porous Structure in

a Two Layer Fluid Flowing over a Step Like Seabed . . . . . . 127Role of Linear Algebra in the Understanding of Wave Impact on Seawall

by a Pair of Asymmetrical Trenches . . . . . . . . . . . . . . . 128Effect of Flexible Porous Breakwater in Mitigating Hydroelastic Re-

sponses of a Very Large Floating Structure . . . . . . . . . . . 129Wave Scattering by a Floating Porous Breakwater over a Rectangular

Bottom-trench . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Wave Diffraction by a Small Base Deformation on a Flexible Bed in an

Ice-covered Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 131Water Wave Scattering by a Flexible Porous Plate in the Presence of

a Submerged Porous Structure . . . . . . . . . . . . . . . . . . 132Water Wave Radiation by an Immersed Cylinder in a Channel with

Flexible Base Surface . . . . . . . . . . . . . . . . . . . . . . . 133Wave Scattering by a Pair of Floating Pontoons having a Submerged

Mesh Cage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Expansion Formula for the Flexural-gravity Waves During Wave Blocking135Diffraction by an Oscillating Water column in Presence of Bottom-

mounted Obstacle in the Channel of Finite Width . . . . . . . 136Water Waves Interaction with Slatted Screens Placed Near a Caisson

Breakwater . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Water wave scattering by a porous breakwater floating over a rectan-

gular bottom-mound . . . . . . . . . . . . . . . . . . . . . . . . 138

7 Minisymposia: Modelling and Simulation of Flow and TransportProcesses in Porous Media 139

Modeling of Fluid-poroelastic Structure Interaction . . . . . . . . . . . 140Upscaling of Flow Models in a Fractured Porous Medium . . . . . . . 141A Two-scale Adaptive Scheme for Mineral Precipitation and Dissolu-

tion in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . 142Unsteady Flow in a Porous Media Considering Variable Permeability

and Local Thermal Non–equilibrium . . . . . . . . . . . . . . . 143Stability of Miscible Rayleigh-Taylor Fingers in Porous Media with

Non-monotonic Density Profiles . . . . . . . . . . . . . . . . . . 145Two-phase Flow Through Highly Heterogeneous Porous Media: Mod-

elling and Upscaling . . . . . . . . . . . . . . . . . . . . . . . . 146Differential Transform Method for Solving One Dimensional Heat Con-

duction Equation With Variable Coefficient . . . . . . . . . . . 147

8 Minisymposia: Recent Advances in the Analysis and Developmentof Numerical Methods for Nonlinear Problems 148

Numerical Solutions of Some Non-Linear Evolutionary Equations byB-Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Iterative Mthods for Solving Equations: Convergence and Dynamics . 150Basic Iterative Methods for Solving Nonlinear Equations and Their

Extensions for Systems . . . . . . . . . . . . . . . . . . . . . . 151On the Semilocal Convergence of a Computationally Efficient Fifth-

Order Method in Banach Spaces under Relaxed Condition . . . 152Multipoint Iterative Methods without Memory for Nonlinear Equation

and its Applications . . . . . . . . . . . . . . . . . . . . . . . . 153Introduction to Numerical Methods and its Applications . . . . . . . . 154Matrix Iterative Methods for Computing Generalized Outer Inverses . 155Convergence of Newton’s Method and its Higher Order Variants in

Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 156A Monotone Domain-Decomposition Technique for Singularly Perturbed

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Numerical Techniques for Solving Systems of Nonlinear Equations and

their Applications . . . . . . . . . . . . . . . . . . . . . . . . . 158The Impact of Aaverage Flow on Forward Sites in Lattice Hydrody-

namic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Method for Solving Nonlinear Multiobjective Integer Optimization Prob-

lem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Optimal Iterative Scheme for Finding Multiple Zeros of Scalar Nonlin-

ear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Studying the Effect of Extraneous Fixed Points on Dynamical Behavior

of Iterative Methods via Basins of Attraction . . . . . . . . . . 162The Role of Information Technology (IT) in Vehicular Dynamics . . . 163Generalized Local Projection Stabilization for Galerkin Approxima-

tions of Darcy and Stokes Problems . . . . . . . . . . . . . . . 164On Complex Dynamics of Some Third Order Methods for Computing

Multiple Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 165The Study of Convergence and Dynamics Analysis of an Iterative Method

Under Weak Conditions . . . . . . . . . . . . . . . . . . . . . . 166A Numerical Approach to Solve Non-Linear Systems With Higher Or-

der of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 167

9 Minisymposia: Stability of Nonlinear Dynamical Systems 168An Invitation to Control Theory of Stochastic Distributed Parameter

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Stability and Stabilizability of Fractional Dynamical Systems . . . . . 170Geometrical Methods for Modeling Complex Systems . . . . . . . . . . 171Stabilization of Burgers Equation by Boundary Control . . . . . . . . 172Mathematical Modeling COVID-19: Forecasting and Analyzing the

Dynamics of the Outbreak in India . . . . . . . . . . . . . . . . 173Robust Feedback Control of Nonlinear PDEs by Numerical Approxi-

mation of High-dimensional Hamilton-Jacobi-Isaacs Equations 174Impulsive Complex-valued Stochastic BAM Neural Networks with Leak-

age and Mixed Time Delays: An Exponential Stability Problem 175Adaptive Controllability of Chaos Generated in Generalized Lotka-

Volterra Biological Model using Projective Combination Dif-ference Synchronization Technique . . . . . . . . . . . . . . . . 176

Impulsive Effects on Complex-valued Neural Networks with Time De-lays: An Asymptotic Stability Issue . . . . . . . . . . . . . . . 177

Non-fragile Event-triggered Control Design for Fuzzy Systems with Ac-tuator Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . 178

FDI Attacks on Various Dynamical Systems: The Known to be Explored179Robust Stability of Fuzzy Genetic Regulatory Networks with Marko-

vian Jumping Parameters . . . . . . . . . . . . . . . . . . . . . 180Synchronization Analysis for a Network of Dynamical Systems Through

Composite Control Approach . . . . . . . . . . . . . . . . . . . 181Mathematical Modelling of Effects of COVID-19 in Poverty . . . . . . 182Jacobi Stability Analysis of Prey-Predator Model . . . . . . . . . . . . 183Water Resource Sharing and its Effects:A Mathematical Modelling Ap-

proach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Modelling Dynamics of Violence Against Women . . . . . . . . . . . . 185Sliding Mode Boundary Control Design for Parabolic Systems . . . . . 186Stochastic Model for the Co-existence of Diabetes and COVID-19 . . . 187

10 Minisymposia: The Use of Block Methods for Solving DifferentialProblems 188

On the Use of Block Methods for Solving Singular Boundary ValueProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Extended Use of One-step Hybrid Block Method on Partial DifferentialEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Adapted Hybrid Second Derivative Block Method for Initial ValueProblems with Oscillating Solutions . . . . . . . . . . . . . . . 191

An Efficient Optimized Hybrid Block Method for Integrating InitialValue Ordinary Differential Systems . . . . . . . . . . . . . . . 192

Block Hybrid Method for the Numerical solution of Fourth order Bound-ary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . 193

An Adaptive Fourth-Derivative Hybrid Block Strategy for IntegratingThird-order IVPs with Applications to Problems in Fluid Dy-namics and Engineering . . . . . . . . . . . . . . . . . . . . . . 194

A High Order Block Hybrid Method for Integrating Oscillatory GeneralSecond Order Initial Value Problems . . . . . . . . . . . . . . 195

Implicit Three-Point Block Numerical Algorithm for Solving Third Or-der Initial Value Problem Directly with Applications . . . . . . 196

Multistep Block Method for Solving Neutral Delay Differential Equations197Solving Differential Systems Y′ = F(x,Y) by Using a Novel Two Pa-

rameter Class of Optimized Hybrid Block Methods . . . . . . . 198

Variable Step-Size Formulation of a Three-Point Fourth-Derivative Hy-brid Block Method for Solving Third-Order IVPs . . . . . . . . 199

11 Minisymposia: Young Researchers in Numerics for EvolutionaryProblems 200

Geometric Analysis of a Phantom Bursting Model . . . . . . . . . . . 201Numerical Stability Analysis of Linear Periodic Delay Equations via

Pseudospectral Methods . . . . . . . . . . . . . . . . . . . . . . 202Convergence of the Piecewise Orthogonal Collocation for Periodic So-

lutions of Retarded Functional Differential Equations . . . . . 203Highly Stable Multivalue Almost Collocation Methods with Structured

Coefficient Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 204Nonreflecting Boundary Conditions for a CSF model of the Fourth

Ventricle - Spinal SAS Dynamics . . . . . . . . . . . . . . . . . 205Virtual Element Method for Bulk-surface Reaction-diffusion Systems

with Electrochemical Applications . . . . . . . . . . . . . . . . 206Numerical Analysis of 1D Blood Flow Coupled with a New Pressure

Area Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 207Perturbative Analysis of the Discretization to Stochastic Hamiltonian

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208Nonlinear Stability Issues in Stochastic Discretizations . . . . . . . . . 209Numerical Preservation Issues for Nonlinear Stochastic Oscillators . . 210Convergence of the Piecewise Orthogonal Collocation for Periodic So-

lutions of Retarded Functional Differential Equations . . . . . . 211Numerical Stability Analysis of Linear Periodic Delay Equations via

Pseudospectral Methods . . . . . . . . . . . . . . . . . . . . . . 212Numerical Preservation Issues for Nonlinear Stochastic Oscillators . . 213Nonreflecting Boundary Conditions for a CSF Model of the Fourth

Ventricle - Spinal SAS Dynamics . . . . . . . . . . . . . . . . . 214Highly Stable Multivalue Almost Collocation Methods with Structured

Coefficient Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 215Perturbative Analysis of the Discretization to Stochastic Hamiltonian

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216Virtual Element Method for Bulk-Surface Reaction-Diffusion Systems

with Electrochemical Applications . . . . . . . . . . . . . . . . 217Geometric Analysis of a Phantom Bursting Model . . . . . . . . . . . 218

12 General Contributed Talk 219Pointwise Error Estimation for Nonhomogeneous Dirichlet Boundary

Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 220Fourier Integral Representation of the R-Function . . . . . . . . . . . 221A New Approximation for Conformable Time Fractional Nonlinear De-

layed Differential Equations via Two Efficient Methods . . . . 222Variable Order Nonlocal Choquard Problem with Variable Exponents 223Approximate Controllability of Semilinear Hilfer Fractional Differential

Equation with Control in the Nonlinear Term . . . . . . . . . . 224Unsteady Flow of Thin Liquid Film Over a Heated Stretching Sheet in

Presence of Uniform Transverse Magnetic Field and ThermalRadiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Numerical Study of Errors Obtained by Combining the Shooting Methodwith the FOM Method in Solving Boundary Value Problems . 226

New Fourth-Order Efficient Numerical Solutions of Heat-Like Equationand Klein-Gordon Equation . . . . . . . . . . . . . . . . . . . . 227

PDE Optimal Control Problems: An Automatic Differentiation Approach228

The Lumped Mass Finite Element Method for Fractional ParabolicIntegro-Differential Equations . . . . . . . . . . . . . . . . . . . 229

An Improvement of Third Order WENO Scheme for Convergence Rateat Critical Points with New Non-linear Weights . . . . . . . . . 230

Port-Hamiltonian Control Approach for CRTBP with Non-ideal Solar-sail and Albedo Effect . . . . . . . . . . . . . . . . . . . . . . . 231

An Exponentially Fitted Numerical Algorithm for a Boundary ValueProblem of Singularly Perturbed Delay Differential Equation . 232

Exponential Spline Method for Two-Point Singularly Perturbed Differential-Difference Equations . . . . . . . . . . . . . . . . . . . . . . . 233

Wave Scattering by a Submerged Circular Porous Membrane . . . . . 234A New Hybrid Cubic B-spline Differential Quadrature Method for a

Class of Convection–diffusion Equations in Extended Domains 235Numerical Methods for Modelling an Oscillating Water Column on a

Porous Sloping Ocean Bed . . . . . . . . . . . . . . . . . . . . . 236Boundary Control Design for Parabolic Systems Using Backstepping

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237Design and Analysis of a Numerical Method for Fractional Neutron

Diffusion Equation with Delayed Neutrons . . . . . . . . . . . . 238Natural Cubic Spline for Singular Boundary Value Problems . . . . . 239Mathematical Modeling of COVID-19 Transmission: The Roles of In-

tervention Strategies and Tockdown . . . . . . . . . . . . . . . 240Output Feedback Boundary Controller for an ODE Coupled to a First

Order Hyperbolic PDE . . . . . . . . . . . . . . . . . . . . . . 241A New Approach for Solving Third-order Partial Differential Equations

via Discrete Spline Technique . . . . . . . . . . . . . . . . . . . 242Hybrid Approach to Find Approximate Analytical Solution of Burger’s

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243Numerical Technique for Singularly Perturbed Delay Differential Equa-

tions Using Gaussion Quadrature . . . . . . . . . . . . . . . . . 244Surfactant-laden Newtonian Falling Film Down a Wavy Channel: Lin-

ear and Nonlinear Stability Analysis . . . . . . . . . . . . . . . 245Long Wave Stability Analysis of a Film Flow in the Presence of an

Insoluble Surfactant . . . . . . . . . . . . . . . . . . . . . . . . 246Wave Scattering by a Pair of Floating Pontoons Having a Submerged

Mesh Cage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247Sonofragmentation of Rectangular Plate-like Crystals: Bivariate Pop-

ulation Balance Modeling and Experimental Validation . . . . 248Lie Group Transformation Analysis of Fractional Partial Differential

Equations with Variable Coefficients . . . . . . . . . . . . . . . 249Analysis of Reflection of Wave at the Free Boundary of a Flexoelectric

Microstructured Half-space . . . . . . . . . . . . . . . . . . . . 250Numerical Approach for Solutions of Delay Differential Equations . . . 251Mathematical Study of Reflection of Plane Waves From the Stress-

free/rigid Surface of a Micro-mechanically Modeled Piezoelec-tric Fiber-Reinforced Composite Half-space . . . . . . . . . . . 252

Mathematical Analysis of Surface Wave Velocity in a Rotating BeddedStructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Numerical Approximation of Time Fractional Telegraph Equation byan RBF Based Meshless Method . . . . . . . . . . . . . . . . . 254

Mathematical Modeling of Elastic Plastic Transitional Stresses in Hu-man Tooth Enamel and Dentine under Pressure using SethsTransition Theory . . . . . . . . . . . . . . . . . . . . . . . . . 255

Impact of Inclined Magnetic Field on Mixed Convection of Nanofluid ina Lid-driven Square Enclosure with Different Heater LocationsUsing Buingirno’s Two-phase Model . . . . . . . . . . . . . . . 256

Radial Miscible Viscous Fingering Induced by an Infinitely Fast Chem-ical Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

Heterogeneous Multiscale Method in Drug Delivery . . . . . . . . . . . 258A Particular Soliton-type Analytic Solution and Computational Mod-

eling of Non-linear Regularized Long Wave Model . . . . . . . 259Water Wave Scattering and Energy Dissipation by Interface Piercing

Porous Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260Substructuring Waveform Relaxation Methods with Time-dependent

Relaxation Parameter . . . . . . . . . . . . . . . . . . . . . . . 261Role of CD8-cells in a HIV-immune Cell System Through Dynamical

Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262Dynamical Analysis of A Mathematical Model with Incubation Time

Delay for COVID-19 . . . . . . . . . . . . . . . . . . . . . . . . 263A New Approach Based on Exponential B-spline Collocation Method

for Solving a Class of Nonlinear Singular Boundary Value Prob-lems with Neumann and Robin Boundary Conditions . . . . . 264

Mathematical Modelling of COVID-19 with the Effects of Quarantineand Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

MHD Natural Convective Flow of a Polar Fluid with Newtonian HeatTransfer in Vertical Concentric Annuli . . . . . . . . . . . . . . 266

A Fourth-order Robust Numerical Method for Fuzzy Integro-differentialEquations under Generalized Differentiability . . . . . . . . . . 267

Bianchi type I Cosmological Model with Ghost Dark Energy in LyraGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

Fractional Order Modeling for Dynamics of Crime Transmission . . . . 269Numerical Solution of Non-linear DPL Model for Analyzing Heat Trans-

fer in Tissue During Thermal Therapy . . . . . . . . . . . . . . 270Bi-variate Extension of an Operator Based on Multivariate q-Lagrange

Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271The Behaviour of Creep in Cylinder and Disc . . . . . . . . . . . . . . 272A Cubic B-Spline Quasi-interpolation Method for Solving Hyperbolic

Partial Differential Equations . . . . . . . . . . . . . . . . . . . 273Toxicity Effect upon the Phytoplankton and Zooplankton Model . . . 274Numerical Simulation of 1D and 2D Multi-term Fractional Wave Model

with Non-linear Source Term . . . . . . . . . . . . . . . . . . . 275A Posteriori Error Analysis of Finite Element Method for Parabolic

Boundary Control Problems: A Reconstruction Approach . . . 276A Finite Element Method for an Elliptic Optimal Control Problem

with Integral State Constraints . . . . . . . . . . . . . . . . . . 277A Stabilized Finite Element Formulation for Numerical Simulation of

Convection-dominated Reactive Models . . . . . . . . . . . . . 278Multi-layer Perceptron Artificial Neural Network Approach for Solving

Sixth-Order Two-Point Boundary Value Problems . . . . . . . 279Acoustic-gravity Wave (AGW) Propagation Under Sea-ice in an Ocean

Having Elastic Floor . . . . . . . . . . . . . . . . . . . . . . . . 280The Performance of Squeeze Film Conical Bearing in Presence of Porous

Wall with Viscosity Variation:Rabinowitsch Fluid Model . . . . 281Numerical Solution of DPL Bio-heat Transfer Model Amidst Hyper-

thermia Treatment . . . . . . . . . . . . . . . . . . . . . . . . . 282A Lobatto Mixed Quadrature of Precision Eleven for Numerical Inte-

gration of Analytic Functions . . . . . . . . . . . . . . . . . . . 283

Numerical Solution of a Time Fractional Mixed Reaction-convectionDiffusion Problem Involving Weak Singularity . . . . . . . . . . 284

Weak Galerkin Finite Element Methods for Parabolic Problems withL2 Initial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Radiative MHD Casson Nanofluid Flow and Heat and Mass Transferpast on Nonlinear Stretching Surface considering Viscous Dis-sipation, Chemical Reaction and Heat Source . . . . . . . . . . 286

Haar Wavelets Collocation Method for a Class of System of Lane-Emden Equations with Four Point Boundary Conditions . . . 287

Water Wave Interaction with a Cylindrical Storage Tank in FiniteOcean Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

Water Wave Scattering by a Pair of Submerged Vertical Porous Barriersover Porous Sea-bed . . . . . . . . . . . . . . . . . . . . . . . . 289

Mathematical Studies on Ureter Smooth Muscle: Modeling Ion Chan-nels and Their Role in Generating Electrical Activity . . . . . . 290

On Solutions of The Fractional Integral Equation . . . . . . . . . . . 291Monotone Iterative Technique for Nonlinear Four Point Neumann BVPs

with Non Well Ordered Upper and Lower Solutions . . . . . . 292On the Stability of Mickens’ Type Nonstandard Finite Difference Schemes

for the Advection Diffusion Reaction Equation . . . . . . . . . 293Analytical Approximate Solution of the Mathematical Models of Effect

of Chemotherapy upon Glioblastoma Tumor Cells Growth inHomogeneous Medium using KVIM . . . . . . . . . . . . . . . 294

A Review of Jaulent-Miodek Hierarchy . . . . . . . . . . . . . . . . . . 295A Finite Element Method for the Equation of Motion Arising in Ol-

droyd Model of Order One with Grad-Div Stabilization . . . . 296Convergence Analysis of Dirichlet-Neumann and Neumann-Neumann

Algorithms for Cahn-Hilliard Equation . . . . . . . . . . . . . . 297Non-Self-Adjoint Eigenvalue Problem for Wave Propagation in Optical

Bent Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . 298On Summation-Integral Type Operators with Their Quantitative Means299On Weakly L-Stable Time Integration Formula with an Application to

Non-Linear Parabolic Partial Differential Equations . . . . . . 300Mathematical Modeling and Numerical Simulation of a Time-Fractional

Porous Medium Equation Arising in Fluid Flow through PorousMedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Application of Chebyshev Polynomial for Numerical Approximation ofSome Real-Life Singular Differential Equation . . . . . . . . . . 302

Nonlinear Impulsive Dynamic Initial Value Problems with NonlocalConditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Dynamics of an Eco-Epidemiological Model with Nonlinear IncidenceRate and Fear Effect . . . . . . . . . . . . . . . . . . . . . . . . 304

Solution of a Differential Equation Using Fixed Point Results in Graph-ical Rectangular Metric Space . . . . . . . . . . . . . . . . . . . 305

Trajectory Controllability of Fractional Dynamical Systems - A Survey 306Natural Convection in MHD Flow Past a Vertical Plate in Presence of

Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . 307The Solution of Volterra Integro-Differential Equation of Second Kind

Using Shehu Transform . . . . . . . . . . . . . . . . . . . . . . 308Numerical Technique for Singularly Perturbed Delay Differential Equa-

tions Using Gaussion Quadrature . . . . . . . . . . . . . . . . . 309Mixed Convection in Four-sided LID-Driven Sinusoidall Heated Porous

Cavity Using Stream Function-Vorticity Formulation . . . . . . 310

Surfactant-laden Newtonian Falling Film Down a Wavy Channel: Lin-ear and Nonlinear Stability Analysis . . . . . . . . . . . . . . . 311

Long Wave Stability Analysis of a Film Flow in the Presence of anInsoluble Surfactant . . . . . . . . . . . . . . . . . . . . . . . . 312

New Approximate Solutions of Time Fractional Klein Fock GordonEquation via HPJTM . . . . . . . . . . . . . . . . . . . . . . . 313

Numerical Solution of System of Nonlinear Equations Associated wihOrdinary Differential Equations . . . . . . . . . . . . . . . . . . 314

Impact of Magnetic Field on Polar Fluid with Newtonian Heat Transferin Vertical Concentric Annuli . . . . . . . . . . . . . . . . . . . 315

A Haar Scale-3 Wavelet Technique for Solving a MEMS Based Frac-tional Differential Equation . . . . . . . . . . . . . . . . . . . . 316

Comparison of Shooting Technique for Boundary Value Problems usingDifferent Runge-kutta Methods . . . . . . . . . . . . . . . . . . 317

Optimization Free Neural Network Approach for Solving Ordinary andPartial Differential Equations . . . . . . . . . . . . . . . . . . 318

A Study on Fractal Theory and its Application in Mathematical Physics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Construction of Bivariate Fractal Interpolation Function and its Frac-tional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

Stability of Double diffusive mixed Convective Flow in Vertical PipeFilled With Porous Medium under LTNE Model . . . . . . . . 321

A New Mathematical Technique for the Analytical Treatment of Dif-ferential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 322

On the Approximate Solutions of a Class of Fractional Order Non-linear Volterra Integro-Differential Initial Value Problems andBoundary Value Problems of First Kind and their ConvergenceAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Stability of Double Diffusive Mixed Convective Flow in Vertical PipeFilled With Porous Medium Under LTNE Model . . . . . . . . 324

Spectral Study of reiner Philippof Fluid Flow from an Inclined Stretch-ing Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

1 Invited Speakers Talk

International Conference on Advances in Differential Equations and Numerical Analysis(ADENA2020)

Indian Institute of Technology Guwahati, October 12 - 15, 2020

Boundary Value Problems for Delay Differential Equations

Ravi P Agarwal,Texas and M–Kingsville,Texas, USA

ABSTRACT

We develop an upper and lower solution method for second order boundary valueproblems for nonlinear delay differential equations on an infinite interval. Sufficientconditions are imposed on the nonlinear term which guarantee the existence of asolution between a pair of lower and upper solutions, and triple solutions betweentwo pairs of upper and lower solutions. An extra feature of our existence theory isthat the obtained solutions may be unbounded. Two examples which show how easilyour existence theory can be applied in practice are also illustrated.

c© Abstracts of ADENA2020



Runge–Kutta methods and B-series

John Butcher University of Auckland, Auckland, New Zealand,[email protected]

ABSTRACT

In 1895 an important discovery was made [1]. It became possible to obtain secondorder Runge–Kutta methods. A few years later Heun [2] and Kutta [3] raised theorder to 3 and 4 and eventually to 5 [4] and 6 [5].

A Runge–Kutta method for a scalar initial value problem

y′(x) = f(x, y(x)), y(x0) = y0, (1)

computes y1 ≈ y(x0 + h) by first computing Yi ≈ y(x0 + hci), i = 1, 2, . . . , s, These,and y1 itself are found in turn from the formulae

Yi = y0 + h∑j<i

aijf(x0 + hcj , Yj), i = 1, 2, . . . , s,

y1 = y0 + h

s∑i=1

bif(x0 + hci, Yi)

The parameters in these formulae are chosen to get a good approximation. For con-venience, they are often arranged in a tableau as follows

c AbT

=

0c2 a21

c3 a31 a32

......

.... . .

cs as1 as2 · · · as,s−1

b1 b2 · · · bs−1 bs

The number of stages is s and the number of free parameters is S := 12s(s+ 1). The

aim of Runge, and the people who followed him, was to make the order p of themethod as high as possible because this gives greater efficiency if a high accuracy isrequired.

Two famous examples are

0

1 112

12

(Runge, s = p = 2)

012

12

12 0 1

216

13

13

16

(Kutta, s = p = 4)


The number of conditions for order p will be written as Q(p). We have benfollowing the lead of Runge and the other the pioneers, and considering a singlescalar problem. However, for a vector-valued problem of arbitrary dimension, therecan be more than Q(p) conditions for order p. Denote the number of conditions inthe vector case by P (p).

Some of this information is shown in Table 1.

Table 1: Order information for a Runge–Kutta method

s 1 2 3 4 5 6 7 8

S(s) 1 3 6 10 15 21 28 36

p 1 2 3 4 5 6 7 8

Q(p) 1 2 4 8 16 31 ? ?

P (p) 1 2 4 8 17 37 85 200

It was once assumed that if S(s) < Q(p) then it is not possible to obtain an orderp method with just s stages. But this is not true.

The value of P (p) is now known to be equal to the number of rooted trees with upto p vertices. To illustrate what trees look like, the 8 trees up to order 4 are shownin Table 2

Table 2: Rooted trees to order 4

The analysis of order is a very simple idea: simply compare the Taylor series forthe solution to the initial value problem in powers of h tand the Taylor series forhe numerical approximation also in powers of h. If the series are identical to withinO(hp+1) then the order is p.

The theory of B-series provides the missing details. The key fact is that each ofthe series we need to compare, can be written in terms which exactly correspond totrees.

References

1. Runge, C.: Uber die numerische Auflosung von Differentialgleichungen. Math.Ann. 46, 167–178 (1895)

2. Heun, K.: Neue Methode zur approximativen Integration der Differentialgleichungeneiner unabhangigen Veranderlichen. Z. Math. Phys. 45, 23–38 (1900)

3. Kutta, W.: Beitrag zur naherungsweisen Integration totaler Differentialgleichungen.Z. Math. Phys. 46, 435–453 (1901)

4. Nystrom, E.J.: Uber die numerische Integration von Differentialgleichungen. ActaSoc. Sci. Fennicae 50 (13), 1–55 (1925)

5. Huta, A.: Une amelioration de la methode de Runge–Kutta–Nystrom pour laresolution numerique des equations differentielles du premier ordre. Acta Fac.Nat. Univ. Comenian. Math. 1, 201–224 (1956)



Invariant Domain Preserving Approximation of Nonlinear ConservationEquations

Prof. Jean-Luc Guermond,Texas A & M University, College Station, USA

ABSTRACT

The objective of this talk is to present a fully-discrete approximation technique for thecompressible Navier-Stokes equations. The method is implicit explicit, second-orderaccurate in time and space,and guaranteed to be invariant domain preserving. Therestriction on the time-step size is the standard hyperbolic CFL condition. To thebest of our knowledge, this method is the first one that is guaranteed to be invariantdomain preserving under the standard hyperbolic CFL condition and be second-orderaccurate in time and space.

Of course there are countless papers in the literature describing techniques to ap-proximate the time-dependent compressible Navier-Stokes equations, but there arevery few papers establishing invariant domain properties. Among the latest resultsin this direction we refer the reader to Grapsas, Herbin, Kheriji, Latche (2016) wherea first-order method using upwinding and staggered grid is developed (see Eq. (3.1)therein). The authors prove positivity of the density and the internal energy (Lem. 4.4therein). Unconditional stability is obtained by solving a nonlinear system involvingthe mass conservation equation and the internal energy equation. One importantaspect of this method is that it is robust in the low Mach regime. A similar tech-nique is developed in Gallouet, Gastaldo, Herbin, Latche (2008) for the compressiblebarotropic Navier-Stokes equations (see §3.6 therein). We also refer to Zhang (2017)where a fully explicit dG scheme is proposed with positivity on the internal energyenforced by limiting. The invariant domain properties are proved there under theparabolic time step restriction.

The key idea of the present talk is to build on Guermond, Nazarov, Popov, Tomas(2019), Guermond, Popov, Tomas (2019) and use an operator splitting technique totreat separately the hyperbolic part and the parabolic part of the problem. The hy-perbolic sub-step is treated explicitly and the parabolic sub-step is treated implicitly.This idea is not new and we refer for instance to Demkowicz et al. (1990) for anearly attempt in this direction. The novelty of our approach is that each sub-step isguaranteed to be invariant domain preserving. In addition, the scheme is conservativeand fully-computable (e.g. the method is fully-discrete and there are no open-endedquestions regarding the solvability of the sub-problems). One key ingredient of ourmethod is that the parabolic sub-step is reformulated in terms of the velocity andthe internal energy in a way that makes the method conservative, invariant domainpreserving, and second-order accurate.




A Well-balanced Positivity-preserving Quasi-Lagrange Moving Mesh DGMethod for the Shallow Water Equations

Weizhang Huang, Department of Mathematics, University of Kansas, Lawrence,Kansas 66045, USA. E-mail: [email protected]

Jianxian Qiu, School of Mathematical Sciences and Fujian Provincial KeyLaboratory of Mathematical Modeling and High-Performance Scientific Computing,

Xiamen University, Xiamen, Fujian 361005, China. E-mail: [email protected] Zhang, School of Mathematical Sciences, Xiamen University, Xiamen, Fujian

361005, China

ABSTRACT

In this talk we will present a high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method for the numerical solution of the shallow waterequations with non-flat bottom topography. The well-balance property is crucial tothe ability of a scheme to simulate perturbation waves over the lake-at-rest steadystate such as waves on a lake or tsunami waves in the deep ocean. The methodcombines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstructiontechnique, and a change of unknown variables. We will discuss the strategies touse slope limiting, positivity-preservation limiting, and change of variables to ensurethe well-balance and positivity-preserving properties. Compared to rezoning-typemethods, the current method treats mesh movement continuously in time and has theadvantages that it does not need to interpolate flow variables from the old mesh tothe new one and places no constraint for the choice of a update scheme for the bottomtopography on the new mesh. A selection of one- and two-dimensional examples arepresented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to featuresin the flow and bottom topography.




Dynamics of the Matryoshka Cavity Generated due to Impact ofHigh-speed Train of Microdrops on a Liquid Pool

Gautam BiswasProfessor and JC Bose National Fellow

Department of Mechanical Engineering, Indian Institute of Technology Kanpur,Kanpur-208016, UP, India

ABSTRACT

When a drop of a liquid impacts on the liquid-air interface of a liquid pool, depend-ing on the size and velocity of the drop, it may coalesce partiallyor completely [1].Based on the shape of the crater and its expansion and contraction time, the finaloutcome can be coalescence, jet formation withor without bubble entrapment andsplashing.Speirset al. [2] demonstrated formation of long slender cavities due to mul-tiple drop impact on a deep liquid pool.Bouwhuis et al. [3] studied the same eventfor microdroplets impacting with frequencies in the range of 10− 30 kHz.

Tongue shaped cavities are seen during the hydrophobic sphere impact, jet im-pact, and impact of a train of microdrops on a deep liquid pool [4]. For the impactof multiple microdrops, the mechanisms, which lead to deep cavity formation andlater bubble entrapment inside the liquid pool, are presented in this work. A train ofhigh-speed microdrops impacting on a liquid pool can create a very deep and narrowcavity, leading to depths more than several hundred times the size of the individual-drops.Seemingly the deep cavity is agglomeration of matryoshka cavities, named afterthe Russiannesting dolls. We analyzed these nested cavities (matryoshka cavities)created by multi-droplet impacts.The investigations were performed in an airwatersystem at large values of Froude numbers, thus having a negligible effect of gravity.Depending on the train length, the capillary wave generating from each drop impactaffects the necking. The temporal variation of the neck radius reveals a power lawbehavior. Pinch-off is observed when the penetration depth of the cavity is more thanthree times the diameter of the cavity.References:[1]. B. Ray, G. Biswas and A. Sharma, Regimes during liquid drop impact on a liquidpool, Journal of Fluid Mechanics, Vol. 768, pp. 492-523, (2015).[2]. N. B. Speirs, Z. Pan, J. Beldenand T. T. Truscott, The water entry of multi-droplet streams and jets, Journal of Fluid Mechanics, Vol. 844, pp. 1084-1111,(2018).[3]. W. Bouwhuis, X. Huang, C.U. Chan, P.E.Frommhold, C.-D. Ohl, D. Lohse, D.,J.H. Snoeuer, and D.van der Meer, 2016, Impact of a high-speed train of microdropson a liquidpool,Journal of Fluid Mechanics,792, 850-868, (2016).[4]. H. Deka, B. Ray, G. Biswas and A. Dalal, Dynamics of tongue shaped cavitygenerated during the impact of high-speed microdrops, Physics of Fluids, Vol. 30,pp. 042103-1- 042103-14, (2018).




Nondegenerate Solitons and Their Collisions in the Two ComponentManakov Nonlinear Partial Differential Equations

M. Lakshmanan, Department of Nonlinear Dynamics, School of Physics,Bharathidasan University, Tiruchirappalli - 620 024, India

ABSTRACT

Nonlinear Schrodinger equation is a well-known soliton possessing nonlinearpartialdifferential equation occurring in many physical contexts. An importantvector gen-eralization of it is the Manakov system for two complex valued functions and in-tegrable by the inverse scattering transform method. Recently, wehave shown thatthe Manakov equation can admit a more general class of non-degenerate vector soli-tons, which can undergo collisions without any intensity re-distribution in generalamong the modes, associated with distinct wave numbers,besides the already knownenergy exchanging solitons corresponding to identicalwave numbers. In my lecture, Iwill discuss in detail the various special features ofthe reported nondegenerate vectorsolitons. To bring out these details, we derivethe exact forms of such vector one-,two- and three-soliton solutions through Hi-rota bilinear method and they are rewrit-ten in more compact forms using Gramdeterminants. The presence of distinct wavenumbers allows the nondegenerate fundamental soliton to admit various profiles suchas double-hump, table-top andsingle-hump structures. We explain the formation ofdouble-hump structure inthe fundamental soliton when the relative velocity of thetwo modes tends to zero.More critical analysis shows that the nondegenerate fun-damental solitons can undergo shape preserving as well as shape altering collisionsunder appropriate conditions. The shape changing collision occurs between the modesof nondegenerate solitons when the parameters are fixed suitably. Then we observethe coexistenceof degenerate and nondegenerate solitons when the wave numbers arerestricted appropriately in the obtained two-soliton solution.In such a situation wefind thedegenerate soliton induces shape changing behavior of nondegenerate solitonduring the collision process. By performing suitable asymptotic analysis we analysethe consequences that occur in each of the collision scenarios. Finally we pointoutthat the previously known class of energy exchanging vector bright solitons,with iden-tical wave numbers, turns out to be a special case of the newly derived nondegeneratesolitons.




Development and Analysis of an Unconditional Stable Method forAcoustic Wave Equations

Prof. Wenyuan Liao, University of Calgary, Calgary, Canada

ABSTRACT

In the field of numerical simulation of seismic wave propagation, the explicit finitedifference scheme is a popular choice due to its high efficiency and simple implementa-tion. However, because of the time-step constraint posed by Courant-Fridrichs-Lewy(CFL) number, the explicit finite difference methods become less efficient for time-domain acoustic wave equation, mainly due to the stability limit, which requires verysmall time step step. The situation is more severe when the wave speed is larger.In this work we focused on the development and analysis of highly accurate and un-conditionally stable numerical schemes for solving acoustic wave equations. Firstly,we derived an unconditionally stable backward difference formula (BDF) for solv-ing second-order ordinary differential equation. The BDF is then applied to solve thesemi-discrete second-order ordinary differential system, which is the result of applyingspatial discretization on the acoustic wave equation. In addition to the conventionalsecond-order finite difference scheme, we also considered the higher order accuracy inspace, which is obtained by the utilization of Pade approximation of the convectionalsecond order central difference.

We tested the new unconditionally BDF on various models through extensivenumerical examples on the accuracy and stability of the new method, such as second-order ordinary differential equation, 1D and 2D acoustic wave equations with constantand variable wave speeds. The new method is compact and fourth-order accurate inspace, while the order of convergence in time can be improved to fourth-order aswell. A rigorous stability analysis has been conducted to show that the new schemeis unconditionally stable. Moreover, the new scheme is very efficient, thus, can findwide applications in various Geophysical inversion areas, such as the full waveforminversion problems.




Computational Data Modelling: Methods and Applications

Prof. Chee Peng Lim, Deakin University, Melbourne, Victoria, Australia

ABSTRACT

Computational intelligence is a broad discipline that encompasses a variety of method-ologies inspired by human and/or animal intelligence. In this talk, the use of com-putational intelligence-based methods for data modelling will be described. The un-derlying algorithms comprising individual and hybrid intelligent data-based models,which include artificial neural networks, fuzzy systems, and evolutionary algorithms,will be explained. In addition, applications of such intelligent data-based models todifferent real-world problems will be demonstrated.




Mathematical Models and Analytical Methods for the HydroelasticResponses of a Very Large Floating Structure

Prof. D.Q. Lu, Shanghai University, Shanghai, China

ABSTRACT

A vast natural ice cover in the polar region and a man-made very large floating struc-ture (VLFS) in the offshore region are usually the idealized as thin elastic platesfloating on an inviscid incompressible fluid. To consider the effects of density stratifi-cation in the ocean, a simple but useful model, namely a multiple-layer fluid, is oftenemployed. For the mathematical formulation, the Laplace equation is taken for thegoverning equation, representing the continuity of the mass. The dynamic conditionon the fluid plate interface indicates the balance among the hydrodynamic pressureof the fluid, the elastic and inertial forces of the plate, and external moving loads,which forms a hydroelastic problem.

Under the assumptions of small-amplitude wave motion and small deflection ofplate, the fluid plate model is established within the linear potential theory. Dynamicresponses of the plate (namely the hydroelastic waves or flexural gravity waves),which are the key concerns of the present study, occur as the structure is subjected toincident ocean waves or an external downward load. For the wave plate interactionproblems, the velocity potentials are expressed by the eigenfunction expansions in thefrequency domain. We introduce some new inner products for the multiple-layer fluidto obtain the expansion coefficients. Thus the wave scattering and plate deflectionare studied. An object moving on or beneath the surface of VLFS can be modeled asa concentrated load singularity, which mathematically involves the Dirac delta func-tion. Far-field hydroelastic responses of the plate due to translating/instantaneoussingularities are analytically investigated with the aid of integral transforms and theasymptotic analysis.

To consider the nonlinear effects on the hydroelastic waves, the convective termin the momentum equation for the fluid motion and the Plotnikov-Toland model forthe elastic structure are employed. Semi-analytical approximation for the propagat-ing characteristics of nonlinear hydroelastic waves is obtained in terms of homotopyanalysis method. For the head-on collision process of two hydroelastic solitary waves,we utilize a singular perturbation method, namely the Poincare Lighthill Kuo (PLK)method of strained coordinates, to obtain the asymptotic solutions analytically. Wemainly examine the effects of important physical parameters, including the density,the thickness and Youngs modulus of the plate, the wave amplitude, larger densityratio or depth ratio of the two-layer fluid, on hydroelastic dynamic characteristics offlexible structures.

This research was sponsored by the National Natural Science Foundation of Chinaunder Grant No. 11872239.




Epidemiological Short-term Forecasting with Model Reduction ofParametric Compartmental Models

(Application to the first pandemic wave of COV ID − 19 in France.)Athmane Bakhta, Thomas Boiveau, Yvon Maday, Olga Mula

ABSTRACT

In this talk, I will present a forecasting method for predicting epidemiological healthseries on a two-week horizon at the regional and interregional level. The approach isbased on model order reduction of parametric compartmental models, and is designedto accommodate small amount of sanitary data.

The efficiency of the method is examined in the case of the prediction of thenumber of hospitalized infected and removed people during the first pandemic waveof COVID-19 in France, which has taken place approximately between February andMay 2020. Numerical results illustrate the promising potential of the approach.




Lax-Phillips Scattering Theory for Simple Wave Scattering

Prof. Mike Meylan, The University of New Castle, Callaghan, New South Wales,Australia

ABSTRACT

Lax-Philips scattering theory is a method to solve for scattering as an expansionover the singularities of the analytic extension of the scattering problem to complexfrequencies. I will show how a complete theory can be developed in the case of simplescattering problems. Even for the simplest case, it requires a non-trivial generalisedeigenfunction transformation to project into the space of analytic functions on thereal line. The scattering operator in this space is simply the complex exponential.I will illustrate how this theory can be used to find a numerical solution, and I willdemonstrate the method by applying it to the vibration of ice shelves.




PDEs and Optimal Control Problems in Domains with Highly OscillatingBoundaries: Asymptotic Analysis

Prof. A.K. Nandakumaran, Indian Institute of Science, Bangalore, India

ABSTRACT

In this talk, we discuss the asymptotic analysis (homogenization) of various optimalcontrol problems defined in domains whose boundary is rapidly (highly) oscillating.Such complex domains appears in many real life applications like heat radiators, flowsin channels with rough boundaries, propagation of electro-magnetic waves in regionshaving rough interface, absorption and diffusion in biological structures, acousticvibrations in medium with narrow channels etc. We present the work which we arecarrying out in my group for the last 10 years. We introduce the so called unfoldingoperators which we have developed for the problems under study through which wecharacterize the optimal controls. Finally, we do a homogenization process and obtainthe limit control problem.




Derivation of Ray Equations of a Polytropic Gas from Fermat’s Principle

Prof. Phoolan Prasad, Indian Institute of Science, Bangalore, India

ABSTRACT

According to Fermat’s principle, a ray going from one point P0to another point Pt

in space chooses a path such that the time of transit is stationary. Given initial po-sition of a wavefront 0, we can use rays to construct the wavefront t at any time t.Huygens’ method states that all points of a wavefront 0 at t = 0 can be considered aspoint sources of spherical secondary wavelets and after time t the new position t ofthe wavefront is an envelope of these secondary wavelets. The equivalence of the twofamous methods of construction of a wavefront t in a medium governed by a generalhyperbolic system of equations does not seem to have been proved and continues toremain open, see

http://math:iisc:ernet:in/prasad/prasad/preprints/130908reprint Huygens Fermat methods General.pdf.

I shall present a proof (by Russo and myself) only a part of the equivalence:

Fermat’s principle =⇒ rays equations of Euler equations, governing the motionof a polytropic gas.




Modeling of Fluid-poroelastic Structure Interaction

Prof. Ivan Yotov, Department of Mathematics, University of Pittsburgh, USA

ABSTRACT

We study mathematical models and their finite element approximations for solvingthe coupled problem arising in the interaction between a free fluid and a fluid ina poroelastic material. Applications of interest include flows in fractured poroelas-tic media and arterial flows. The free fluid flow is governed by the Navier-Stokesor Stokes/Brinkman equations, while the poroelastic material is modeled using theBiot system of poroelasticity. We present several approaches to impose the continu-ity of normal flux, including an interior penalty method and a Lagrange multipliermethod. A dimensionally reduced fracture model based on averaging the equationsover the cross-sections will also be presented. Stability, accuracy, and robustness ofthe methods will be discussed.




Semantic Technologies in a Decision Support System

Prof. Marcin Paprzycki, Systems Research Institute of the Polish Academy ofSciences, Warshaw, Poland

ABSTRACT

The aim of our work was to design a decision support system based on ontologicalrepresentation of domain(s) and semantic technologies. Specifically, we consideredthe case when Grid / Cloud users describe their requirements regarding a ”resource”as a semantic expression (based on domain capturing ontology), while the instancesof (the same) ontology represent available resources.

The aim of the presentation is to discuss in what way semantic technologies canand in what way they cannot be used in a decision support system.




Recent Advances in Numerical Methods for Singular PDEs

Tim Sheng, Center for Astrophysics, Space Physics and Engineering Research(CASPER), Baylor University. Waco, Texas, USA

ABSTRACT

In this talk, we will start with some interesting singular reaction-diffusion problems inmultiple scientific applications. An outline of a theoretical background of exponentialsplitting approaches will then be introduced. We will continue on typical quenching-combustion equations via decomposed finite difference approaches. Straightforwardnumerical analysis on the monotonicity, convergence and linear stability will be dis-cussed. The latest exponential evolving grid development inspired by moving gridstrategies will be proposed. Several experimental results will be given. The generalidea of adaptative splitting can be extended for solving other multiphysics equationsin particular those in biophysics, oil pipeline decay preventions and laser-materialsinteractions. Potentials of research collaborations will be explored.




Stability of Time Discretizations for Semi-discrete High Order Schemesfor Time-dependent PDEs

Chi-Wang Shu, Division of Applied Mathematics, Brown University, Providence, RI02912, USA

ABSTRACT

In scientific and engineering computing, we encounter time-dependent partial differ-ential equations (PDEs) frequently. When designing high order schemes for solvingthese time-dependent PDEs, we often first develop semi-discrete schemes paying at-tention only to spatial discretizations and leaving time t continuous. It is then impor-tant to have a high order time discretization to main the stability properties of thesemi-discrete schemes. In this talk we discuss several classes of high order time dis-cretization, including the strong stability preserving (SSP) time discretization, whichpreserves strong stability from a stable spatial discretization with Euler forward, theimplicit-explicit (IMEX) Runge-Kutta or multi-step time marching, which treats themore stiff term (e.g. diffusion term in a convection-diffusion equation) implicitly andthe less stiff term (e.g. the convection term in such an equation) explicitly, for whichstrong stability can be proved under the condition that the time step is upper-boundedby a constant under suitable conditions, and the explicit Runge-Kutta methods, forwhich strong stability can be proved in many cases for semi-negative linear semi-discrete schemes. Numerical examples will be given to demonstrate the performanceof these schemes.




The Numerical Solution of Time-fractional Initial-boundary ValueProblems

Prof. Martin Stynes, Beijing Computational Science Research Center, Beijing, China

ABSTRACT

An introduction to fractional derivatives and some of their properties will be pre-sented. The regularity of solutions to Caputo fractional initial-value problems is thendiscussed; it is shown that typical solutions have a weak singularity at the initial timet=0. This singularity has to be taken into account when designing and analysing nu-merical methods for the solution of such problems. To address this difficulty we usegraded meshes, which cluster mesh points near t=0, and answer the question: howexactly should the mesh grading be chosen? Finally, initial-boundary value problemsare considered, where the time derivative is a Caputo fractional derivative. (This isa fractional-derivative generalisation of the classical parabolic heat equation.) Onceagain a weak singularity appears at t=0, and the mesh in the time coordinate shouldbe graded to compute satisfactory numerical solutions.




Stability of Nature-Inspired Algorithms Using Dynamical System Theory

Xin-She Yang, Middlesex University, London, United Kingdom

ABSTRACT

Nature-inspired algorithms such as the particle swarm optimization, bat algorithmand firefly algorithm have been used to solve optimization problems quite efficiently.However, it lacks some in-depth mathematical analysis of these algorithms. This talksummarizes the latest developments, and provide some analysis of stability of thesealgorithms using dynamical system theory. Some challenges and open problems willalso be highlighted.




Godunov Type Solvers for Hyperbolic Systems Admitting

Prof. G.D. Veerappa Gowda, Tata Institute of Fundamental Research - CAM,Bangalore, India

ABSTRACT

Discontinuous flux based numerical schemes for the class of hyperbolic systems ad-mitting non-classical δ shocks are proposed, by extending the theory of discontinuousflux for non-linear conservation laws. It is shown that the numerical scheme convergesto the solution which preserves the physical properties such as positive density andbounded velocity. The numerical results are compared with the existing literatureand the schemes are shown to capture the solution efficiently. This is a joint workwith Aekta Aggarwal and Ganesh Vaidya.




Turnpike Control and Deep Learning

Enrique Zuazua, Friedrich-Alexander-Universitt Erlangen-Nrnberg, Germany,Deusto Foundation, Bilbao, Spain, Universidad Autnoma de Madrid, Spain

ABSTRACT

The tunrpike principle asserts that in long time horizons optimal control strategiesare nearly of a steady state nature.In this lecture we shall survey on some recentresults on this topic and present some its consequences on deep supervised learning.

This lecture will be based in particular in recent joint work with C: Esteve, B.Geshkovski and D. Pighin.




Higher Order PDE Based Image Processing:Theory, Computation &Application

B.V.Rathish Kumar, Department of Mathematics & Statistics, IIT Kanpur, India

ABSTRACT

Image processing is one of the interesting topicsof research in mathematics and en-gineering.In last few decades partial differential equation (PDE) based image pro-cessinghas attracted the researchers because of the sound theoretical and numericalbackground ofPDEs. The PDE models give the insight into the physical phenomenaand help to comeup with new models and effective numerical methods to solve it.Most of the PDE modelsof initial days are of lower order but they have some draw-backs such as blocky effectin denoising, failure with large gap in inpainting. Higherorder PDE models have shownpromise to overcome these defects. So the idea is tolook for appropriate higher orderPDE models to deal with theproblems that occurredin the field of image processing. Inthis talk, we will focus on three different typesof image processing problems namely image denoising, inpaintingand segmentationvia higher order PDE modelsand will share with you the developments which wehave made on theoretical and computational fronts towards better PDE based imageanalysis.


2 Minisymposia: Recent Trends in ComputationalMethods for Singularly Perturbed Differential Equa-tions

Organizers:

• Prof. Kaushik Mukherjee, Department of Mathematics, Indian Institute ofSpace Science and Technology (IIST), Thiruvanthapurm-695547, India. Email:[email protected]

• Prof. Jugal Mahapatra, Department of Mathematics, National Institute of Tech-nology Rourkela-769008, India. Email: [email protected]

Invited Speakers Talk



Revisiting the Slow Manifold of the Lorenz-Krishnamurthy Quintet

A. S. Vasudeva Murthy,TIFR Centre For Applicable Mathematics Bangalore-560065,Karnataka, INDIA,

email: [email protected]

ABSTRACT

A system of five nonlinear ODEs has been proposed by Ed. Lorenz and V. Krish-namurthy in the mid 80s as a model for the interaction of slow and fast time scalewaves in the atmosphere. Slow manifolds are solutions that are devoid of fast waves.The precise way of constructing these manifolds has been controversial. We present aslow manifold based on minimising evolution rate: a technique proposed by SharathGirimaji in 1999.




Isogeometric Analysis for Singularly Perturbed Problems

Christos Xenophontos,Department of Mathematics & Statistics,

University of Cyprus, P.O. BOX 20537, Nicosia 1678, Cyprusemail: [email protected], www.mas.ucy.ac.cy/∼xenophon

ABSTRACT

Since its introduction by Cottrell et. al. [1], Isogeometric Analysis (IgA) has beenused in avariety of settings. Even though a lot of emphasis was placed on convec-tion dominated problems, general convection-reaction-diffusion problems have beenneglected.

In this study, we consider singularly perturbed convection-reaction-diffusion prob-lems whose solution contains boundary layers and apply IgA for their numerical ap-proximation. We report results on the choice of the appropriate knot vector for theconstruction of the B-spline basis functions, as well as present the error analysis foran IgA Galerkin method yielding robust, exponential convergence for p refinement,under the assumption of analytic input data [2, 3, 4, 5]. Several numerical exampleswill also be presented.

Key Words: Isogeometric Analysis, Singularly Perturbed Problems, Boundary Lay-ers, Robust Exponential Convergence

References[1] J. A. Cottrell, T. R. Hughes and Y. Basilevs, Isogeometric Analysis: Towards integration of

CAD and FEA, Wiley and Sons, 2009.

[2] K. Liotati and C. Xenophontos, Isogeometric analysis for singularly perturbed problems in 1-D:a numerical study. Lecture Notes in Computational Science and Engineering 135, Springer,DOI 978-3-030-41800-7-15, pp. 231 - 243 (2020).

[3] C. Xenophontos and I. Sykopetritou, Isogeometric analysis for singularly perturbed problemsin 1-D: error estimates. Electronic Transactions in Numerical Analysis, Vol. 20, pp.1-25 (2020).

[4] C. Xenophontos,, Isogeometric analysis for singularly perturbed high-order, two-point bound-ary value problems of reaction-diffusion type, in press in Computers and Mathematics withApplications (2020). https://doi.org/10.1016/j.camwa.2020.05.011

[5] C. Xenophontos, Isogeometric analysis for two-dimensional singularly perturbed problems. Inpreparation, 2020.


International Conference on Advances in Differential Equations and Numerical Analysis(ADENA2020)Indian Institute of Technology Guwahati, October 12 - 15, 2020

Global Accuracy for Singularly Perturbed Reaction-Diffusion Problemswith Non-smooth Data

† J.L. Gracia, Institute of Mathematics and Applications (IUMA) & Department of AppliedMathematics, University of Zaragoza, Spain,


ABSTRACT

Singularly perturbed parabolic reaction-diffusion problems with non-smooth boundary-initial dataare considered. The non-smooth data can be either a function containing a layer or a discontinu-ity [1, 2, 3, 4, 5, 6]. Estimates for the derivatives of the solution are given and they are used toconstruct uniformly convergent globally accurate numerical methods on Shishkin meshes. In thecase of discontinuous data, a numerical/analytical approach [3, 4] is also examined by identifying ananalytical function associated with the discontinuity in the data. In this case, parameter-uniformnumerical approximations to the difference between the analytical function and the solution of thesingularly perturbed problem are generated. Numerical experiments are given to illustrate the the-oretical results. This research has been carried out in collaboration with Prof. Eugene O’Riordan(Dublin City University, Ireland.)

References[1] J.L. Gracia and E. O’ Riordan, A singularly perturbed parabolic problem with a layer in the

initial condition, Appl. Math. Comput., 219 (2012) 498–510.

[2] J.L. Gracia and E. O’Riordan, A singularly perturbed reaction-diffusion problem with incom-patible boundary-initial data. Proceedings NAA Conference 2012, Lectures Notes in Comput.Sci., 8236 (2013) 303-310.

[3] J.L. Gracia and E. O’Riordan, Parameter-uniform numerical methods for singularly perturbedparabolic problems with incompatible boundary-initial data, Appl. Numer. Math., 146 (2019)436-451.

[4] J.L. Gracia and E. O’Riordan, Singularly perturbed reaction-diffusion problems with disconti-nuities in the initial and/or the boundary data. Accepted in J. Comput. Appl. Math.

[5] P. Hemker and G.I. Shishkin, Discrete approximation of singularly perturbed parabolic PDEswith a discontinuous initial data, Comp. Fluid Dynamics J, 2, (1994) 375–392.

[6] G.I. Shishkin, Grid approximation of singularly perturbed parabolic reaction-diffusion equationswith piecewise smooth initial-boundary conditions, Math. Model, Anal., 12 (2007) 235-254.

c© Abstracts of ADENA2020†Research partially supported by IUMA, the projects PID2019-105979GB-I00 and PGC2018-

094341-B-I00 and the Diputacion General de Aragon (E24-17R)


FEM and SDFEM on Graded Meshes for a Problem with CharacteristicLayers

Ljiljana Teofanov, Faculty of Technical Sciences, University of Novi Sad, Serbia,email:[email protected]

Mirjana Brdar, Faculty of Technology, University of Novi Sad, SerbiaGoran Radojev, Department of Mathematics and Informatics, Faculty of Sciences, University of

Novi Sad, Serbia

ABSTRACT

In this talk we consider singularly perturbed convection-diffusion problem on unite square whose so-lution may have exponential and characteristic boundary layers. The problem is solved numericallyby a finite element method with piecewise bilinear elements on a graded mesh. We give results onalmost uniform convergence and superconvergence of this method in an energy norm. Furthermore,we consider a streamline-diffusion version of the method and give results on superconvergence prop-erty in the corresponding SD norm. Our analysis offers a choice of SD parameters which improvesstability.



Solution of Singularly Perturbed Boundary Layer Problems

Manoj Kumar,Department of Mathematics,

Motilal Nehru National Institute of Technology Allahabad-211004(U.P) India,email: dr [email protected]

ABSTRACT

In physics and fluid mechanics, a boundary layer is an important concept and refers to the layer offluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant.Boundary layers are also a common feature of singular perturbed systems. In these cases higherorder derivatives disappear in the unperturbed equations which lead to the cancellation of degreeof freedom of the system and finally in small regions where the system changes rapidly. The maindifficulty of solving such problems is due to non-smooth solutions at the end points. In this paper,Multiple-scales method is presented for solving second and third order singularly perturbed prob-lems with the boundary layer at one end either left or right. The original second and third orderordinary differential equations are transformed to partial differential equations. These problems havebeen solved efficiently by using Multiple-scales method and numerical simulations are performed onstandard test examples to justify the robustness of the proposed method.



Numerical Methods for Singularly Perturbed Delay DifferentialEquations

N. Ramanujam,Department of Mathematics,

Srimad Andavan Arts and Science College Tiruchirappalli,Tamilnadu-620005, INDIA,


ABSTRACT

The main objective of my talk is to discuss a few numerical methods for singularly perturbed delaydifferential equations. Before presenting them some well-known simple mathematical models repre-sented by non delay and delay differential equations are discussed. Then a few notions, propertiesand a method of solving delay differential equations are briefly presented. A brief introduction aboutnon delay and delay singularly perturbed problems is given. Finally numerical methods for singularlyperturbed delay problems are discussed in detail.



New Finite Difference Schemes for a Class of Singularly PerturbedDifferential-Difference Equations

P. Pramod Chakravarthy, Visvesvaraya National Institute of Technology, Nagpur, 440010, India,email: [email protected]

ABSTRACT

The theory of singular perturbations is now a mathematical area with fairly long history and having agood amount of applications in the fields of science and engineering. Singular perturbation problemsare frequently encountered in fluid mechanics, fluid dynamics, aero dynamics, magneto hydro-dynamics, plasma dynamics, rarefied gas dynamics,elasticity, oceanography and other domains. Afew notable examples are boundary layer problems, convective heat transport problems with largePeclet numbers, the modeling of steady and unsteady viscous ow problems with large Reynoldsnumbers, magneto-hydrodynamics duct problems at high Hartman numbers, etc. The solution ofthese problems varies rapidly or jump abruptly in some parts and varies slowly or behaves regularlyin some other parts of the domain. Large oscillations may arise, if we use the classical numericalmethods to solve these type of problems and the solution may be polluted in the entire domain ofintegration due to the boundary layer behavior.

A singularly perturbed differential-difference equation is a differential equation in which thehighest order derivative is multiplied by a small parameter and the equation involves at least onedelay or/and advance term. Singular perturbation problems are generally the first approximationof the considered physical model. Hence in such cases, more realistic model should include someof the past and the future states of the system and hence, a real system should be modelled bydifferential equations with delay or advance. Such type of equations arise frequently in the mathe-matical modeling of various practical phenomena, for example, in the modeling of the human pupil-light re ex, model of HIV infection, the study of bistable devices in digital electronics, variationalproblem in control theory, first exist time problems in modeling of activation of neuronal variability,immune response, evolutionary biology, dynamics of networks of two identical amplifiers, mathemat-ical ecology, population dynamics, the modeling of biological oscillators and in a variety of modelsfor physiological processes. Since standard finite difference schemes fail to capture the layer regionsperfectly, we proposed new stable finite difference schemes on the uniform mesh for solving singu-larly perturbed differential-difference equations. The proposed schemes have different advantages.They give oscillation free solution on a uniform mesh. Results are more accurate than conventionalmethods. These schemes can keep convergence order stable much better than conventional methodsfor very small values of perturbation parameter ε: Prior information about the location and width ofthe layer is not required. These methods are easily adaptable on special meshes like Shishkin meshor Bakhvalov mesh. Error estimates are calculated and the difference schemes are shown to convergeto the continuous solution uniformly with respect to the perturbation parameter ε and is illustratedwith numerical results. These methods are also extendable for higher-dimensional problems.



Numerical Methods for Option Pricing in Jump DiffusionFinancial Models

Mohan K Kadalbajoo,Department of Mathematics,

The LNM Institute of Information Technology, Jaipur-302031, (Rajasthan) INDIA,email:[email protected]

ABSTRACT

In this talk, we shall cover a very broad overview of some popular numerical methods for solvingoption pricing problems under jump diffusion models(PDEs & PIDEs) involving both European styleoptions and American options. The authors contributions in this direction shall be listed towardsthe end.



Numerical Method for a Weakly Coupled System of SingularlyPerturbed Parabolic Convection-Diffusion Equations with Non-smooth

Source Terms

S. Chandra Sekhara Rao, Department of Mathematics, Indian Institute of Technology Delhi, HauzKhas, New Delhi-110016, India, email:[email protected]

ABSTRACT

Singularly perturbed initial-boundary and boundary value problems often arise in the study of severalphysical phenomena, for example, in the classical theory of heat-mass transfer processes at mechanicalworking of materials, in rolling hot materials, in the plastic shear of materials, in the modeling ofbi-molecular reactions with small diffusion or fast reaction rates, in the aerodynamics of airfoilsand wings, in the viscous flow at high and low Reynolds numbers (from low-speed viscous flows tohypersonic flows), in the fluid flow through porous media, etc. Therefore, it is essential to studythe numerical aspects of such problems. A great deal of research work has been devoted to thenumerical solution of problems having boundary layers (see [8] and references therein). Limitedworks are available related to the numerical solution of problems having interior layers. Interiorlayers may appear due to several reasons, for example, due to non-linearity of the problem [9], lackof compatibility conditions [8, Part II], presence of turning point [10], non-smooth boundary and/orinitial data [3], non-smooth reaction coefficient and/or convection coefficient and/or source term[4, 5, 1, 7, 6, 2].

In this work, we analyze a numerical method for a time-dependent weakly coupled system oftwo singularly perturbed convection-diffusion equations. The highest order spatial derivatives in thefirst and second equations are multiplied by the perturbation parameters ε and µ, respectively. Theparameters 0 < ε, µ ≤ 1 could be arbitrarily small and assumed to be different in magnitude. Thesource terms in both equations have discontinuities in the spatial variables along the line x = d,d ∈ Ω = (0, 1). In the solution, boundary and interior layers appear in small regions of the domain.The solution is decomposed into regular and singular components according to the boundary andinterior layers, and precise bounds on the solution and its derivatives are given. Away from thediscontinuities, the problem is discretized using an upwind central difference method in space andbackward Euler in time, and along the lines of discontinuities, a special finite difference scheme.The domain is discretized using an appropriate Shishkin mesh. Parameters-uniform error estimatesin ”maximum norm” (which is strong enough to capture the sharp layers and singularity that mayoccur in the solution of singular perturbation problems) have been obtained. It is proved here thatthe method is parameters-uniformly convergent of first-order in time and almost first order in spaceconcerning both perturbation parameters. Numerical experiments are conducted to demonstrate theefficiency of the method.

References[1] I. Boglaev and S. Pack. A uniformly convergent method for a singularly perturbed semilinear

reaction-diffusion problem with discontinuous data. Appl. Math. Comput., 182:244–257, 2006.

[2] I. A. Brayanov. Numerical solution of a two-dimensional singularly perturbed reaction-diffusionproblem with discontinuous coefficients. Appl. Math. Comput., 182(1):631–643, 2006.

[3] J. L. Gracia and E. O’Riordan. A singularly perturbed reaction-diffusion problem with incom-patible boundary-initial data. Lecture Notes in Comput. Sci., 8236:303–310, 2013.

[4] S. C. S. Rao and S. Chawla. Numerical solution for a coupled system of singularly perturbedinitial value problems with discontinuous source term. Springer Proc. Math. Stat., 143:753–764,2015.

[5] S. C. S. Rao and S. Chawla. Numerical solution of singularly perturbed linear parabolic systemwith discontinuous source term. Appl. Numer. Math., 127:249–265, 2018.


[6] S. C. S. Rao and S. Chawla. Parameter-uniform convergence of a numerical method for acoupled system of singularly perturbed semilinear reaction–diffusion equations with boundaryand interior layers. J. Comput. Appl. Math., 352:223–239, 2019.

[7] S. C. S. Rao and S. Chawla. The error analysis of finite difference approximation for a systemof singularly perturbed semilinear reaction-diffusion equations with discontinuous source term.Lecture Notes in Comput. Sci., 11386:175–184, 2019.

[8] H. G. Roos, M. Stynes and L. Tobiska. Robust numerical methods for singularly perturbeddifferential equations. Springer Series in Computational Mathematics, Berlin, 2008.

[9] M. Stynes and N. Kopteva. Numerical analysis of singularly perturbed nonlinear reaction-diffusion problems with multiple solutions. Comput. Math. Appl., 51:857–864, 2006.

[10] R. Vulanovic and L. Teofanov. A uniform numerical method for semilinear reaction-diffusionproblems with a boundary turning point. Numer. Algorithms, 54:431–444, 2010.


Generation of Layer-adapted Meshes using Mesh PDE methods: aComputational Study

Niall Madden, National University of Ireland Galway, Ireland, email: [email protected]

Roisın Hill, National University of Ireland Galway, Ireland.

ABSTRACT

Much of the mathematical literature on the robust numerical solution of singularly perturbed prob-lems is concerned with the construction of layer-adapted meshes. The celebrated piecewise uniform“Shishkin” mesh [8] is the best known. It is predated by the graded mesh of Bakhvalov [2], but ismore popular because it is easier to implement, analyse, and generalise, even though it yields lessaccurate approximations.

As originally presented, the Bakhvalov mesh for a one-dimensional problem is implemented bysolving a (relatively simple) nonlinear equation. Linß has observed that it is equivalent to solving anequidistribution problem: one generates a mesh that equidistributes a so-called “monitor function”that represents pointwise bounds on the ODE’s solution’s derivatives [7].

The concept of equidistribution is usually associated with a posteriori error estimation andmesh generation. It entered the mainstream of research on singularly perturbed problems with thework of Beckett and Mackenzie [3], and Kopteva and Stynes [6]. It continues to find application inincreasingly complex settings; see, e.g., [4] which considers (as we do) equidistribution of a priorisolution derivative estimates, as well as a posteriori ones.

But standard equidistribution methods are mainly restricted in their applications to problemsthat are one-dimensional (in space), or which can be adequately resolved on tensor (Cartesian)product grids. To generalise them, one can use that equidistribution problems can also be understoodas differential equations, which is the basis for the idea behind (moving) mesh partial differentialequations (MPDEs) [5].

In this talk, we will present recent results in applying MPDE methods to generating meshesfor the robust solution, by finite element methods, of various classes of singularly perturbed ODEsand PDEs. While the results are largely computational, they show the approach is quite promising.Our numerical results are implemented in FEniCS [1], and the source code is soon to be publiclyavailable.

References[1] M. S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J Ring,

M. E. Rognes, and G. N. Wells. The FEniCS project version 1.5. Archive of Numerical Software,3(100), 2015.

[2] N. S. Bakhvalov. On the optimization of the methods for solving boundary value problems inthe presence of a boundary layer. Z. Vycisl. Mat i Mat. Fiz., 9:841–859, 1969.

[3] G. Beckett and J. A. Mackenzie. On a uniformly accurate finite difference approximation of asingularly perturbed reaction-diffusion problem using grid equidistribution. J. Comput. Appl.Math., 131(1-2):381–405, 2001.

[4] S. Gowrisankar, Srinivasan Natesan. An efficient robust numerical method for singularly per-turbed Burgers equation.Applied Mathematics and Computation, 346 (2019), 385–394.

[5] Weizhang Huang and Robert D. Russell. Adaptive Moving Mesh Methods, volume 174 of AppliedMathematical Sciences. Springer, New York, 2011.

[6] N. Kopteva and M. Stynes. A robust adaptive method for a quasi-linear one-dimensionalconvection-diffusion problem. SIAM J. Numer. Anal. 39 (2001), no. 4, 14461467.

[7] Torsten Linß. Layer-Adapted Meshes for Reaction-Convection-Diffusion Problems, volume 1985of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2010.


[8] G. I. Shishkin. Grid approximation of singularly perturbed boundary value problems withconvective terms. Soviet J. Numer. Anal. Math. Modelling, 5(2):173–187, 1990.


Numerical Treatment of Singularly Perturbed Differential-DifferenceEquations

Y. N. Reddy,Department of Mathematics,

National Institute of Technology WarangalWarangal-506004, INDIA,


ABSTRACT

In general, an ordinary differential equation in which the highest order derivative is multiplied bya small positive parameter and containing at least one delay/advance is known as singularly per-turbed differential-difference equation. These equations arise in the modelling of various practicalphenomena in bioscience, engineering, control theory, such as in variational problems in control the-ory, in describing the human pupil-light reflex, in a variety of models for physiological processes ordiseases and first exit time problems in the modelling of the determination of expected time for thegeneration of action potential in nerve cells by random synaptic inputs in dendrites. To solve theseproblems, perturbation methods such as Matched Asymptotic Expansions, WKB method are usedextensively. These asymptotic expansions of solutions require skill, insight and experimentation.Further, the Matching Principle: matching of the coefficients of the inner and outer regions solutionexpansions is also a demanding process. Hence, researchers started developing numerical methods.If we use the existing numerical methods with the step size more than the perturbation parameters,for solving these problems we get oscillatory solutions due to the presence of the boundary layer.Existing numerical methods will produce good results only when we take step size less than theperturbation parameters. This is very costly and time-consuming process. Hence, the researchersare concentrating on developing the methods, which can work with a reasonable step size. In fact,the methods should be parameter independent. The efficiency of a numerical method is determinedby its accuracy, simplicity in computing the numerical solution and its sensitivity to the parametersof the given problem. With this motivation, we, present here some simple, easy and efficient nu-merical methods which are readily adaptable for computer implementation with a modest amountof problem preparation. Several model example are solved to demonstrate the applicability of thesemethods. The solutions are tabulated and compared with the exact solutions. It is observed thatthe present method approximate the exact solution very well.


Contributed Speakers Talk


Parameter-uniform Convergence Analysis for Numerical Approximationof System of Singularly Perturbed Time-dependent Convection-diffusion

Problems

Sonu Bose, Department of Mathematics, Indian Institute of Space Science and Technology,Trivandrum-695547, India, email: [email protected], [email protected]

Kaushik Mukherjee, Department of Mathematics, Indian Institute of Space Science andTechnology, Trivandrum-695547, India.

ABSTRACT

In this work, we deal with a coupled system of singularly perturbed delay parabolic convection-diffusion initial-boundary-value problems exhibiting boundary layers. For numerical approximation,we discretize the time-derivative using the backward-Euler method for the temporal discretizationand a monotone finite difference method is proposed for the spatial discretization. The method isanalyzed on uniform mesh in the temporal direction and an appropriate layer-resolving mesh in thespatial direction. The parameter-uniform error estimate is derived for the numerical solution andalso for the solution derivatives. Finally, numerical results are presented to validate the theoreticalresults.



Richardson Extrapolation Technique for 2D Singularly Perturbed DelayParabolic Partial Differential Equations

Abhishek Das, VIT University, Vellore-632014 , India, email:[email protected]

ABSTRACT

In this article, we propose a higher order efficient numerical method for 2D singularly perturbed delayparabolic convection-diffusion problem defined on the domain G = D × Ωt, D = Ix × Iy = (0, 1)2,Ωt = [0, T ]:

ut + Lεu(x, y, t) = −c(x, y)u(x, y, t− τ) + f(x, y, t), (x, y, t) ∈ G,

u(x, y, t) = φb(x, y, t), (x, y, t) ∈ Γb = D× [−τ, 0],

u(x, y, t) = 0, (x, y, t) ∈ ∂D× Ωt,

(2)

whereLεu = −ε∆u+ a(x, y).∇u+ b(x, y)u,

0 < ε 1 and τ > 0. The coefficients a = (a1, a2), b, and c are sufficiently smooth and boundedfunctions that satisfies

ai(x, y) ≥ 2αi > 0, i = 1, 2, b(x, y) ≥ 0, onD.

The solution of (2) has a regular boundary layer of width O(ε) along the sides x = 1 and y = 1 anda corner layer in the neighborhood of the corner (1, 1). It is well-known that the classical numericalmethods (standard finite difference, finite element methods or finite volume methods) give inaccuratesolution of (2) on uniform meshes. In the proposed method, first, we time-discretize the problem insuch a way so that it yields a set of 1D problems. The convergence rate of the upwind scheme onShishkin mesh applied on the resulted 1D problems is almost first order, i.e., O(N−1 lnN + ∆t),where N is the number of mesh-intervals in the spatial direction and ∆T is the step size in thetemporal direction. To enhance the convergence rate, we use Richardson extrapolation by which weare able to increase the order of convergence to almost second order, i.e., O(N−2 ln2N + ∆t2). Tovalidate the theoretical results, numerical experiments are carried out.



The ADI type Operator Splitting SDFEM for the Singularly Perturbed2D Parabolic PDE

Avijit Das, Department of Mathematics, Indian Institute of Technology, Guwahati, India, 781039,email: [email protected]

S. Natesan, Department of Mathematics, Indian Institute of Technology, Guwahati, India, 781039.

ABSTRACT

A class of singularly perturbed 2D parabolic convection-diffusion-reaction initial-boundary-valueproblem (IBVP) is numerically solved by the alternating direction implicit (ADI) type operatorsplitting streamline-diffusion finite element method (SDFEM). The proposed scheme alleviates thecomputational complexity and the high storage requirement for high-dimensional problems. Theoverall stability of the two-step method is established. A piecewise-uniform Shishkin mesh is usedfor the spatial domain discretization. An efficient error estimation has carried out with an appropriatechoice of the stabilization parameter. Some numerical simulations are provided for the validation ofthe theoretical proofs.



A Second Order Finite Difference Scheme for a Class of SingularlyPerturbed Mixed Parabolic-elliptic Problem

L. Govindarao, Department of Mathematics, National Institute of Technology Rourkela, India,email: [email protected]

J. Mohapatra, Department of Mathematics, National Institute of Technology Rourkela, India.

ABSTRACT

We consider the following singularly perturbed mixed type problem involving both parabolic andelliptic types on a domain G− ∪ G+:

ut(x, t) + L1ε,xu(x, t) = f(x, t), (x, t) ∈ G−,

L2ε,xu(x, t) = f(x, t), (x, t) ∈ G+,

u(x, 0) = θ0(x), u(0, t) = θ1(t), u(1, t) = θ2(t), 0 ≤ t ≤ T, x ∈ F[u]= 0,

[∂u∂x

]= 0, at x = d,

(3)

where, L1ε,xu = −εuxx + b(x, t)u, L2

ε,x = −εuxx − a(x, t)ux + b(x, t)u, F = (0, 1), F = [0, 1],

F− = (0, d), F+ = (d, 1), G− = F−× (0, T ], G+ = F+× (0, T ], G = F × (0, T ]. Also, 0 < ε 1 is aperturbation parameter. [u] is represents a discontinuity point of u. Braianov [1] solved the problem(3) by using an inverse-monotone finite volume method on layer-adapted meshes. Mukherjee andSrinivasan [2] used implicit Euler scheme for time derivative and second ordered scheme for spatialvariables for solving (3). These articles show that the rate of convergence is almost second order upto a logarithm factor in spatial direction and first order in time direction.

To achieve optimal second order convergence, we use the central difference scheme in the firstsub-domain F− and a hybrid finite difference scheme in the second sub-domain F+ on Shishkin-typemeshes (standard Shishkin mesh, Bakhvalov-Shishkin mesh). First, use the implicit Euler scheme ontime derivative on uniform mesh. Again, to obtain the second order accuracy with respect to time,we use the implicit trapezoidal for time discretization. The article [2] dealt with standard Shishkinmesh only whereas we extended the proposed schemes for both standard Shishkin mesh and theBakhvalov-Shishkin mesh for solving (3).

References[1] I. A. Braianov. Uniformly convergent difference scheme for singularly perturbed problema of

mixed type. Electron. Trans. Numer. Anal., 23:288–303, 2006.

[2] K. Mukherjee and S. Natesan. Uniform convergence analysis of hybrid numerical scheme forsingularly perturbed problems of mixed type. Numer. Methods Partial Differential Equations,30(6): 1931–1960, 2014.



Discontinuous Galerkin Method with Interior Penalties for SingularlyPerturbed Convection-Diffusion Problems on Equidistributed Mesh

S. Gowrisankar, National Institute of Technology Patna, Patna, INDIA, email:[email protected]

ABSTRACT

In this talk, we consider singularly perturbed convection-diffusion problem

−ε4u+ a · ∇u+ cu = f in Ω = (0, 1)× (0, 1), (4)

u = 0 on Γ = ∂Ω.

where ε > 0 is a small parameter. a, c and f be sufficiently smooth functions. Under suitable con-tinuity and compatibility condition of the data, the convection-diffusion problem (4) has a uniquesolution u(x, y). Boundary layers occur in the solution when ε→ 0. These boundary layers are neigh-bors of the boundaries of the domain, where the solution varies rapidly, while away from the layersthe solution changes slowly, and smoothly. Numerical treatment to the problem is difficult becauseof the presence of boundary layers in its solution. In particular, classical finite difference methodsfail to yield satisfactory numerical results on uniform meshes, and to obtain uniform convergencewith respect to perturbation parameter one has to reduce step-sizes in relation with ε.

Here, we propose numerical scheme consists of discontinuous Galerkin method with interiorpenalties and layer adaptive equdistributed mesh. The mesh is generated through equidistribution ofsuitable monitor function. The proposed method converge uniformly with respect to the perturbationparameter ε. Uniform error estimates are obtained. Numerical experiments are carried out to validatethe theoretical results.



A Nonstandard Method for a Coupled System of Singularly PerturbedDelay Differential Equations

Dr. Trun Gupta, Visvesvaraya National Institute of Technology, Nagpur, India,email:[email protected]

Prof. P. P. Chakravarthy, Visvesvaraya National Institute of Technology, Nagpur, India.

ABSTRACT

A coupled system of singularly perturbed delay differential equations are solved via nonstandardmethod. The denominator function of second order derivative is obtained systematically from theproperties of governing equations. The proposed method is found to be uniformly convergent inconnection with the perturbation parameter and the error estimates are derived. Numerical resultsare analysed and it is found the results are agreeing with the theoretical results.



Hybrid Method for Two Parameter Singularly Perturbed EllipticBoundary Value Problems

Anuradha Jha, Department of Mathematics, IIIT Guwahati, GuwahatI, India, email:[email protected]

M.K. Kadalbajoo, Department of Mathematics, The LNM Institute of Information Technology,Jaipur, India.

ABSTRACT

We consider a two parameter singularly perturbed elliptic problem posed on a unit rectangle, whereboth convection and diffusion terms are multiplied with two small parameters. Presence of smallparameters gives rise to both boundary and corner layers. A hybrid scheme [1] on Shishkin meshis used to solve the problem. This hybrid scheme comprises of central difference method in layerregion which helps in resolving the layers more accurately and upwind method in regular region. Arestriction on coefficient of convection term and number of mesh points is placed to maintain themonotonicity of the resultant matrix. The method is analysed for convergence and is shown to befirst order uniformly convergent with respect to both the parameters. It produces better results thanupwind scheme [2] where the convergence is deteriorated due to presence of logarithmic factor. Thenumerical results are also carried out to validate the theoretical estimates obtained and claims made.

References[1] T. Linss; M. Stynes, A hybrid difference scheme on a Shishkin mesh for linear convection-

diffusion problems, Appl. Numer. Math., 1999, 31(3), 255–270.

[2] E. O’Riordan; G. I. Shishkin; M. L. Picket, Numerical methods for singularly perturbed ellipticproblems containing two perturbation parameters, Math. Model. Anal., 2006, 11(2), 199-212.



Numerical Approximation of a Two-parameter Singularly PerturbedParabolic Weakly Coupled System with Discontinuous Convection and

Source Terms

Aarthika K, National Institute of Technology, Tiruchirappalli, India.email: [email protected]

V Shanthi, National Institute of Technology, Tiruchirappalli, India.

ABSTRACT

We are concerned in this article with the numerical approximation of a two-parameter singularlyperturbed weakly coupled system involving boundary layers. The presence of the layers is observedfor small values of the perturbation parameters and also due to the existence of the discontinuousconvection and source terms. To obtain appropriate point-wise accuracy, we have considered a centralfinite-difference scheme defined on a spatial piecewise uniform Shishkin mesh and the implicit Eulerscheme on a time uniform mesh. Some numerical experiments have been performed to confirm thetheoretical first-order accuracy and error bounds.



The Numerical Solution of Two-Dimensional Singularly PerturbedConvection-Diffusion Problems

Kamalesh Kumar, Visvesvaraya National Institute of Technology, Nagpur, India,email:[email protected]

P. Pramod Chakravarthy, Visvesvaraya National Institute of Technology, Nagpur, India.

ABSTRACT

An introduction to two-dimensional singularly perturbed convection-diffusion problems and theirsolution behavior will be presented. The solution has exponential boundary layers at the sides x = 1and y = 1. A new finite difference scheme is developed. Due to the presence of layers, mesh in x andy coordinate should be graded to resolve the layer behavior. We have used the piecewise-uniformShishkin mesh in x-direction and y-direction, condensing at x = 1 and y = 1 respectively.



Fitted Special Finite Difference Scheme for Delay Differential Equationwith Dual Boundary Layers

Diddi Kumara Swamy, Indian Institute of information Technology, Sonepat,India.email:[email protected]

ABSTRACT

In this paper we have proposed a fitted special finite difference method for the solution of delaydifferential equations with dual boundary layers. The delay differential equation transformed to anasymptotically equivalent singular perturbation problem using the Taylors series expansion. Then,a fitted special finite difference scheme is described to get accurate solution to the problem. Themethod is demonstrated by implementing on several model examples by taking various values forthe delay parameter and perturbation parameter . To show the effect of delay on the boundary layeror oscillatory behaviour of the solution, several numerical examples are carried out in this paper. Todemonstrate the effect on the layer behaviour, the solution of the problems are shown graphically. Weobserved that when the order of the coefficient of the delay parameter is of o(1), the delay affects theboundary layer solution but maintains the layer behaviour and as the delay increases, the thicknessof the left boundary layer decreases while that of the right boundary layer increases.



An SDFEM for a Singularly Perturbed Fourth Order OrdinaryDifferential Equation with Mixed Boundary Conditions

T. Lalithasree, Department of Mathematics, Schoool of Engineering, AMRITA VishwaVidyapeetham, Coimbatore - 641 112, India, email:[email protected]

A. Ramesh Babu, Department of Mathematics, Schoool of Engineering, AMRITA VishwaVidyapeetham, Coimbatore - 641 112, India.

ABSTRACT

In this paper, Streamline-Diffusion Finite Element Method(SDFEM) for singularly perturbed fourthorder ordinary differential equations containing two parameters with mixed boundary conditions ispresented. Regularity of a solution is established under some specific conditions on the coefficientsof the problem. A scheme is constructed on layer adapted meshes like Shishikin, B-Shishkin andValonovic type meshes in order to resolve the layer phenomena and prove that the scheme is uni-formely convergent with respect to purturbation parameter. Finally, we present the computationalresults which support our theoretical findings.

AMS Mathematics Subject Classification: 65L10, CR G1.7Key words:Singularly perturbed problem,Fourth order differential equation, Energy norm, Streamline-diffusion, Finite element method, Boundary value problem, Layer adapted mesh.



Parameter-Uniform Numerical Method for Singularly Perturbed 2DElliptic Convection-Diffusion Problem with Boundary and Interior Layers

Anirban Majumdar, National Institute of Technology Nagaland, Chumukedima, India, email:[email protected]

Srinivasan Natesan, Indian Institute of Technology, Guwahati, India.

ABSTRACT

In this article, we develop uniformly convergent numerical scheme for solving singularly perturbedtwo-dimensional elliptic convection-diffusion problem with non-smooth convection coefficients andsource term of the form:

ε4u+ a(x, y)ux − b(x, y)u = f(x, y), in D = D \ Γ±x ,

u(x, y) = 0, on ∂D,

ux(x−, y)− ux(x+, y) = 0, on Γ±x ,

where

a(x, y) ≥ α > 0, ∀(x, y) ∈ D,

|a(x−, y)− a(x+, y)| ≤ C on Γ±x ,

b(x, y) ≥ β > 0, ∀(x, y) ∈ D,

|f(x−, y)− f(x+, y)| ≤ C, on Γ±x ,

and Ω−x = (0, ξ), Ω+x = (ξ, 1), Ωx = Ω−x ∪ Ω+

x ∪ x = ξ, Ωy = (0, 1), D = Ωx × Ωy , Γ±x = x =ξ × Ωy .

Due to the the discontinuity of a (positive throughout the domain) and f on Γ±x , the solutionof this kind of problem typically exhibits interior layer along x = ξ and boundary layer along x = 0.Also, since the coefficient of uy is zero, the solution of the model problem is also having boundarylayers near y = 0 and y = 1. To capture the interior and boundary layers, the piecewise-uniformShishkin mesh is considered in x and y directions. We apply upwind method on piecewise-uniformShishkin mesh to construct an ε-uniform scheme. Theoretically, we prove that the proposed methodis ε-uniformly stable and ε-uniformly convergent. Numerical results are presented to demonstratethe theoretical estimates.



Robust Numerical Method for a Partially Singularly Perturbed ParabolicSystem

Manikandan Mariappan, Department of Mathematics, Bharathidasan University, Tiruchirappalli -620 024, Tamil Nadu, India, email: [email protected]

Ayyadurai Tamilselvan, Department of Mathematics, Bharathidasan University, Tiruchirappalli -620 024, Tamil Nadu, India,email: [email protected]

ABSTRACT

In this article, a coupled system of m(m < n) singularly perturbed and n − m non-perturbedparabolic equations of reaction-diffusion type is considered. A uniform mesh in temporal directionand a piecewise uniform Shishkin mesh in spatial direction are constructed and used together withthe standard finite difference operators to formulate a new computational method for solving thesystem. The method is proved to be first order convergent in time and second order convergent inspace uniformly with respect to the perturbation parameter. Numerical experiments support thetheoretical results.

Acknowledgment: The first author wishes to acknowledge the financial assistance extendedthrough DSKPDF by the UGC, Government of India. And both authors wish to thank DST,Government of India, for the computational facilities under DST-PURSE phase II scheme.



The Bakhvalov Mesh: a History, Recent Results and Some OpenQuestions

Thai Anh Nhan, Holy Names University, 3500 Montain Blvd., Oakland, CA 94619, USA, email:[email protected]

ABSTRACT

For singularly perturbed differential equations, the use of layer-adapted meshes is one of the mostfrequently applied approaches to obtain parameter-uniform convergence for finite difference or finiteelement methods. Two popular layer-resolving meshes are the Shishkin mesh and the Bakhvalovmesh. Although the latter appeared about two decades before the former, the Bakhvalov mesh—considered as the the first layer-adapted—has still gained less attention than the Shishkin mesh.

In this talk, we look back Bakhvalov’s phenominal idea of the mesh construction, discuss chal-lenges in its analysis, as well as answer some open questions in the numerical analysis of singularlyperturbed problems on the Bakhvalov mesh [1,2]. We conclude the talk by reporting some remainingchallenges related to the mesh [3].

1. T.A. Nhan, R. Vulanovic, Analysis of the truncation error and barrier-function technique fora Bakhvalov-type mesh, ETNA 51 (2019), 315–330.

2. T.A. Nhan, R. Vulanovic, The Bakhvalov mesh: a complete finite-difference analysis oftwo-dimensional singularly perturbed convection-diffusion problems, Numer Algor (2020).https://doi.org/10.1007/s11075-020-00964-z

3. H.-G. Roos, M. Stynes, Some open questions in the numerical analysis of singularly perturbeddifferential equations, Comput. Meth. Appl. Math. 15 (2015), 531–550.



Discrete Approximation for a Two Parameter Singularly Perturbed TwoPoint Boundary Value Problem with a Turning Point

T Prabha 1, SRM TRP Engineering College, Tiruchirappalli, India,[email protected]:

M Chandru 2, Vellore Institute of Technology, Vellore, India.N Geetha 3, Bishop Heber College, Tiruchirappalli, India.

ABSTRACT

In this paper singularly perturbed boundary value problem with two parameters (ε, µ) that multiplythe diffusion coefficient and the convection term with a turning point is considered. The solutionto the problem exhibits boundary and interior layers with distinct layer widths. The presence ofthese layers make the solution to the problem more arduous compared to the non-turning pointproblems. Sharp bounds on the solution and their derivatives are derived. A finite difference schemewith carefully chosen piecewise uniform mesh is applied to discretise the numerical problem. Arigorous error analysis shows that the constructed method is uniformly convergent of order almostone. Numerical examples are considered to elucidate the theoretical findings.



High-Order Finite Difference Scheme for a System of SingularlyPerturbed Convection-Diffusion Equations

Dr. R. Mythili Priyadharshini, National Institute of Technology Karnataka, Surathkal, Mangalore,India - 575025, email:[email protected]

ABSTRACT

Singular perturbation problems may arise from viscous flow, edge effects in certain shell problems andthe concentration or thermal layers in mass and heat transfer problems. Singularly perturbed InitialValue Problems / Boundary Value Problems in Ordinary Differential Equations are characterizedby the presence of a small parameter that multiplies the highest derivative term. Solution of suchproblems exhibits sharp boundary and/or interior layers when the small parameter is much smallerthan 1. The numerical solution of such problems exhibits significant difficulties, particularly when thediffusion coefficient is small. Therefore, the interest in developing and analyzing efficient numericalmethods for singularly perturbed problems has increased enormously.

Most of this work has concentrated on problems involving a single differential equation. Onlya few authors have developed robust parameter-uniform numerical methods for system of singularlyperturbed ordinary differential equations. While many finite difference methods have been proposedto approximate such solutions, there has been much less research into the finite difference approxi-mations of their derivatives, even though such approximations are desirable in certain applications(flux or drag).

In this paper two hybrid difference schemes are proposed to approximate the solution and itsscaled first derivative of a weakly coupled system of two singularly perturbed second-order ordinarydifferential equations of convection diffusion type subject to mixed type boundary conditions. Here,bounds on the errors in approximating the first derivative of the solution with a weight in thefine mesh and without a weight in the coarse mesh are also obtained. We prove that the methodhas almost second order convergence in the supremum norm independent of the diffusion parameter.Error bounds for the numerical solution and also the numerical derivative are established. Numericalresults are provided to illustrate the theoretical results.

AMS Mathematics Subject Classification: 65L10, CR G1.7Key Words: Singular perturbation problems, weakly coupled system, mixed type boundary

conditions, piecewise uniform meshes, scaled derivative, mid-point scheme, Cubic spline scheme.



Higher-Order Difference Scheme for Parabolic Singularly PerturbedProblem with Time delay

Sanjay Ku Sahoo, The LNM Institute of Information Technology, Jaipur, India, email:[email protected]

Sanjay Ku Sahoo, The LNM Institute of Information Technology, Jaipur, India.Vikas Gupta, The LNM Institute of Information Technology, Jaipur, India.

ABSTRACT

In this work, we study numerical approximation for a singularly perturbed convection-diffusionproblem with time delay. A priori bounds for the classical solution and its derivatives are studied,which are helpful for the ε − uniform error estimates of the numerical method are given. Theproblem is discretized by an implicit Euler method on a uniform mesh in the time direction, andHigher-Order Difference approximation with Identity Expansion (named as HODIE) scheme on aPiece-wise uniform Shishkin mesh in the space direction. Then Richardson extrapolation scheme isused to enhance the order of convergence in the time direction. The resulting scheme is second-orderaccurate in both the space (with a logarithmic factor) and time direction.



Numerical Investigation for Time Delay Singularly Perturbed ParabolicProblems Involving Space Shifts

S. R. Sahu, Department of Mathematics, National Institute of Technology Rourkela, India -769008, email:[email protected]

J. Mohapatra, Department of Mathematics, National Institute of Technology Rourkela, India -769008,email:[email protected]

ABSTRACT

In this work, a time dependent singularly perturbed differential difference equation involving timedelay and general small space shifts (both positive and negative) is considered. The shift terms inspace are expanded by the Taylor series approximation. A hybrid scheme which consists the midpointupwind scheme and the second order central difference scheme for the spatial derivatives on Shishkinmesh and backward Euler scheme on an uniform mesh in the time derivative is proposed. It isshown that the proposed scheme converges uniformly with respect to the perturbation parameterwith almost second order for Shishkin mesh. Numerical results are presented, which illustrates ourtheoretical findings.

Key words: Singular perturbation, Shishkin mesh, Space and Time delay, Hybrid scheme, Uniformconvergence.

References[1] A. R. Ansari, S. A Bakr and G. I. Shishkin, A parameter-robust finite difference method for

singularly perturbed delay parabolic partial differential equations, Journal of Computational andApplied Mathematics, Vol. 205 No. 1, pp. 552-566, 2007.

[2] K. Bansal, P. Rai and K. K. Sharma, Numerical treatment for the class of time dependentsingularly perturbed parabolic problems with general shift arguments, Differential Equationsand Dynamical Systems, Vol. 25 No. 2, pp. 327-346, 2017.

[3] S. Gowrisankar and S. Natesan, ε-Uniformly convergent numerical scheme for singularly per-turbed delay parabolic partial differential equations, International Journal of Computer Mathe-matics, Vol. 94 No. 5, pp. 902-921, 2017.

[4] C. G. Lange and R. M. Miura, Singular perturbation analysis of boundary-value problems fordifferential-difference equations, SIAM Journal on Applied Mathematics, Vol. 42 No. 3, pp.502-531, 1982.



Uniformly Convergent Computational Method for a System of SingularlyPerturbed Parabolic Reaction-diffusion Initial Boundary Value Problems

Using Moving Mesh Refinement

Deepti Shakti, School of Advanced Sciences, Vellore Institute of Technology, Vellore, India,email:[email protected] and [email protected]

Jugal Mohapatra, Department of Mathematics, National Institute of Technology Rourkela, IndiaPratibhamoy Das, Department of Mathematics, Indian Institute of Technology, Patna, India.

Jesus Vigo-Aguiar, Department of Applied Mathematics, University of Salamanca, Salamanca,Spain

ABSTRACT

In this work, we consider a parabolic system of boundary layer originated reaction diffusion problemswith arbitrary small diffusion terms. Under sufficient smoothness and compatibility conditions, thisproblem exhibits a unique solution and has regular boundary layers at x = 0, 1, [1, 3].

The main aim of the work is to develop a parameter uniform optimal order computational methodfor the system of reaction diffusion problems using the mesh equidistribution approach. The problemis discretized by a modified implicit-Euler scheme in time direction and spatial derivatives by centraldifference scheme. The adaptive mesh generation is implemented in space due to the singularlyperturbed nature of the problem [2, 4]. To generate such a mesh, a positive monitor function is usedwhose equidistribution will move the mesh points toward the boundary layers. Parameter uniformerror estimates are derived to show that the convergence rate is optimal with respect to the problemdiscretization. Several numerical experiments are presented to support the theoretical findings andconfirm the efficiency and accuracy of the proposed method.

References[1] C. Clavero, J. L. Gracia and F. Lisbona, Second order uniform approximations for the solution of time

dependent singularly perturbed reaction-diffusion systems, Int. J. Numer. Anal. Model., 7(3), 428-433,(2010)

[2] P. Das and J. Vigo-Aguiar, Parameter uniform optimal order numerical approximation of a class ofsingularly perturbed system of reaction diffusion problems involving a small perturbation parameter, J.Comput. Appl. Math., 354, 533-544, (2019)

[3] J. L. Gracia, F. J. Lisbona and E. O. Riordan, A coupled singularly perturbed parabolic reaction–diffusion equations. Adv. Comput. Math, 32, 43–61, (2010).

[4] D. Shakti and J. Mohapatra, Numerical simulation and convergence analysis for a system of nonlinearsingularly perturbed differential equations arising in population dynamics, J. Differ. Equations Appl.,24, 1185-1196, (2018).



Uniformly Convergent Quadratic B-spline Collocation Method forSingularly Perturbed Parabolic Partial Differential Equation with Two

Small Parameters

Meenakshi Shivhare, Visvesvaraya National Institute of Technology, Nagpur-440010, India email:

[email protected]

P. Pramod Chakravarthy, Visvesvaraya National Institute of Technology, Nagpur-440010, IndiaDevendra Kumar, Birla Institute of Technology and Science, Pilani, Rajasthan-333031, India

ABSTRACT

We construct a parameters-uniform numerical scheme to solve the singularly perturbed parabolicpartial differential equation whose solution exhibits boundary layers at both the lateral surfaces ofthe rectangular domain. The method comprises an implicit Euler scheme on a uniform mesh inthe temporal direction and quadratic B-spline collocation scheme on an exponentially graded meshin the spatial direction. The exponentially graded mesh is generated by choosing an appropriatemesh generating function which adapts the mesh points in the boundary layers. We prove theparameters-uniform convergence of the proposed numerical scheme and the method is shown to beof O(N−2

x + ∆t) where Nx denotes the number of mesh points in the space direction and ∆t is themesh step size in the temporal direction. To support the obtained theoretical estimates test examplesare considered numerically.



Superconvergence Error Estimates of the NIPG Method for SingularlyPerturbed 2D Elliptic BVPs

Gautam Singh, IIT Guwahati, Assam, India, email:[email protected] Srinivasan, IIT Guwahati, Assam, India.

ABSTRACT

Superconvergence properties of the discontinuous Galerkin method for singularly perturbed 2D el-liptic BVPs are studied. In order to discretize the domain, here, we use the layer-adapted piecewiseuniform Shishkin mesh. Then we applied the NIPG method to obtain numerical solution and es-tablished the error bound in the discrete energy norm. We show that the proposed method isparameter-uniformly convergent with the order almost (k + 1) on the Shishkin mesh in the discreteenergy norm, where k is the order of the polynomials. Numerical experiments are carried out tovalidate the theoretical findings.



Superconvergence Study of Galerkin FEM for Singularly PerturbedSystems with Multiple Scales

Maneesh Kumar Singh, Department of Computational and Data Sciences, IISc Bangalore, India,[email protected]

Gautam Singh, Department of Mathematics, IIT Guwahati, India.Natesan Srinivasan, Department of Mathematics, IIT Guwahati, India.

ABSTRACT

In this talk, we discuss the superconvergence finite element analysis for the singularly perturbedcoupled system of both reaction-diffusion and convection-diffusion types.

Singularly perturbed coupled system of reaction-diffusion BVPs: −E2~u′′(x) +B(x)~u(x) = ~f(x), in Ω = (0, 1),

~u(0) = ~u(1) = ~0.

We assume that the reaction coefficient matrix B = bkj2k,j=1 is a strongly diagonally dominant

matrix which satisfies‖b12‖∞‖b11‖∞

< ξ,‖b21‖∞‖b22‖∞

< ξ, ξ ∈ (0, 1).

Singularly perturbed coupled system of convection-diffusion BVPs: −E~u′′(x)−A(x)~u′(x) +B(x)~u(x) = ~f(x), in Ω = (0, 1),

~u(0) = ~u(1) = ~0,

We assume that the convection coefficient matrix A = diag(a1, a2) and reaction matrix B =bkj2k,j=1 satisfy

ak ≥ αk > α > 0,

(∥∥∥∥bkk +a′k2

∥∥∥∥∞

)≥ |bkj |+ η, j 6= k, k = 1, 2,

with bkk > 0, k = 1, 2 and η is a positive constant. We also assume that the reaction, convectionterms and the source functions are sufficiently smooth.

The superconvergence study of Galerkin FEM on Shishkin mesh for both the coupled modelproblems is carried out and it is shown that the optimal error estimate is achieved under the suitablediscrete energy norm. Numerical experiments are conducted which validate the theoretical results.



Uniformly Convergent Numerical Scheme for Singularly PerturbedParabolic Partial Differential Difference Equations

Mesfin Mekuria Woldaregay, Jimma University, Jimma, Ethiopia, email: [email protected]

Gemechis File Duressa, Jimma University,Jimma, Ethiopia.

ABSTRACT

In this paper, singularly perturbed parabolic partial differential difference equations having smalldelay and advance on the spatial variable of the reaction terms are considered. The solution of theproblem exhibits a boundary layer behaviour on left or right side of the domain depending on thesign of the convective term. The term with the delay and advance are approximated using Taylorseries approximation. The resulting singularly perturbed parabolic partial differential equationsare treated using Crank Nicolson method in the temporal discretization with exponentially fittedoperator finite difference method in the spatial discretization. The developed scheme satisfies themaximum principle and the uniform stability of the scheme investigated using barrier function andsolution bound. The uniform convergence analysis of the scheme is carried out with the order ofconvergence one in space and two in time direction. Richardson extrapolation technique is applied inspatial direction for accelerating the rate of convergence of the scheme to order two. Test examplesand numerical results are considered to validate the theoretical analysis of the scheme.



Investigation of an Efficient Numerical Method for Singularly PerturbedTime-dependent Convection-diffusion Problems with Non-smooth Data

Narendra Singh Yadav, Department of Mathematics, Indian Institute of Space Science andTechnology(IIST), Trivandrum-695547, India email:[email protected],

[email protected]

Kaushik Mukherjee, Department of Mathematics, Indian Institute of Space Science andTechnology(IIST), Trivandrum-695547, India.

ABSTRACT

The work mainly focuses on the numerical investigation of singularly perturbed parabolic convection-diffusion initial-boundary-value problems with discontinuous convection term exhibiting strong inte-rior layers. Aiming to get better numerical approximation to the solutions of this class of problems,an efficient numerical method, which is composed of the backward-Euler method for the tempo-ral discretization together with a new hybrid finite difference scheme for the spatial discretization,is prosed and analyzed. To construct the numerical method, we use uniform mesh in the tempo-ral direction and an appropriate layer-resolving mesh in the spatial direction. The stability of theproposed method is discussed and the parameter-uniform error estimate is established. Numericalresults are presented to support the theoretical results both for linear and semi-linear parabolic prob-lems. Further, the proposed numerical method is extended for solving singularly perturbed parabolicconvection-diffusion problems with non-smooth data exhibiting both boundary and interior layers.



A Finite Difference Scheme to Solve Convection-Reaction Equation witha Delay Parameter in Convection and Reaction term

Ch. Lakshmi Sireesha, National Institute of Technology, Warangal, India,email: [email protected]

ABSTRACT

In this paper, a finite difference scheme is used to solve Convection-Reaction equations with delayparameter in convection term and also in the reaction term. The original convection-reaction equa-tion is replaced by an asymptotically equivalent singular perturbation problem and a finite differencescheme is employed to solve the SPP. To validate the applicability of the method, model exampleswith boundary layer have been solved for different values of delay parameter and perturbation pa-rameter. Maximum absolute errors are calculated and tabulated. Convergence of the scheme isestablished.



Challenges and Advantages of Deep Learning for Solving SingularlyPerturbed Partial Differential Equations

Sangeeta Yadav, Prof. Sashikumaar GanesanComputational and Data Science, Indian Institute of Science, Bangalore, India,

[email protected]

ABSTRACT

Singularly Perturbed PDEs has posed critical challenges for the scientific computing community fordecades. Conventional techniques produce spurious oscillations in the numerical solution. Gen-erally stabilization techniques are employed for reducing spurious oscillations. Streamline Up-wind/PetrovGalerkin(SUPG) is one such popular residual based stabilization technique very fre-quently used in research for solving SPDEs. In this method, the extra term depends on a user-definedparameter called stabilization parameter. Finding the optimal value of the stabilization parameteris a challenge. In this work, we have considered the prediction of stabilization parameter for SUPGusing deep neural network. It’s a semi-supervised network. In this research work, we have focused onconvection diffusion equations and a detailed deep learning based function approximation techniquefor predicting optimal stabilization parameter for SUPG is proposed. The feature set consists ofequation coefficients, mesh size and the spatial location of oscillations/interior layers. In this talk,the initial performance of this technique and the challenges observed will be discussed.

References1. A. N. Brooks and T. J. Hughes, Streamline upwind/petrovgalerkin formulations for convectiondominated flows with particular emphasis on the incompressible navier-stokes equations, ComputerMethods in Applied Mechanics and Engineering, vol. 32, no. 1, pp. 199 259, 1982.

2. Roos, H.-G.; Stynes, M.; Tobiska, L.: Robust Numerical Methods for Singularly PerturbedDifferential Equations – Convection-Diffusion-Reaction and Flow Problems. Springer Series in Com-putational Mathematics , Vol. 24, 2nd ed., 2008, 616 pages, ISBN: 978-3-540-34466-7



A Higher Order Numerical Method for Singularly Perturbed ParabolicConvection-diffusion Problem with Interior Turning Point

Swati Yadav, University of Delhi, Delhi-110007, India, email: [email protected]

Pratima Rai, University of Delhi, Delhi-110007, India, email: [email protected]

ABSTRACT

In this work, we have studied a class of singularly perturbed parabolic problems of convection-diffusion type with interior turning point. The exact solution of the problem exhibits two exponentialboundary layers. We have carried out the analytical study of the exact solution of the problem. Thesolution is approximated by using the implicit Euler method on a uniform mesh for time discretizationand a hybrid scheme on a generalized Shishkin mesh for spatial discretization. The proposed hybridscheme is a proper combination of the central difference scheme and the midpoint upwind scheme.The proposed numerical scheme is proved to be ε-uniformly convergent with order of convergence twoupto a logarithmic factor in space variable and one in time variable. Finally, some numerical resultsare presented to demonstrate the high accuracy and convergence rate of the proposed numericalscheme over the simple upwind scheme on a standard Shishkin mesh.



A Finite Difference Scheme to Solve Convection-Reaction Equation witha Delay Parameter in Convection and Reaction term

Ch. Lakshmi Sireesha, National Institute of Technology, Warangal, India,email: [email protected]

ABSTRACT

In this paper, a finite difference scheme is used to solve Convection-Reaction equations with delayparameter in convection term and also in the reaction term. The original convection-reaction equa-tion is replaced by an asymptotically equivalent singular perturbation problem and a finite differencescheme is employed to solve the SPP. To validate the applicability of the method, model exampleswith boundary layer have been solved for different values of delay parameter and perturbation pa-rameter. Maximum absolute errors are calculated and tabulated. Convergence of the scheme isestablished.


3 Minisymposia: Advances in Computational Mul-tiphase Flows

Organizers:

• Prof. Anugrah Singh, Department of Chemical Engineering, IIT Guwahati, India. Email:[email protected]

• Prof. Raghvendra Gupta, Department of Chemcial Engineering, IIT Guwahati, India. Email:[email protected]


Analysis of Particulate Flow Through Screw Conveyors

Prabhu R Nott, IISc Bangalore, Bangalore-560065, Karnataka, INDIA

ABSTRACT

Screw conveyors are widely employed in industry for transporting particulate materials. Despite theirimportance, there is yet no mechanics-based model to determine the discharge rate as a functionof the material and system properties. In this presentation, I will first discuss a simple modelthat significantly simplifies the kinematics of the material. I will show that under certain limitingconditions, the discharge rate can be obtained without knowledge of the stresses on the boundingsurfaces. The discharge rate is found to reach a maximum at a particular value of the ratio ofscrew pitch to barrel radius. I will then present a detailed computational study of the flow usingthe Discrete Element Method, and compare the continuum kinematic fields with those assumed inthe simple model. This allows us to relax the limiting conditions employed in the model, therebydetermining the connection between friction at the walls and the kinematics of flow. I will showthat the variation of the discharge rate with pitch to barrel diameter ratio is qualitatively similar tothat obtained from the simple model. I will end by discussing a constitutive model we have recentlyproposed for dense granular flows that can be employed to model such complex flows without recourseto simplifying assumptions.



Avalanche Flows of Grains on Minor Planets

Ishan Sharma, IIT Kanpur

ABSTRACT

Minor planets asteroids, small moons, comets are observed to have surface deposit of grains, calledregolith. Given the low-gravity on these rotating bodies it is possible to easily mobilize regolith.Changes in the rotational motion due to close planetary encounters can also initiate regolith flow. Thesubsequent surface granular flow called avalanches or landslides may lead to changes of shape andeven mass loss which, in turn, may then affect the bodys rotational motion because of conservation ofangular momentum. I will present some recent work on modeling avalanching processes on granularminor planets, and put it in context of recent space missions to asteroids such as Itokawa, Ryuguand Bennu.



On the Effect of Interfacial Tension and Wettability on Pore-ScaleTwo-phase Flow Mechanisms in a Three-Dimensional Porous Medium

using Pore-Resolved Volume-of-Fluid Simulations

Aniket S. Ambekar, Department of Chemical Engineering, Indian Institute of Technology Delhi,New Delhi 110016, India

Vivek V. Buwa, Department of Chemical Engineering, Indian Institute of Technology Delhi, NewDelhi 110016, India, E-mail: [email protected]

ABSTRACT

Two-phase flow through porous media is important to development of improved secondary andtertiary oil recovery processes. In the present work, we have simulated oil recovery process in a pore-resolved three-dimensional medium using Volume-of-Fluid method. The effects of water-floodingvelocity, interfacial tension (IFT) and wettability on two-phase flow mechanisms are investigatedusing pore-scale events, oil-phase morphology, forces acting on oil ganglia surfaces and oil recoverycurves, for Capillary numbers (Ca) in the range of 1.2 × 10−3 to 6 × 10−1. We found that thetwo-phase flow through oil-wet medium is governed by pore-by-pore filling mechanism dominatedby the Haines-jump events. We show that a decrease in the IFT results in the change of pore-scalemechanism from pore-by-pore filling to viscous fingering. The viscous fingering; characterized byearly breakthrough, swelling and growth of invading-phase fingers post-breakthrough and absence ofHaines-jump events; leads to an increase in the oil recovery. Further, the change in the wettabilityfrom oil- to water-wet results into the change of pore-by-pore filling mechanism to a combination ofco-operative pore filling and corner flow. The dynamics of two-phase flow through weakly water-wetporous medium is dominated by co-operative pore filling and leads to an increase in the oil recovery.On the other hand, the two-phase flow dynamics through strongly water-wet medium is found to begoverned by corner flow events resulting in low oil recovery. Finally, a two-phase flow mechanismmap is proposed in terms of Ca and contact angle that help to relate underlying flow mechanismswith the oil recovery.



Dynamics of Rayleigh Breakup of Charged Droplets

Neha Gawande and Rochish M Thaokar

ABSTRACT

A spherical charged droplet, subjected to small amplitude shape deformations, can exhibit instabilitywhen the charge on the droplet exceeds a certain limit and undergoes breakup by emitting a fractionof charge and mass in the form of jets. Lord Rayleigh, in a seminal work, performed a linearstability analysis on the charged conducting droplets to predict the maximum critical charge adroplet can withstand. Although the expression for the critical charge is known for more than acentury, the deformation dynamics of such critically charged droplets and the associated jet emissionis the complex fluid dynamical phenomena which cannot be explained by linear order theory. Thus,to understand the detailed mechanism of Rayleigh instability and the breakup process, a non-linearelectrohydrodynamic model is solved numerically using the axisymmetric boundary element method.

Our numerical analysis indicates that a critically charged viscous droplet, when assumed to beperfectly conducting, deforms into self-similar conical tips and cannot predict the jet ejection or theassociated charge and mass loss due to the occurrence of numerical singularity. It is found that dueto the imposed condition of instantaneous charge relaxation, the surface of the perfectly conductingdrop remains iso-potential as it deforms, and the charge density diverges at the conical tip. Thus,it is not possible to numerically resolve the shape near the tip and the simulations cannot proceedfurther.

To overcome this limitation, the electrostatic model is modified by invoking surface charge dy-namics that account for finite time required for charge relaxation on the deforming drop surface.Due to the singularly fast dynamics of the process and involvement of diverse length scales (rangingover two orders of magnitude), the conservation of total charge during Rayleigh breakup is a nu-merically challenging problem. Thus, the problem is solved by using adaptive mesh generation andtime-stepping algorithms. The numerical results for this modified model indicate that as the dropdeforms and approach singularity, the charge transport to the tip is restricted by the convectioneffects, and the condition of the iso-potential surface becomes invalid. This gives rise to tangen-tial electric stresses which are responsible for the emergence of a jet that subsequently breaks intoprogeny droplets by Rayleigh-Plateau instability.



Entropic Lattice Boltzmann Model

Santosh Ansumali, JNCASR, Bangalore

ABSTRACT

The lattice Boltzmann method constructs a simplified kinetic picture on a lattice designed to capturethe physics of macroscopic flow through simple local micro-scale operations. A number of limitationsof this method have been attributed to the loss of thermodynamic consistency while transitioningfrom the continuum to the discrete dynamics. The entropic lattice Boltzmann models restore Boltz-manns H theorem (a generalization of the second law) to the discrete dynamics by numericallysolving a nonlinear equation. I present construction of closed form analytical solutions to this non-linear equation, thus, substantially reducing the computational requirement. This guarantees thenumerical stability of the model in a computationally efficient manner. Finally, I will discuss multi-phase lattice Boltzmann method and thermodynamic issues associated with it. I would argue thatnumerical instabilities and spurious current in diffuse interface methods are related to breakdown ofGibbs-Duhem relation in numerical simulation. A possible way to enforce Gibbs-Duhem relation atdiscrete level will be discussed.

References[1] Atif et al, Essentially entropic lattice Boltzmann model. Phys Rev Lett, 119: 240602, 2017.

[2] Kolluru et al, Lattice Boltzmann model for weakly compressible flows. Phys Rev E, E 101:013309, 2020.



Thin Film Drainage Between Bubbles Colliding with Bubbles and Solids

Rogerio Manica, Bo Liu and Qingxia Liu, Department of Chemical and Materials Engineering,University of Alberta, Edmonton, Canada

ABSTRACT

Interactions involving bubbles, droplets, particles and surfaces are typically encountered in mineralflotation, and in oil and gas extraction. The success of these applications depends on the outcome ofcollisions between different components, where the boundary conditions at the air-liquid interface areof critical importance. Our group has made significant progress in understanding bubble collision byusing experiments from our unique Dynamic Force Apparatus [1]. Specifically, we showed that thecoalescence between bubbles can occur in milliseconds in clean water system with mobile boundaryconditions, several orders of magnitude faster than in systems containing surfactants or impuritiesthat takes over a second. Such multiphase flow systems can be studied using different theoreticallytechniques such as computational fluid dynamics, but also simplified theories that are applicable tospecific problems. By assuming the system is axisymmetric and using lubrication theory, the Navier-Stokes equations can be simplified and solved numerically in a personal laptop in a few seconds. Bycomparing experiments and theory, we have been able to validate some approaches that assumedmobile boundary conditions. Such conditions have been difficult to achieve in many of the literatureexperiments because even minor amounts of contaminants can immobilize the bubble surface. Whena bubble is freshly generated, it is clean and mobile and by performing bubble collision experimentsafter different elapsed times from bubble generation, the adsorption rate of surfactants could beestimated [2]. In this presentation, we will provide details of the theory, numerical techniques andcomparisons with our experimental data to highlight the progress made and lay down directions offurther research

References[1] B. Liu, R. Manica, Q. Liu, E. Klaseboer, Z. Xu and G. Xie, Phys Rev Lett, 122: 194501, 2019.

[2] B. Liu, R. Manica, Q. Liu, E. Klaseboer, Z. Xu and G. Xie, Phys Chem. Lett. , 2019.



Principles of Particle Technology and Energy Utilisation in Mineral,Metallurgical and Energy Related Industries

Geoffrey Evans, Subhasish Mitra, Roberto Moreno Atanasio, Elham Doroodchi and TomHoneyands, School of Engineering, University of Newcastle, Callaghan NSW 2308 Australia

ABSTRACT

Complex interaction between fluids and particles, bubbles and droplets underpin the performanceof many processes used by the minerals, metallurgical and petrochemical industries. Such interac-tions involve a combination of forces acting at the phase interface, including those associated withinterfacial tension, pressure and fluid drag. For liquid systems involving solids and either bubblesor other immiscible liquid droplets, then wetting behavior of the solid surface plays an importantrole in how each of the phases interact with each other. Turbulence in the liquid phase, generatedeither within the liquid itself and/or resulting from the presence of the dispersed phase, is also adetermining parameter in mixing (dispersion), collision interaction, and/or breakup and coalescenceof the bubbles, droplets or particles.

In large-scale industrial processes the flows are almost always turbulent, whilst for micro-scaleoperations the flow will be laminar. Each condition provides its own challenge, and for this rea-son there is on-going experimental and theoretical investigation in order to optimise operationalconditions, especially in terms of energy dissipation rate, that maximises performance. The pre-sentation is focused on visualising, quantitative measurement and theoretical and computational(CFD and DEM) modelling of the interactions between bubbles, droplets and particles in liquids inmotion. Particular emphasis is placed on simplified constructs of complex turbulent flow in orderto reduce computational requirement often encountered in multiphase systems- the fluidised beds,bubble columns, novel mineral flotation approaches, and high temperature reactors.



CFD Simulations of Bubble Column Bioreactors: Closure models,Numerics and Validation

Ege Ertekin1,2, Dale D. McClure1, John M. Kavanagh1 and David F. Fletcher1, 1-Department ofChemistry, Technical University of Munich, Germany.

2-School of Chemical and Biomolecular Engineering, University of Sydney, Australia.

ABSTRACT

There is increasing interest to use bioreactors to produce a variety of high value products suchas recombinant proteins, enzymes, therapeutics, food ingredients and fine chemicals. As with allindustrial processes, there is scope for process optimisation via changes to vessel design and operatingconditions. Computational Fluid Dynamics models have the potential to optimise this process byproviding detailed information on the flow field, mass transfer rates, mixing times and the impactof environmental changes on the cell physiology and product formation. In this talk we describe amathematical model that has been developed progressively over the last eight years to make use ofthe latest physical models and numerical approaches. We discuss the important issues of numericalstability and performance. The talk is concluded with a presentation of some recent validation workand proposes future research directions.



Chaotic Orbits of Ellipsoids

Rama Govindrajan, ICTS, Bangalore.

ABSTRACT

Kozlov and Onischenko in 1982 proved that ellipsoids in inviscid fluid can display chaotic orbitsunder certain conditions. We explore these conditions and show that ellipsoids of rotation havean additional integral of motion, which makes their dynamics integrable, as opposed to the chaoticdynamics of general ellipsoids. We explore what consequences these findings have for the viscousdynamics of these bodies.



Particle Migration in a liquid-liquid Stratified Flow in a Microchannel

T. Krishnaveni, T. Renganathan, S. Pushpavanam, Department of Chemical Engineering, IITMadras, Chennai 600036, India.

ABSTRACT

Inertial focusing is a separation technique where particles in an axial flow migrate laterally to equi-librium positions in the presence of finite inertia. These equilibrium positions form mainly due to thebalance of two lift forces, namely wall lift force (which repels the particle away from the wall) andshear gradient lift force (which directs the particle towards the wall). These equilibrium positionscan be altered by changing the shear gradient profile. One possible way to change the shear gradientprofile is by sending two liquids in the form of a stratified flow. Recent experimental studies haveshown that particle transfer across streamlines can be controlled passively using stratified flows ofco-flowing streams at finite Reynolds number. We analyze particle migration in a stratified flowconsidering two cases: (i) miscible and (ii) immiscible liquids. In a stratified flow of two miscibleliquids, the particle can migrate from one fluid to the other solely due to inertial force, whereas inimmiscible stratified flow, the interfacial force plays an important role. The particle may attach/detach from the interface, and the interface can also become unstable. A numerical approach usingan immersed boundary and level set method is employed to study the particle dynamics. The effectof the viscosity ratio, flowrate ratio, Reynolds number, Weber number and particle size on focusingposition are analyzed.



A Consistent and Well-Balanced Immersed Boundary Method forIncompressible Multi-Component Flows

Ganesh Natrajan1, Jitendra Kumar Patel2

1-IIT Palakkad.2-KIIT Bhubaneswar.

ABSTRACT

Multi-component flows are ubiquitous in engineering and numerical simulations of such flows requirea robust and accurate numerical framework. We discuss our efforts in developing a finite volumeframework for incompressible multiphase flows using the diffuse-interface immersed boundary methodon unstructured meshes. In particular, we focus on three aspects,

1. A consistent formulation to handle high density ratio flows2. A well-balanced scheme to ensure that there are no spurious currents in surface tension and

gravity dominated flows3. A volume-of-body immersed boundary method that can be seamlessly integrated with volume-

of-fluid based flow solver to handle moving body problems.Numerical simulations highlighting these aspects and their application to canonical multiphase

problems will be presented to demonstrate the efficacy and versatility of the proposed framework,which can be implemented with relative ease in existing legacy solvers



Control Mechanisms of Miscible Viscous Fingering Instability

Manoranjan Mishra, Department of Mathematics, Indian Institute of Technology Ropar, INDIA

ABSTRACT

Viscous fingering (VF) is one of the hydrodynamic instabilities, which is observable while displacinga less-mobile fluid by another more-mobile fluid through porous media, and it is pervasive to thetransport phenomena in several porous media flows application. Moreover, instabilities at the in-terface of two distinct fluids for enhanced oil recovery processes such as polymer flooding, remain asignificant challenge. On the other hand, this instability is likely to be detrimental to the separationefficiency in the chromatographic separation process. It can improve mixing in non-turbulent systemsand micro-fluidic devices. The fact that depending on the application, either a stable or an unstablepattern, is desirable, and therefore it is essential to control interfacial fingering instabilities. Such acontrol mechanism of instabilities in fluid-fluid systems can be achieved by manipulating the variousphysio-chemical properties of the underlying fluids as well as the porous medium. In this talk, acouple of control mechanism of VF will be discussed based upon the solute adsorption on the porousmatrix and the competition between advection and diffusion. Numerical simulations result, basedon a hybrid scheme of finite difference and spectral methods, in comparison to the correspondingexperimental investigation, will be presented.



Coalescence Dynamics of Unequal Sized Drops

Amaresh Dalal, IIT Guwahati.

ABSTRACT

The interaction and coalescence of drops in a flow have been the subject of research due to theirfundamental importance in a variety of multiphase flow systems. It has been unveiled that dependingon the impact condition, the drop may either coalesce with the receiving liquid or it may splash. Thecoalescence may be partial with the formation of a secondary drop or may be complete. In addition,the drop may also bounce off or float on the liquid surface. Although the satellite drop formationhas been widely studied for the case of a drop interacting with a flat surface, the spatio-temporalevents concerning the dynamical evolution of the coalescence of two unequal sized drops remain farfrom being comprehensively addressed. The present investigation consists of a numerical study on theidentification of various coalescence events, with emphasis on the conditions and mechanisms leadingto partial coalescence and generation of secondary drops during the coalescence of two unequal sizeddrops. The critical values of appropriate non-dimensional parameters for which transition fromcomplete coalescence to partial coalescence occurs have been explored.



Simulation of Blast Waves and its mitigation using Smoothed ParticleHydrodynamics (SPH)

Dr. S. S. Prasanna Kumar and Prof B. S. V. Patnaik, Department of Applied Mechanics, IndianInstitute of Technology Madras, Chennai 600036

ABSTRACT

A variety of explosions occur due to the forces of nature as well as technology. The sudden releaseof energy in an explosion, results in the rapid mechanical expansion leading to the formation ofa strong compression wave known as a blast wave. These blast effects are often aggravated byhigh temperature gaseous products which are of concern to combustion and fire safety analysts[1]. However, the primary destruction comes from the blast wave itself, which is essentially anunsteady moving shock wave, with its strength decaying over space and time. Any Blast WaveMitigation (BWM) strategy mainly addresses reduction in the intensity of peak overpressure of theblast wave and the impulse of the blast wind on the target structure [2]. Different strategies areused to dissipate/absorb or reflect/deflect the energy away from the target structure. Methodsthat absorb or dissipate energy necessitate the design of sacrificial structures that can deform anddevelop plastic strain in its material. Hence, cellular materials such as, polymer foams, aqueousfoams, metallic foams etc., are in vogue [3]. Although foam materials are suitable as a protectivecoating on the target structure, undesirable amplification of loads have been reported under someloading conditions and foam thickness. This effect is known as the shock enhancement [4] whichoccurs due to foam compaction against the target structure. The reflection-based BWM strategieswhich rely on diverting the incident energy of the blast wave are also popular. In this case, an arrayof rigid obstacles placed along the flow path offers blast protection [5]. This talk would introducethe multiphase and multi-component nature of the simulations using smooth particle hydrodynamics(SPH).

References[1] K. Ramamurthi. Explosion and Explosion Safety. Tata McGraw-Hill Education, New Delhi,

2011.

[2] O. Igra, J. Falcovitz, L. Houas and G. Jourdan. Review of methods to attenuate shock/blastwaves, Progress in Aerospace Sciences , 58:1-35, 2013.

[3] S. S. Prasanna Kumar, K. Ramamurthi, and B. S. V. Patnaik. Numerical study of foam-shocktrap based blast mitigation strategy, Physics of Fluids, 30(8):086102, 2018.

[4] F. Zhu, C. C. Chou and K. H. Yang. Shock enhancement effect of lightweight composite struc-tures and materials, Composites Part B: Engineering, 42(5):1202-1211, 2011.

[5] S. S. Prasanna Kumar, B. S. V. Patnaik and K. Ramamurthi. Prediction of air blast mitigationin an array of rigid obstacles using smoothed particle hydrodynamics., Physics of Fluids,30(4):046105, 2018.



Turbulence Collapses at a Threshold Particle Loading in a DiluteParticle-gas Suspension

V. Kumaran1, P. Muramalla2, A. Tyagi and P. S. Goswami,1Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India.

2Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 400 076,India.

ABSTRACT

Two mechanisms are considered responsible for the turbulence modification due to suspended parti-cles in a turbulent gas-particle suspension. Turbulence augmentation is due to the enhancement offluctuations by wakes behind particles, whereas turbulence attenuation is considered to result fromthe increased dissipation due to the particle drag. In order to examine the turbulence attenuationmechanism, Direct Numerical Simulations (DNS) of a particle-gas suspension are carried out at aReynolds number of about 3333 based on the average gas velocity u, channel width h, and the gaskinematic viscosity. The particle Reynolds number based on the particle diameter dp, gas kinematicviscosity and the flow velocity u is about 42 and the Stokes number is in the range 9.5−−377. Theparticle volume fraction is in the range 0− −2× 103, and the particle mass loading is in the range0−−9. As the volume fraction is increased, a discontinuous decrease in the turbulent velocity fluc-tuations is observed at a critical volume fraction. There is a reduction, by one order of magnitude,in the mean square fluctuating velocities in all directions and in the Reynolds stress. Though thereis a modest increase in the energy dissipation due to particle drag, this increase is smaller than thedecrease in the turbulent energy production; moreover, there is a decrease in the total energy dis-sipation rate when there is turbulence collapse. Thus, turbulence attenuation appears to be due toa disruption of the turbulence production mechanism, and not due to the increased dissipation dueto the particles. There is a discontinuous collapse in the turbulence intensities at a critical particleloading, instead of the continuous decrease as the particle loading is increased.



Secondary Atomization: Breakup of a Liquid Drop in a High Speed Flow

Prof. Gaurav Tomar, IISc Bangalore

ABSTRACT

Spray cooling is an efficient mechanism for thermal management. Coolant is sprayed at high speedsonto the heat load and phase change is used as the dominant mode of heat transfer. At high speeds,a drop in a spray may undergo further breakup. Droplets in high-speed flows breakup via differentmodes depending upon the nondimensional number, the aerodynamic Weber number, defined bythe ratio of the dynamical forces to the capillary forces. Initially the droplets deform into flat discsand at high Weber numbers (> 80), the flat disc-shaped droplets are sheared at the periphery andsmall droplets are generated from the edge of the drop. In contrast, at moderate Weber numbers(between 10 to 80), droplets are inflated by the flow into a balloon-shaped bag with a thicker rim andsubsequently the rupture of the bag results in the formation of smaller droplets. Droplet breakupmodes determine the final droplet size distribution. In this study, we perform a large set of volumeof fluid based simulations and show that the density ratio and the Reynolds number significantlyaffect the transition Weber numbers for different breakup mechanisms.



Simulation of Droplet Impact at High Dynamics using a Diffuse InterfacePhase-Field Methods in OpenFOAM

H. Marschall1, M. Bagheri1, N. Samkhaniani2, A. Stroh2, B. Frohnapfel2, M. Worner31Computational Multiphase Flow, Technical University Darmstadt, Alarich-Weiss Str. 10, 64287

Darmstadt, Germany2Karlsruhe Institute of Technology (KIT), Institute of Fluid Mechanics, Kaiserstr. 10, 76131

Karlsruhe, Germany3Karlsruhe Institute of Technology (KIT), Institute of Catalysis Research and Technology,

Engesserstr. 20, 76131 Karlsruhe, Germany

ABSTRACT

We have developed a unified solver framework for two-phase flow based on diffuse interface phase-field methods [1], which is to be released within the FOAM-extend project. In contrast to standardsharp interface model approaches, phase-field methods rely on diffuse interface models. As theirname suggests, these methods allow for diffusion of the phase constituents in a thin interfacial regionof well-defined thickness, thus, promoting a smooth but rapid transition of phase properties suchas density and viscosity. Particularly, capillary-dominated two-phase flow can be dealt with athigh accuracy, i.e. parasitic currents are found to be low and consistently converging under meshrefinement [2].

The present work focus on droplet impact and impingement scenarios at high dynamics. Recentsimulations for both droplet impact on thin liquid films of the same fluid, and droplet impingementand bouncing on a heated hydrophobic surface show very good agreement with experiments (see Fig.1). The talk will detail on necessary method enhancements to achieve this.

Figure 1: Image sequence of bouncing droplet (d0 = 2.3 mm, We = 20, Td;0 = 20C)on the smooth hydrophobic surface (θe = 120, Ts = 60C). Top: experiment [3],bottom: simulation.

References[1] X. Cai, H. Marschall, M. Worner, and O. Deutschmann, Numerical Simulation of Wetting Phe-nomena with a Phase field Method using OpenFOAM, Submitted, 2015.[2] F. Jamshidi, H. Heimel, M. Hasert, X. Cai, H. Marschall, and M. Worner, On suitability ofphase-field and algebraic Volume-Of-Fluid OpenFOAM solvers for gas-liquid microfluidic applica-tions, Comput. Phys. Commun., vol. 236, pp. 7285, 2019.[3] C. Guo, D. Maynes, J. Crockett, and D. Zhao, Heat transfer to bouncing droplets on superhy-drophobic surfaces, International Journal of Heat and Mass Transfer, vol. 137, pp. 857-867, Jul.2019.



Thin Film Drainage Between Colliding Bubbles

Rogerio Manica, Department of Chemical and Materials Engineering, University of Alberta,Edmonton, Canada, email: [email protected]

Bo Liu, University of Alberta, Canada.Qingxia Liu, University of Alberta, Canada.

ABSTRACT

Interactions involving bubbles, droplets, particles and surfaces are typically encountered in mineralflotation, and in oil and gas extraction. The success of these applications depends on the outcome ofcollisions between different components, where the boundary conditions at the air-liquid interface areof critical importance. Our group has made significant progress in understanding bubble collision byusing experiments from our unique Dynamic Force Apparatus [1]. Specifically, we showed that thecoalescence between bubbles can occur in milliseconds in clean water system with mobile boundaryconditions, several orders of magnitude faster than in systems containing surfactants or impuritiesthat takes over a second. Such multiphase flow systems can be studied using different theoreticallytechniques such as computational fluid dynamics, but also simplified theories that are applicable tospecific problems. By assuming the system is axisymmetric and using lubrication theory, the Navier-Stokes equations can be simplified and solved numerically in a personal laptop in a few seconds. Bycomparing experiments and theory, we have been able to validate some approaches that assumedmobile boundary conditions. Such conditions have been difficult to achieve in many of the literatureexperiments because even minor amounts of contaminants can immobilize the bubble surface. Whena bubble is freshly generated, it is clean and mobile and by performing bubble collision experimentsafter different elapsed times from bubble generation, the adsorption rate of surfactants could beestimated [2]. In this presentation, we will provide details of the theory, numerical techniques andcomparisons with our experimental data to highlight the progress made and lay down directions offurther research.

References[1] B. Liu, R. Manica, Q. Liu, E. Klaseboer, Z. Xu and G. Xie, Phys. Rev. Lett. 122, 194501

(2019).[2] B. Liu, R. Manica, Q. Liu, E. Klaseboer, Z. Xu and G. Xie J. Phys. Chem. Lett. (2019).



Velocity and Thermal Slip Effects on MHD Convective Two Phase Flowin an Asymmetric Convergent Channel

S. Ramaprasad, M.S.Ramaiah Institute of Technology, Bangalore, India email:[email protected]

S.H.C.V.Subba Bhatta1, M.S.Ramaiah Institute of Technology, Bangalore, IndiaB. Mallikarjuna3, BMS College of engineering, Affiliate to VTU, Belagavi, Bangalore, India

ABSTRACT

The motive of this paper to explore the influence of velocity and temperature slip effects on buoyancy-driven hydromagnetic two-phase flow in an asymmetric convergent channel. Suitable transformationsare employed to obtain ordinary differential equations from the governed partial differential equationsand solved numerically. The influence of emerging parameters on velocity and temperature profilesare adequately delineated through graphs and tables. A grid-independent test is done to validate thecode. A comparison is established with existing literature to support our results and good agreementis found which corroborates our work. A decline in fluid temperature is witnessed with a rise in thetemperature slip parameter. It is also observed that as the velocity slip parameter increases the fluidvelocity inclines in the left part and declines in the right part of the channel. These types of flowshave multifarious range applications like fluidized beds, gas cooling systems, polishing heart valvesand internal cavities, etc.


4 Minisymposia: Advances in Computational Sci-ence and Parallel Computing

Organizer:

• Prof. Sashikumaar Ganesan, Department of Computational and Data Sciences, IISc Banga-lore.


Scalable Asynchronous Solvers for Partial Differential Equations

Konduri Aditya, Indian Institute of Science, Bengaluru, India, email: [email protected] J. Matthew, Indian Institute of Science, Bengaluru, India Shubham K. Goswami, Indian

Institute of Science, Bengaluru, India

ABSTRACT

Recent advances in computing technology have made numerical simulations an indispensable re-search tool in understanding fluid flow phenomena in complex conditions at a great detail. Due tothe nonlinear nature of the governing Navier-Stokes equations, simulations of high Reynolds num-ber turbulent flows are computationally very expensive and demand extreme levels of parallelism.The current state-of-the-art turbulent flow simulations are routinely being performed on hundreds ofthousands of processing elements (PEs). At this extreme scale, communication and synchronizationbetween PEs significantly affect the scalability of solvers. Indeed, communication and data synchro-nization pose a bottleneck in scalability as simulations advance towards exascale computing. In thistalk, we present an overview of a novel approach based on widely used

finite-difference schemes in which computations are carried out in an asynchronous fashion,i.e. synchronization of data among processing elements is not enforced and computations proceedregardless of the status of communication. This drastically reduces the CPU idle time and resultsin much larger computation rates and scalability. We show that while standard schemes are ableto remain stable and consistent, their accuracy is significantly reduced. New asynchrony-tolerantschemes, which can maintain accuracy under relaxed synchronization conditions, are introduced.Wepresent computational performance results on homogeneous and heterogeneous architectures. Also,we briefly illustrate the extension of this approach to the discontinuous Galerkin (DG) method.



Parallel Algorithms for Optimal Vehicle Routing

Deepak N. Subramani,Department of Computational and Data Science, Indian Institute of Science Bangalore, India,

email: [email protected] Chowdhury, Department of Computational and Data Science, Indian Institute of Science

Bangalore, India.Chennam Revanth, Department of Computational and Data Science, Indian Institute of Science

Bangalore, India.

ABSTRACT

Optimal path planning of marine vessels such as ships and autonomous underwater vehicles instochastic and dynamic environments is important for a variety of application areas including com-merce, oil and gas, security, disaster management and environmental conservation. We develop asuite of distributed CPU and GPU algorithms for optimal marine vehicle routing. In the first part, wedevelop a computationally efficient algorithm for optimal routing of ships in deterministic dynamicwaves and currents, solving the optimal planning Hamilton Jacobi PDE using parallel narrow bandschemes. In the second part, we model the planning problem as a Markov Decision Process and ob-tain an optimal policy through a new GPU algorithm for fast exact dynamic programming solution.Idealized and realistic cases are showcased and multi-objective optimal paths are exemplified.



Parallel Algorithms for Centrality Computations

Kishore Kothapalli,International Institute of Information Technology, Gachibowli, Hyderabad, India, email:

[email protected]

ABSTRACT

Graph algorithms are popular owing to their applications to problems from a variety of domainsincluding the biology, engineering, optimization, and the like. An important problem with graphsfrom these domains is computing the various centrality metrics. Centrality metrics assign a numericalvalue to the nodes/edges of a graph. Computing these metrics is usally very time consuming andrequire time/work in O(m · n) for a graph of n nodes and m edges.

In this talk, we will highlight some of the challenges faced by parallel algorithms for computingcentrality metrics. we show how one can make use of techniques from algorithm engineering to addressthese challenges. These techniques enable algorithms to reduce the volume of computation and reusecomputation. Applying these techniques requires the development of algorithmic techniques thatextend the existing algorithms.

We will show examples of the above by considering closeness-centrality and betweenness- central-ity in static and dynamic graphs. We will show how the algorithms perfrom on modern architecturessuch as GPUs and CPUs.



Parallel and Space-time Adaptive Numerical Simulation of 3D CardiacElectrophysiology

Nagaiah ChamakuriInstitute of Applied Mathematics and Statistics, University of Hohenheim, Stuttgart, Germany,


ABSTRACT

The bidomain equations form the state-of-the-art model of cardiac electrophysiology and describenormal or pathological propagation of the excitation wave through cardiac tissue. If based on mech-anistic cell models and applied to anatomically realistic geometries, their discretization requires veryfine resolutions both in space and time and renders the numerical solution an extremely challengingcomputational problem. In this presentation, we address this challenge by combining space-timeadaptive discretization with dynamic load balancing for parallel computing. In order to further re-duce the computational costs, we exploit the sparsity of local matrices during the assembly of globalFEM matrices which ensures an optimal usage of memory. In numerical tests, we demonstrate thefeasibility of our approach, by which a substantial reduction of computing time with good parallelefficiency can be achieved.



A Scalable, Hybrid Fourier-Compact Finite-Difference Large EddySimulation Code for Wind Energy Applications

Niranjan S. Ghaisas,IIT Hyderabad, India, email: [email protected]

Aditya S. Ghate, Stanford University, USASanjiva K. Lele, Stanford University, USA

ABSTRACT

We describe a highly scalable, high-order accurate incompressible Navier-Stokes code (PadeOps-igrid). The code uses a strongly stability preserving 4th-order Runge-Kutta time-advancementscheme that allows for the use of time steps with CFL numbers of order 1. The code computesderivatives in the horizontal (periodic) directions using Fourier collocation while derivatives in thevertical (non-periodic) direction are computed using 6th-order staggered compact finite-dfferenceschemes.

Two key features of the code that enable good scalability will be described. The first featureis the use of the 2DECOMP&FFT library [4] that allows decomposition of the Cartesian domaininto 2D ‘pencils’, rather than the more common 1D ‘slab’ decomposition. The second feature isan efficient, direct solver for solving the pressure Poisson equation [2]. Development of this directsolver is complicated by the fact that for the staggered compact schemes, used in the vertical, thedivergence of the gradient (which is distinct from the Laplacian) leads to a full matrix. The directsolver uses a modi

ed wavenumber approach along with a cosine transform and an analytical method to incorporateinhomogeneous Neumann boundary conditions. This direct solver ensures a deterministic time tosolution and leads to discrete conservation of mass on the vertically staggered grid.

Scaling tests conducted on different architectures will be described. Specifically, on the TACC’sStampede2 supercomputer employing Intel Knights Landing nodes connected in a Intel Omnipathfat-tree topology, the code shows a strong scaling efficiency of 74% on 16, 384 cores for a granularityof roughly 65, 000 grid points, and an efficiency of 90% on 16, 384 cores for a granularity of roughly262, 000 grid points.

A few examples [1, 3] of the application of the PadeOps-igrid code towards large eddy simulationsof processes occurring in wind turbine wakes in large wind farms will be discussed.

References[1] N. S. Ghaisas, A. S. Ghate, and S. K. Lele. Effect of tip spacing, thrust cofficient and turbinespacing in multi-rotor wind turbines and wind farms. Wind Energy Science, 5:51-72, 2020.[2] A. Ghate. Gabor mode enrichment in large eddy simulation of turbulent flows. PhD thesis,Stanford University, 2018.[3] A. Ghate, N. S. Ghaisas, A. S. Towne, and S. K. Lele. Interaction of small-scale HomogeneousIsotropic Turbulence with an Actuator Disk. In 36th Wind Energy Symposium, AIAA ScitechForum, 2018.[4] N. Li and S. Laizet. 2DECOMP&FFT– A highly scalable 2D decomposition library and FFTinterface. http://www.2decomp.org/, 2010.



Mixed-precision Subspace Iteration Algorithm for Large-scale NonlinearEigenvalue Problems Towards Quantum-mechanical Modeling of

Materials.

Phani Motamarri,Indian Institute of Science, Bangalore, India email: [email protected]

ABSTRACT

Quantum mechanical modeling of materials have played a significant role in determining a widevariety of material properties. In particular, Kohn-Sham density functional theory (DFT) calcula-tions involving the solution of a non-linear eigenvalue problem, have been instrumental in providingmany crucial insights into materials behavior (mechanical, chemical, electronic and optical proper-ties), and occupy a sizable fraction of world’s computational resources today. However, the stringentaccuracy requirements required to compute meaningful material properties, in conjunction with theasymptotic cubic-scaling computational complexity of the underlying eigenvalue problem, demandhuge computational resources for accurate DFT calculations. Thus, these calculations are routinelylimited to material systems with at most few thousands of electrons.

In this talk, I will present Chebyshevltered subspace iteration algorithm for solving the underlying large-scale nonlinear DFT eigen-

value problem that has enabled fast, scalable and accurate DFT calculations on material systemswith tens of thousands of electrons. This has been facilitated by (i) the development of efficient andaccurate spatially adaptive discretization strategies using higher-order finite-element discretization;(ii) developing efficient and scalable algorithms in conjunction with mixed-precision strategies forthe solution of DFT nonlinear eigenvalue problem; (iii) implementation innovations, both on manycore and hybrid architectures, that significantly reduce the data movement costs and increase arith-metic intensity. These developments have resulted in providing a time-to solution that is an orderof magnitude faster than the state-of-art plane-wave based methods for similar accuracy. Further-more, a sustained performance of 53 PFLOPS has been demonstrated on a nonlinear DFT eigenvalueproblem involving the solution of 50, 000 lowest eigenvectors using 3800 GPU nodes of ‘Summit’, thecurrent fastest supercomputer in the world. This sustained performance recorded is unprecedentedfor DFT codes and 17x greater than that of any previously reported DFT code. The reportedadvance discussed in this talk has wide ranging implications in tackling critical scientific and tech-nological problems by making used of the predictive capability of DFT calculations for large-scalematerial systems. This work is jointly with Dr. Sambit Das and Prof. Vikram Gavini at Universityof Michigan, Ann Arbor.



Intra- and Interconnect Contention-aware Data Movement

Preeti Malakar,Indian Institute of Technology Kanpur, Kanpur, UP, India email: [email protected]

ABSTRACT

In this talk, we will first describe a few strategies to optimize data movements in the intraconnects(network-on-chip) of compute nodes and the interconnect of high performance computing systems,followed by communication-aware job scheduling. Some applications require producer-consumermode of communication, such as the in situ simulation-analysis codes and collective MPI I/O. Withincreasing disparity between the compute power and network bandwidth, in situ analysis is beingpopularly adopted in the HPC simulation codes. Collective MPI I/O is a popular I/O scheme usedin many parallel codes for faster read/write speeds. We studied various process mappings for suchproducer-consumer communication patterns. By default, the consumer processes (such as the MPIaggregator processes in case of collective writes) may be several hops away depending on the MPIruntime and the nodes allocated by the job scheduler. This may lead to several other communi-cations interfering with the producer-consumer communications within the application. Therefore,to reduce the interference, a group of producer and consumer processes may be placed within thesame node. This depends on the ratio of producer to consumer processes. Further, with increasingcore count per node, we demonstrate that the placement of processes within a node itself also affectsthe communication times. This is because the intraconnect topology within a node may lead tocongestion due to intranode data movements.

Next, we will describe optimizing data movement due to I/O. High-performance I/O is impor-tant for scaling up data-intensive HPC applications on large-scale systems. With complex systeminterconnects such as high-dimensional tori and Dragon-y, it is important to efficiently route the I/Opackets in the interconnect. It is also important to consider the impact of network contention onI/O performance because the I/O packets traverse several hops in the interconnect before reachingthe I/O nodes or the shared file system. In this talk, we show the utility of contention-aware I/Orouting.

Finally, we will briefly elucidate the possible role of job schedulers in improving the executiontimes of jobs by allocating nodes considering the impact of communication of neighborhood jobs. Wepropose algorithms that minimize network contention in shared switches and shared network links.We demonstrate a reduction of 326 hours of execution time on an average for tree-based topologiesusing three job logs.



Turbulent ows in Turbomachines: Role of High Performance Computing

Nagabhushana Rao Vadlamani,Indian Institute of Technology Madras, India, email: [email protected]

ABSTRACT

The architecture of a gas turbine engine, which propels an aircraft, is highly complex. Air flowingthrough the engine experiences high pressures (≈ 30−50Patm), temperatures (≈ 1500−1800 K) andhigh levels of unsteadiness due to a series of stators and rotors in the fan, compressor and turbine(rotating at ≈ O(1000rpm)). The flow field is inherently turbulent, comprising of a wide range ofspatio-temporal scales. Predicting the unsteady flows is crucial to design efficient components by re-ducing the losses, weight and fuel burn [1]. The transitional/turbulent flows can be modelled/resolvedby solving tightly coupled non-linear Navier-Stokes (NS) equations in complex geometries. With theadvent of High Performance Computing (HPC), strategies to predict such complex flows are rapidlyevolving. The gas-turbine industry is trending towards multi-objective optimization at a system levelrather than a single-objective optimization at a component level. However, due to the non-lineardynamics of the fluid flow, the inaccuracies of the low-order models at a component level accumulateand consequently alter the system level optimization. There is an increasing interest in the commu-nity to carry out high-fedelity eddy resolving simulations like Large Eddy Simulations (LES)/HybridLES-Reynolds Averaged Navier Stokes (RANS). Such simulations are computationally much moredemanding that the steady RANS simulations and require exascale computing to reasonably simu-late the entire engine [3]. With the current computing power, the use of eddy resolving simulationsis restricted to component-level/stage-level computations. These simulations can a) the detailedflow physics and statistics to improve the accuracy of the low-order models and b) be coupled withlow-fidelity methods to accurately build frameworks to handle multi-components.

Current talk will provide a broad overview of the high fidelity simulations to capture turbulentflows in gas-turbines using HPC. Importance of accelerating the CFD applications on distributedmemory (MPI), shared memory (GPU), and Multi-GPU platforms [2]; and the need for high orderschemes (preferably with data locality), will be highlighted.

References[1] Tyacke, J., Vadlamani, N. R., Trojak, W., Watson, R., Ma, Y., & Tucker, P. G. (2019). Turbo-machinery simulation challenges and the future. Progress in Aerospace Sciences, 110, 100554.[2] Bres, G. A., & Lele, S. K. (2019). Modelling of jet noise: a perspective from large-eddy simula-tions. Philosophical Transactions of the Royal Society A, 377(2159), 20190081.[3] Gourdain, N., Sicot, F., Duchaine, F., & Gicquel, L. (2014). Large eddy simulation of flows inindustrial compressors: a path from 2015 to 2035. Philosophical Trans- actions of the Royal societyA: Mathematical, Physical and engineering sciences, 372(2022), 20130323.



Bulk-Synchronization Avoiding Algorithms in Graph Applications andIterative Solvers

Sathish Vadhiyar,Department of Computational and Data Sciences Indian Institute of Science Bangalore, India


ABSTRACT

Avoiding bulk synchronization and communication between parallel processes will play a major rolefor performance on large-scale systems including future exascale systems. This talk will cover ourrecent works on avoiding global synchronization steps, one in graph applications and the other foriterative solvers. In the area of graph applications, we propose a divide-and-conquer model wherewe partition the input graph into parts for CPUs and GPUs, invoke the original algorithm on theCPU and GPU devices and then merge the results. This is shown to have very less communicationsbetween the devices when compared to thepopular Bulk Synchronous Parallel (BSP) approaches.We have demonstrated large performance benefits with this approach for different graph applicationsincluding community detection.

In the area of iterative solvers, we have proposed a pipelined preconditioned Conjugate Gradient(PCG) solver that reduce thenumber of Allreduces to only one per two iterations. This involvedrearrangement of the computations and introduction ofadditionalSpMVs that are highly amenablefor parallelism. The resulting approach is shown to provide large-scale performance improvementsover state-of-art methods in PETSc.



Higher Order and Power Efficient Numerical Methods for ModelingContinuum Phenomena on Modern Computer Architectures

S. Gopalakrishnan,Department of Mechanical Engineering, Indian Institute of Technology Bombay, [email protected]

ABSTRACT

Most physical processes in nature can be described as sets of partial differential equations. Thenon-linear nature of these complex systems from various disciplines, such as computational fluiddynamics, computational electromagnetics, geophysics etc, make the numerical solution of thesesystems inevitable. Traditional numerical methods such as the Finite Difference Method (FDM),Finite Element Method (FEM) and the Finite Volume Method (FVM) suffer from lower ordersof spatial accuracies and lack of scalability of large supercomputers. The current talk will focuson High-order continuous Galerkin (CG) and discontinuous Galerkin (DG) methods which haveenjoyed much success in obtaining the higher spatial accuracy desired in numerical simulations whileresolving complex physical phenomena. The success of CG and DG methods is due to their high-order accuracy and their impressive scalability on massively parallel (multi-core) computers; localhigh-order methods are perfectly suited for multi-core computing because the on-processor workloadis large while the communication stencil is small. Both CG and DG methods are currently ableto achieve efficiencies in the terascale and possibly to the petascale ranges. However, a multi-coreonly approach will not be able to reach the exascale range. To reach the exascale range will requirebeing able to exploit hybrid computing strategies using both central processing units (CPUs) andaccelerator systems. The higher convergence rates of the higher order methods also give an advantagein terms of effort to reach solution accuracy. In this talk, the power efficiency of this class of numericalmethods will be discussed and compared with traditional numerical methods.



Prediction of Data Distributions Using Deep Learning

Suja A. Alex, St. Xaviers Catholic College of Engineering, Nagercoil, India,email:[email protected]

ABSTRACT

Existing machine learning models can be trained for classification and regression but data generationis impossible. Deep learning is a modern machine learning technique which works based on the data.The Markov Chain Monte Carlo plays an important role in Deep learning. The goal of this paperis to develop a Deep learning model which performs data generation for the given input data bydetermining its probability distributions. Hence a new probability model called Deep Belief Networkis designed using Restricted Boltzmann Machine. The prediction performance of the model alsoevaluated using RME.

References[1] William Bolstad, Comparing the Bayesian and Likelihood Approaches to Inference: A Graphical

Approach, Eighth International Conference on Teaching Statistics , ICOTS8, July, 2010.

[2] Mohamed, Abdel-rahman, Geoffrey Hinton, and Gerald Penn, Understanding how deep beliefnetworks perform acoustic modelling, IEEE International Conference on Acoustics, Speech andSignal Processing (ICASSP), IEEE, 2012.

[3] Fischer, Asja, and Christian Igel, An introduction to restricted Boltzmann machines, Iberoamer-ican Congress on Pattern Recognition. Springer Berlin Heidelberg, 2012.

[4] Karakida, R.M. Okada, and S. Amari, Analyzing feature extraction by contrastive divergencelearning in RBM, NIPS Workshop on Deep Learning, 2014.

[5] Hinton, Geoffrey.E, Deep belief networks, Scholarpedia 4.5, 2009.



Parallel Numerical Solution of Bates PIDE for Pricing Options on a GPU

Abhijit Ghosh, Indian Institute of Technology Ropar, Ropar, India, email:[email protected]

Chittaranjan Mishra, Indian Institute of Technology Ropar, Ropar, India.

ABSTRACT

Since the onset of the current pandemic, financial markets have become extremely volatile. Pricingof financial options in the present scenario with the classical Black-Scholes framework does notproduce accurate market dynamics. There is a pressing need to consider stochastic volatility withjumps in the pricing model, such as Bates jump-diffusion model [1]. This leads one to a PartialIntegro-Differential Equation (PIDE) whose solution determines the option price. Although robustnumerical solutions of this PIDE exist in the literature, see [3], they are often slow to compute forpractical purposes. This is mainly because, like in many other scientific problems, we encounter aproblem to solve huge numbers of large tridiagonal and pentadiagonal systems together. This hasmotivated us to look for a high-speed parallel computation of the numerical solution.

However, obtaining a parallel algorithm to solve the said problem is just not straight-forwardsince there is no ready-made algorithm that can be directly applied to solve the problem as such.Designing a suitable algorithm to tackle the problem is very challenging and employing one optimallyon a specific hardware is still tougher.

Graphics processing units (GPUs) are increasingly becoming popular for scientific computationsdue to their unique hardware architecture, computing power and affordability. We have designeda new parallel cyclic reduction algorithm for pentadiagonal systems that has fine-grain parallelism[2]. The parallel system solver optimally exploits the architectural benefit as well as the computingpower of a modern GPU. We present the algorithm, discuss its implementation on an NVIDIA TeslaK80 GPU using CUDA in order to harness its full computing potential and employ the solver tosolve the Bates PIDE where we have achieved significant speed-up over a CPU computation usingMATLAB.

References[1] D. Bates, Jumps and stochastic volatility: the exchange rate processes implicit in Deutsche

mark options, Rev. Financ. Stud. 9 (1996) 69–107.

[2] A. Ghosh & C. Mishra, A parallel cyclic reduction algorithm for pentadiagonal systems withapplication to convection-dominated Heston PDE, (under revision - SIAM).

[3] K. J. in ’t Hout & J. Toivanen, ADI schemes for valuing European options under the Batesmodel, Appl. Num. Math. 130 (2018) 143–156.


5 Minisymposia: Delay, Functional and DynamicEquations with Applications

Organizers:

• Prof. Syed Abbas, School of Basic Sciences, IIT Mandi, India. Email: [email protected]


Qualitative Theory of Dynamic Equations on Time Scale

Syed AbbasSchool of Basic Sciences, Indian Institute of Technology Mandi, Mandi, 175005, H.P., India.

email:[email protected]; [email protected]

ABSTRACT

In this talk, we discuss the theory of dynamic equations on time scale. This theory unifies the theoryof continuous and disceete calculus under cetain conditions. We discuss the existence of almostperiodic and some more general kind of solutions of such systems.



Bifurcation Results for Fractional Laplace Equation

Jagmohan Tyagi, IIT Gandhinagar, Gandhinagar, India, [email protected]

ABSTRACT

In this talk, I will discuss the bifurcation results for fractional Laplace equations.



Fractional Differential Operators: Different definitions and variousapproaches

Jehad AlzabutDepartment of Mathematics and General Sciences, Prince Sultan University

Riyadh, Saudi Arabia, email:[email protected]

ABSTRACT

This talk is designed to provide an informative platform about some newly defined fractional dif-ferential operators. We will present some recent definitions of fractional derivatives and integralsand discuss their properties. A comparison with classical definitions will be conducted and furthera discussion in terms of their advantageous and disadvantageous will be addressed. For the sake ofcompleteness, we will expose some recent works handled by the new definitions.

References[1] A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Application of Fractional Differential

Equations (North Holland Mathematics Studies 204, 2006).

[2] F. Jarad, T. Abdeljawad, J. Alzabut: Generalized fractional derivatives generated by a class oflocal proportional derivatives, Eur. Phys. J. Special Topics 226 (16-18) (2017), 3457-3471.

[3] D. R. Anderson, D. J. Ulness: Newly defined conformable derivatives, Adv. Dyn. Sys. App.10(2) 109–137 (2015).

[4] M. Caputo, M. Fabrizio: A new definition of fractional derivative without singular kernel,Progr. Fract. Differ. Appl. 1, 73–85 (2015).

[5] R. Khalil, M. Al Horani, A.Yousef, M. Sababheh: A new Definition Of Fractional Derivative,J. Comput. Appl. Math. 264 (2014), 6570.

[6] A. Atangana: New fractional derivatives with nonlocal and non-singular kernel: Theory andapplication to heat transfer model, Thermal Science 2016, DOI: 10.2298/TSCI160111018A.

[7] R. Almeida: A Caputo fractional derivative of a function with respect to another func-tion,Communications in Nonlinear Science and Numerical Simulation 44 (2016), DOI:10.1016/j.cnsns.2016.09.006.



Extended Fractional Derivative Operators and Applications

Praveen AgarwalAnand International College of Engineering, Jaipur, India, email:[email protected]

ABSTRACT

In this talk, our principal aim to discus on the extended fractional derivative operators and theirapplications. As we all known many authors extended the fractional derivatives operators by usingthe generalized special functions. In this talk, we highlight the extended Riemann-Liouville andCaputo type fractional operators involving generalized hypergeometric functions and their properties.



Wavelet Preconditioned Method for the Numerical Solution of StochasticDifferential Equations

Mounesha H. Kantli,Department of Mathematics, Biluru Gurubasava Mahaswamiji Institute of Technology,

Mudhol-587313, India,email: [email protected]

ABSTRACT

In this paper, wavelet preconditioned method is used for the numerical solution of stochastic dif-ferential equation. The proposed method is the robust technique for faster convergence with lowcomputational cost which is acceptable through error (L∞) and CPU time. It is concluded that thewavelet preconditioned technique easily outperforms over existing standard classical preconditionedmethods.


Conditional and Unconditional Finite Difference Methods for a CoupledSystem of Hyperbolic Delay Differential Equations

S. Karthick, Department of Mathematics, Faculty of Engineering and Technology,SRM Institute ofScience and Technology, Kattankulathur-603 203, Tamilnadu, India.


V. Subburayan, Department of Mathematics, Faculty of Engineering and Technology,SRMInstitute of Science and Technology, Kattankulathur-603 203, Tamilnadu, India

ABSTRACT

In this article system of first order hyperbolic delay differential equations are considered. A condi-tionally stable and an unconditionally stable finite difference methods with linear interpolation aresuggested to solve the problem. It is proved that, the methods are consistent, stable and convergent.Numerical illustrative examples are given to validate the theoretical results.


Stability Analysis of a Spatially Coupled Model with Delayed Dispersal

Binandita Barmana, National Institute of Technology Meghalaya, Shillong, Meghalaya, INDIA,email: [email protected]

Bapan Ghoshb, Indian Institute of Technology IndoreSimrol, Indore 453552, Madhya Pradesh, INDIA.

ABSTRACT

Dispersal of species from one place to another in search of food or due to competition, survival etc. isan indispensable phenomenon in ecology. Many literatures can be found which studies the predator-prey interactions subjected to dispersal between patches. It is mostly accepted that instantaneousdispersal does not alter the system stability. In this paper we consider a two-patch predator-preymodel with prey dispersal. In absence of predator, the movement of prey is density independent.Since predators affect the movement of prey, the predator-influenced dispersal of prey is taken intoaccount. Traveling time (time delay) linked with the movement process among the prey communityis incorporated. The positivity and boundedness of the solutions in the spatially coupled systemare established. Stability analysis due to delay parameter has been investigated. It is found that,unlike the case of instantaneous dispersal, delayed prey dispersal can potentially alter the stabilityand even cause stability switching around the steady state equilibrium. The system is unstableprior to incorporation of delay. However, after some consecutive changes in stability, the equilibriumundergoes instability for larger delay. Numerical examples are considered to explain the results.



Existence of Hydra Effect in Unstructured Predator-prey Model

Prabir Das Adhikary, Department of Mathematics, National Institute of Technology Meghalaya,Bijni Complex, Shillong-793003, Meghalaya, India email:[email protected]

Saikat Mukherjee, Department of Mathematics, National Institute of Technology Meghalaya, BijniComplex, Shillong-793003, Meghalaya, India.

Bapan Ghosh, Discipline of Mathematics, Indian Institute of Technology Indore, Khandwa Road,Simrol 453552, Madhya Pradesh, India.

ABSTRACT

We investigate predator-prey model under constant rate harvesting of populations. Several bifurca-tions are identified and the patterns of the stock size under population harvesting are shown. It isproved that hydra effect occurs at a stable state, when predator species is culled. On the contrary,prey harvesting does not induce any hydra-effect at stable state. Single Hopf-bifurcation, multi-ple Hopf-bifurcation and multiple saddle-node bifurcation occur when predator species is harvested.Multiple hydra effect also occurs with increasing harvesting effort of predator. The basin of at-traction of the equilibrium corresponding to the upper predator stock is expanded due to predatorharvesting. Maximum sustainable yield exists and appears at globally asymptotically stable state.



Stage Structured Prey Predator Model with Maturation and GestationDelay for Predator using Holling Type 2 Functional Resonse

Vandana Kumari, Amity Institute of Applied Sciences/ Amity University, Noida, U.P, India,[email protected]

Sudipa Chauhan, Amity Institute of Applied Science/ Amity University, Noida, India,Nisha Sharma Amity Institute of Applied Science/ Amity University, Noida, India,

Sumit Kaur Bhatia Amity Institute of Applied Science/ Amity University, Noida, India,Joydip Dhar ABV-Indian Institute of information Technology and Management, Gwalior, M.P,

India.

ABSTRACT

In this paper, we have proposed a prey predator model with stage structuring in predator. We haveincorporated maturation and gestation delay in predator class. We have studied the existence ofboundary and interior equilibrium points and the positivity of the model. Next, we have discussedthe local stability of the trivial, boundary and interior equilibrium point. We have also discussedthree cases for the local stability analysis of E∗ :Case 1: τ1 = τ = 0Case 2:τ1 = 0, τ > 0Case 3: τ1 > 0, τ = 0Case 4: τ1 > 0, τ > 0From the local stability of Case 2,3 and Case 4, we have obtained the bifurcating parameter τ10 andτ0 = τ . We have shown that E∗ is asymptotically stable if the maturation delay 0 < τ1 < τ∗1 and hasHopf bifurcation when τ1 > τ∗1 . Further, we have also analyzed that if τ1 > 0 and τ < τ∗ then E∗isasymptotically stable and if τ ≥ τ∗ then, Hopf bifurcation occurs. The permanence of the systemusing comparison theorem is also obtained by evaluating the least upper bound and greatest lowerbound of the populations. At last, numerical example has been given to validate our theoreticalresults using MATLAB software.

References:

[1] K.Jatav and J. Dhar, Global behaviour and hopf bifurcation of stage structured prey predatormodel with maturation delay for prey and gestation delay for predator, Journal of Biological System23(2015), 57–77.

[2] S.K Golam, Mortojan, P. Panja and M.K. Shyamal, Dynamics of a Predator-Prey model withstage-structuring on both species and anti predator behaviour, Informatics in Medicine Un-locked10(2018),50–57.


6 Minisymposia: Mathematical Aspects of WaterWaves and Applications

Organizers:

• Prof. S. C. Martha, Department of Mathematics, IIT Ropar, Rupnagar, India. Email: [email protected]


Bragg Scattering of Long Waves by an Array of Floating Flexible Platesin the Presence of Multiple Submerged Breakwaters

P. Kar, IIT Kharagpur, India. email: [email protected]

T. Sahoo, IIT Kharagpur, India. email: [email protected]

ABSTRACT

One of the greatest challenges due to global warming is the rise in sea level which would contributeto flooding, storm surges, coastal erosion and inundation of the low-lying coastal regions. As anenvironmentally friendly and sustainable technological solution to meet the demand for land spaceand address some of the challenges due to sea level rise, the concept of floating structure was intro-duced for utilization of ocean space. These structures are planned to be kept in position with thehelp of an appropriate mooring system that will restrict the horizontal movement. Due to the largesurface area and small thickness, these structures deform elastically due to wave-induced load. Toreduce the wave loads on such various coastal infrastructures and to create a tranquillity zone in themarine environment in a cost-effective and environmentally friendly manner, various types of peri-odic bottom bed profiles such as submerged breakwaters have been proposed. During the last twodecades, for various serviceability conditions, several techniques have been proposed for mitigatingwave-induced structural responses on floating structures due to periodic sea-bed undulation.In the present study, an investigation is carried out to analyze the Bragg scattering of surface gravitywaves by an array of floating flexible structures in the presence of multiple bottom-standing sub-merged breakwaters in two-dimensions under the assumption of linear long wave theory and smallamplitude structural response. To account for the structural flexibility, the structure is modelledusing Euler-Bernoulli beam having free edges. The solution of the associated mathematical problemis obtained using the method of eigenfunction expansion method and matching the velocity andpressure at the interface boundaries of the breakwaters and floating plates. Matrix multiplication isused to derive explicit expression for the reflection and transmission coefficients. Various configura-tions and locations of the submerged breakwaters are considered to find the optimum position of thefloating structure for minimizing the wave-induced hydroelastic structural responses. Various resultssuch as reflection and transmission coefficients, plate deflection, bending moment and shear forcesacting on the structures will be computed and analyzed for various wave and structural parametersand breakwater position and configurations. Known results in the literature are reproduced to checkthe accuracy of the computational results.

ReferencesP. Kar, T. Sahoo, and M. H. Meylan (2020) Bragg scattering of long waves by an array of floatingflexible plates in the presence of multiple submerged trenches, Physics of Fluids, 32(9), 096603.



Localized Structures in Integrable Models

Sudhir Singh, National Institute of Technology, Tiruchirappalli, India, email:[email protected]

ABSTRACT

In this work, two integrable models, namely the Benjamin-Ono equation in (1+1)-dimension andBoussinesq equation in (2+1)-dimension, is studied. The Hirota’s bilinear approach and appropriatetest functions are utilized for the construction of several localized structures, including rogue wavesof bright and dark nature along with interaction solutions, are obtained. Moreover, their dynamicsis explored for various choices of free parameters.



Flexural Gravity Wave Blocking in Coupled Plate System

Susam Boral, IIT Kharagpur, India email: [email protected]

Trilochan Sahoo, IIT Kharagpur, India email: [email protected]

ABSTRACT

One of the biggest challenge for mankind is the scarcity of land space for various developmentalactivities. As 2/3rd of the earth surface is covered by ocean, there is a need to utilise the oceanspace for various human activities by developing large platforms which can float on the sea surfacelike a giant structure. During the last three decades, there is a significant interest in the study ofhydroelastic analysis of very large floating structures. A similar class of problems arise in the studyof wave-ice interaction problems in which the floating ice sheet is modelled as an elastic plate. Therole of axial forces with/without current on the hydroelastic analysis of these structures has lead tothe study on blocking dynamics of flexural gravity waves. The phenomenon of blocking is leadingto a class of eigenvalue problem in which the eigen-system possesses non-distinct eigenvalues. As aresult, finding the solution of the associated boundary value problem has becomes very complex andinvestigating such physical problems during blocking has brought in new challenges for analysis.

In the present study, a complete mathematical model of surface gravity wave interaction with acoupled arrangement of very large flexible structures is developed, in which one is floating on the freesurface and the other is fixed at the sea bed. The physical problem is analysed under the assumptionof linear water wave theory and small amplitude structural response. Euler-Bernoulli beam equationis used for structural analysis. Because of the presence of floating structure with flexible seabed,two flexural gravity wave modes are generated and thus wave blocking occurs in both the modesfor certain frequencies. For some particular frequency between primary and secondary blockingpoints, there exist four real roots of the dispersion relation under the deep water approximationwith coalescence of roots occurring during blocking. The study reveals that the stopping velocity,buckling limit of compressive force corresponding to the floating plate and the bottom fixed platedepend on the compressive force, current speed and flexural rigidity of the corresponding plate.



Linear Water Wave Interaction With a Composite Porous Structure in aTwo Layer Fluid Flowing over a Step Like Seabed

Koushik Kanti Barman

ABSTRACT

The present work is concerned with the interaction of oblique surface gravity waves by a simpleand composite porous block of finite width placed on a multi-step bottom in a two-layer fluid. Theocean depth is taken to be finite and its bed impermeable. The problem is studied by employinglinearized water wave theory and eigenfunction expansion. The dispersion relations and their rootsare analysed which give a clear understanding of the phenomenon. The cases of simple and compositeinterface-piercing structures are taken up separately to investigate the impact of porosity in waveattenuation for surface and interface modes. Waves propagate through the porous structure withdistinct eigenvalues. The appropriateness of structures of various configurations on the scatteringof surface waves is investigated by examining the reflection coefficients for waves in surface andinterface modes as well as their effects on the free surface and interface elevations, the wave-loads onthe structure and the rigid wall supporting the structure at one end. Further, as a special case, thesea-bed preceding the step bottom is considered to be porous and its effect on reflection is examined.The investigation establishes that for a suitable configuration of the porous structure, an optimumwidth can be ascertained to design a breakwater of reasonable efficiency possessing characteristics ofboth reflection and dissipation processes. The problems are solved analytically and the results arepresented in graphical form. This kind of study is likely to have immense significance for designingof different types of coastal structures with respect to reflection and dissipation of wave energy atcontinental shelves which is influenced by a stratified fluid, which is modeled in this work as a two-layer fluid for convenience. Comparison of present results with available results show good agreementand this points towards the effectiveness of the model described in this work.



Role of Linear Algebra in the Understanding of Wave Impact on Seawallby a Pair of Asymmetrical Trenches

Amandeep Kaur, Indian Institute of Technology Ropar, Punjab, India,Email: [email protected]

S. C. Martha, Indian Institute of Technology Ropar, Punjab, India.A. Chakrabarti, Indian Institute of Science, Banglore, India.

ABSTRACT

The study involving propagation of surface water waves over an uneven bottom is important dueto their significant applications in the field of ocean and marine engineering. During the last fewdecades, different kinds of bottom topography have been investigated to examine the characteristicsof wave energy transformation. In this context, the problems of propagation of surface water wavesover bottom undulation in a finite depth of water were studied by various researchers (ref. Kirbyand Dalrymple [1], Xie and Liu [2], Chakraborty and Mandal [3]). In the present work, the problemof wave diffraction by a pair of asymmetrical trench in the presence of seawall is examined for itssolution with the aid of solution of an overdetermined system of linear algebraic equations. Theresulting overdetermined system of linear algebraic equations is solved approximately, by using theleast square method giving rise to the numerical results of the important practical quantities ofinterest such as force experienced by seawall, surface wave elevation. These quantities are analysedthrough different graphs to study the transformation of wave energy by a pair of trenches for variousvalues of system parameters. It is observed that the said algebraic method is very quick and simple,provides the best solution to the overdetermined system of equations arising in the said problem.Moreover, it is observed that the creation of a pair of trenches imposed on the seabed helps to reducethe wave load on seawall.

References[1] Kirby, J. T., Dalrymple, R.A. Propagation of obliquely incident waves over a trench. Journal of

Fluid Mechanics, 133 (1983) 47-63.

[2] Xie, J.J., Liu, H.-W. An exact analytic solution to the modified mild-slope equation for wavespropagating over a trench with various shapes. Ocean Engineering, 50 (2012) 72-82.

[3] Chakraborty, R., Mandal, B.N. Oblique wave scattering by a rectangular submarine trench.The ANZIAM Journal, 56 (2015) 286-298.

[4] Amandeep Kaur, S. C. Martha, A. Chakrabarti, Linear algebraic method of solution for theproblem of mitigation of wave energy near seashore by trench-type bottom topography, Journalof Engineering Mechanics 2020, 146 (11): 04020125.



Effect of Flexible Porous Breakwater in Mitigating HydroelasticResponses of a Very Large Floating Structure

Sofia Singla, Indian Institute of Technology Ropar, Rupnagar, India,email:[email protected]

S. C. Martha, Indian Institute of Technology Ropar, Rupnagar, India.T. Sahoo, Indian Institute of Technology Kharagpur, Kharagpur, India.

ABSTRACT

Very large floating structures (vlfs) are proposed as alternatives for coping with the shortage ofspace for various human activities. Protecting these vlfs from wave action in the ocean is highlyimportant. In this context, the reduction of the hydroelastic response of a vlfs by flexible porousbarrier over a uniform finite depth of water is examined. The mathematical problem is formulatedbased on the assumption of linearized water wave theory and small amplitude structural response.The eigenfunction expansion method is used to solve the boundary value problem involving twodimensional Laplaces equation and mixed boundary conditions. The effectiveness of the flexibleporous barrier for mitigating the structural response of the very floating structure is studied byanalyzing the physical quantities such as reflection and transmission coefficients which highlight theglobal characteristics of wave energy transformation. In addition, results on the force and strain onvlfs are analyzed for various structural and wave parameters. The study shall be useful to oceanengineers engaged in the design and protection of vlfs.



Wave Scattering by a Floating Porous Breakwater over a RectangularBottom-trench

N.M. Prasad, IIT(ISM) Dhanbad, Jharkhand, India, email:[email protected]. Nikhil , IIT(ISM) Dhanbad, Jharkhand, India.

R.B. Kaligatla, IIT(ISM) Dhanbad, Jharkhand, India.

ABSTRACT

Trench configurations in ocean bathymetry often encounter and have high reflective property forwaves. Chakraborty and Mandal (2014) reported that much deeper trenches are accounted forthe reflection of more wave energy. The higher reflected waves have to be controlled for the safenavigation of submarines and the other under water vehicles. In view of this necessity, a modelproblem of oblique wave scattering by a porous sheet of finite width, floating over a rectangulartrench is studied. Matched eigenfunction expansion method is utilized as a solution procedure. Theresults reveal that the floating porous breakwater reduces the reflected waves significantly. Further,the forces on the breakwater for different values of porous parameter and breakwater’s width, arealso computed here.



Wave Diffraction by a Small Base Deformation on a Flexible Bed in anIce-covered Fluid

Sagarika Khuntia, Veer Surendra Sai University of Technology, Sambalpur, India, email:[email protected]

Smrutiranjan Mohapatra, Veer Surendra Sai University of Technology, Sambalpur, India.

ABSTRACT

The diffraction of surface waves by an obstacle or a geometrical disturbance at the bottom of anocean is important for its possible applications in the area of coastal and marine engineering, andas such it is being studied with interest for a long time. The diffraction of normal incident waterwaves with a small base distortion on a flexible bed in an ice-covered fluid is examined analyticallyhere within the framework of linear water wave theory. The ice-cover surface is modelled as anelastic plate of very small thickness. The expression for first-order potential and, henceforth, thereflection and transmission coefficients up to first-order are acquired by the strategy in view ofGreen’s function method. A patch of sinusoidal ripples is considered as an example and the relatedcoefficients are determined. The main advantage of the present study is that the results for the valuesof reflection and transmission coefficients obtained are found to satisfy the energy-balance relationalmost accurately.



Water Wave Scattering by a Flexible Porous Plate in the Presence of aSubmerged Porous Structure

Uma Vinod Kumar, Dayananda Sagar University, Bangalore, India,email:[email protected]

S.Saha, SAS, VIT, Vellore, India.S.N.Bora, IIT Guwahati, Guwahati, India.

ABSTRACT

A matched eigenfunction method is employed to investigate the scattering of oblique surface gravitywaves by a system consisting of a thin floating porous-elastic plate and a rectangular submergedporous structure, in finite water depth, with usual assumptions of linear theory for water wavesand structure response. The flexible plate is modelled based on thin plate theory, neglecting com-pressive forces and incorporating a porous effect parameter while the flow through the submergedstructure follows the Sollitt and Cross model. A matched-eigenfunction method is employed to setup a system of linear equations that is solved to obtain the hydrodynamic scattering coefficients forvarious physical parameters associated with the wave and structure. The roots of the associatedcomplex dispersion relation are analysed using contour plots. Reflection, transmission and dissipa-tion coefficients are calculated to understand the effect of various wave and structural parameterson wave-scattering by this composite system. Validity of the conclusions has been confirmed bycomparing with established results for a submerged rectangular porous structure that are availablein the literature.It is found that energy dissipation due to porosity of plate and structure cause appreciable reduc-tion in transmission. Porous effect of the plate material dominates the scattering property of theplate/structure system, contributing to an exponential decrease in transmission coefficient with in-crease in relative width, whereas variation in the material characteristics of the submerged porousstructure has little effect on wave reflection and transmission for the values of the parameters con-sidered in this study. The results of this study can be used in the design of a composite breakwateror for analysing the performance of a floating breakwater in the presence of inhomogeneous bottomtopography.

REFERENCES1. Koley, S., Mondal, R. and Sahoo, T. [2018] “Fredholm integral equation technique for hydroelasticanalysis of a floating flexible porous plate,” Eur. J. Mech. B-Fluid 67, 291-305.2. Koley, S. and Sahoo, T. [2018] “An Integro-differential equation approach to study the scatteringof water waves by a floating flexible porous plate,” Geophysical and Astrophysical Fluid Dynamics112(5), 345-356.3. Losada, I. J., Silva, R. and Losada, M. A. [1996] “3-D non-breaking regular wave interaction withsubmerged breakwaters,” Coastal Engineering 28, 229-248.



Water Wave Radiation by an Immersed Cylinder in a Channel withFlexible Base Surface

Lopamudra Das, Veer Surendra Sai University of Technology, Odisha, India, email:[email protected]

Smrutiranjan Mohapatra, Veer Surendra Sai University of Technology, Odisha, India.

ABSTRACT

A three-dimensional problem of water wave radiation by an immersed horizontal circular cylinderin a single layer fluid streaming in a channel is investigated within the framework of linear wavetheory. The upper surface of the channel is assumed to be bounded by a rigid lid, while the lowerone is bounded by a flexible base surface. The flexible base surface is considered as a thin elasticplate. In such a situation, the time-harmonic waves propagate with one wavenumber only, unlikethe case the waves propagating at two different wavenumbers. The method of multipole expansiontechnique is used to find the solution of the above boundary value problem. This method is found tobe an extremely powerful method for solving this problem. It eliminates the need to use large andcumbersome numerical packages for the solution of such radiation problem. The added mass anddamping coefficient for both heave and sway motions are studied.



Wave Scattering by a Pair of Floating Pontoons having a SubmergedMesh Cage

V. Venkateswarlu, Department of Civil Engineering, Bapatla Engineering College,Mahatmajipuram, Bapatla, Andhra Pradesh-522102, India

C.S. Nishad, Department of Mathematics, School of Technology, Pandit Deendayal PetroleumUniversity, Gandhinagar, Gujarat-382007, India

K.G. Vijay, Oceaneering International Services Limited, Chandigarh-160101, IndiaT. Sahoo, Department of Ocean Engineering and Naval Architecture, IIT Kharagpur, West

Bengal-721302, India

ABSTRACT

In recent decades, recurrence of storm surges is on the rise due to global warming which has resultedin a significant threat to coastal facilities and infrastructure. Often temporary floating structuresprovide a cost-effective and feasible solution which is because these structures are independent ofwater depth, reusable and easy to install and decommission. In the present study, scattering of gravitywaves by a pair of floating pontoons having a mesh cage is studied under the assumptions of smallamplitude wave theory. The waves past the mesh cage are assumed to follow nonlinear pressure dropboundary condition to capture the effect of non-breaking wave height on the energy dissipation. Tosolve the boundary value problem, a numerical model based on the dual boundary element method isdeveloped. The code is initially validated with known results in the literature. For different wave andstructural parameters such as relative water depth, relative spacing and different barrier porositiesvarious physical quantities of interests like scattering coefficients, horizontal and vertical forces arecomputed and analyzed. The study will be useful in the design of effective breakwaters as waveabsorption systems.

Keywords: Floating pontoons, mesh cage, non-linear pressure drop, dual boundary elementmethod.



Expansion Formula for the Flexural-gravity Waves During Wave Blocking

Sunil Chandra Barman , Indian Institute of Technology Kharagpur, West Bengal, India, email:[email protected]

Trilochan Sahoo, Indian Institute of Technology Kharagpur, West Bengal, India,

ABSTRACT

During the last two decades, there is a significant growth of knowledge in the understanding of flexuralgravity waves which are generated due to the interaction of surface gravity waves with large floatingice sheet as well as very large floating structures. In these studies, both ice sheet and very largefloating structures are modelled as a thin elastic plate and often modelled using Euler-Bernoulli beamequation in one-dimensions. Under the assumption of linear water wave theory and small amplitudestructural response, the boundary condition on the flexible structure becomes 5th order for theboundary value problem in which the governing equation is the two-dimensional Laplace equation.As a result, the boundary value problems associated with this class of problems are non-Sturm-Liouville type in nature and the associated eigenfunctions are not orthogonal in the usual sense. Thestudy has become more interesting in the context of blocking dynamics of flexural gravity waves.Within the limits of flexural gravity wave blocking and buckling, the coalescence of roots of dispersionrelation occurs. Thus, in the context of blocking dynamics, the expansion formulae associated withflexural gravity waves in a homogeneous fluid having ice-covered sea surface are revisited in both thecases of finite and infinite water depths. These expansion formulae and characteristics of associatedeigensystem are obtained using Greens function technique and the use of Cauchy integral theorem.As an application of the expansion formulae, flexural gravity wave scattering due to a crack ina floating ice-sheet in the presence of compression is investigated. Various physical quantities ofinterest such as the reflection and transmission coefficients are obtained analytically and analysednumerically to understand the role of compressive during blocking and buckling.



Diffraction by an Oscillating Water column in Presence ofBottom-mounted Obstacle in the Channel of Finite Width

Dr. M. Hassan,North Eastern Regional Institute of Science and Technology, Arunachal Pardesh, India,

email:[email protected]

P. Borah, M. Hassan North Eastern Regional Institute of Science and Technology, ArunachalPardesh, India

ABSTRACT

In this work, we investigated the problem of diffraction in water waves by an oscillating water columnin a channel of finite width. It consists of a partially submerged foating hollow cylinder place above afixed coaxial bottom-mounted obstacle. With the help of matched eigenfunction expansion, channelmultipole and variables separation methods, the analytical expression of potentials are obtained inrespective regions. By using these analytical expressions of diffracted velocity potentials, we canderive the expressions of exciting forces exerted by the cylinders. The effect of various parameters,viz, draft of the hollow cylinder, radius of the cylinders, the gap between the cylinders and the widthof the channel walls on the exciting forces have been investigated. It has found that no oscillation

occurs for those value of r2(radius of lower cylinder) when r2 >1

2d, where d refers width of the

channel walls. Also we observe oscillation for higher value of r2, e.g. r2 = 0.2H and 0.3H in which

r2 ≤1

2d. As concerned the width of the channel walls, we observe that the values of the exciting

force increases while the values of width of the channel walls decreases.

References[1] Bhatta DD, Rahman M (2003) On scatering and radiation problem for a cylinder in water of

finite depth. Int. J. of Engg. Sci. 41(9):931-967

[2] Buffer BP, Thomas GP (1993) The diffraction of water waves by an array of circular cylindersin a channel, Ocean Engg. 20:296-311.

[3] Hassan M, Bora SN (2014) Hydrodynamic coefficients for a radiating hollow cylinder placedabove a coaxial cylinder at finite ocean depth. J. Mar. Sci. Technol., 19:450-461.

[4] Mclver P, Bennett GS (1993) Scattering of water waves by axisymmetric bodies in a channel. J.of Engg. Math. 27:1-29.

[5] Yeung RW, Sphaier SH (1989) Wave-interference effects on a truncated cylinder in a channel. J.of Engg. Math. 23:95-117.

[6] Zhu SP, Mitchell L (2009) Diffraction of ocean waves around a hollow cylindrical shell structure.Wave Motion, 46:78-88.



Water Waves Interaction with Slatted Screens Placed Near a CaissonBreakwater

Kottala Panduranga, Department of Mathematics, Birla Institute of Technology and Science-Pilani,Hyderabad Campus, Telangana, India-500078 email:[email protected]

Santanu Koley, Department of Mathematics, Birla Institute of Technology and Science-Pilani,Hyderabad Campus, Telangana, India-500078. email:[email protected]

Trilochan Sahoo, Department of Ocean Engineering and Naval Architecture, Indian Institute ofTechnology Kharagpur, West Bengal, India-721302. email:[email protected]

ABSTRACT

Breakwaters are one of the most commonly used coastal structures to protect the harbor, anchorage,or shorelines from the extreme wave loads. In this regard, slatted breakwater/slatted screens arethe most efficient and widely used structures for practical coastal engineering purposes as it caneffectively reflect and dissipate the incident wave energy. In general, slatted screens consist of aseries of slots or holes. When wave impinges upon slatted wave screens, jet-like flows are createdthrough the screens perforations, and as a result, the incident wave energy is dissipated. In thepresent study, numerical solutions are developed to analyze the scattering of surface gravity wavesby the multiple slatted screens placed near a porous caisson breakwater. A non-linear pressure dropcondition, which includes both inertial and drag effects is considered to model the waves past aslatted wall. Further, an iterative boundary element method is adopted to handle the non-linearpressure drop condition. The wave energy dissipation, wave loads, wave reflection, and transmissionare analyzed for various values of perforation-effect KeuleganCarpenter (KC) number. The influenceof the incident wavenumber, perforation-effect KC-number, and the gap between the slatted screenson the wave reflection and transmission, wave energy dissipation and wave loads are illustrated indetail. It is concluded that the reflection coefficient, wave energy dissipation and wave forces stronglydepend on the perforation-effect Keulegan-Carpenter KC number and the gap length between theslatted screens. Appropriate optimization of these two parameters can significantly enhance thedurability of the caisson breakwater.



Water wave scattering by a porous breakwater floating over arectangular bottom-mound

R.B. Kaligatla, IIT(ISM) Dhanbad, Jharkhand, India, email:[email protected]. Agrawal, IIT(ISM) Dhanbad, Jharkhand, India.

N.M. Prasad, IIT(ISM) Dhanbad, Jharkhand, India.

ABSTRACT

A model problem of linear oblique wave scattering by a thin horizontal porous breakwater, floatingover a rectangular bottom-mound in sea, is studied. The breakwater is assumed to have a finitewidth and infinite length. In practical applications, the bottom-mound is viewed as a submergedstorage tank (Mei and Black (1969)). A porous boundary condition is used from Yu (1995). Theeigenfunction expansion method is employed to Helmholtz equation for its solution. Wave reflectionand transmission coefficients and wave forces on the mound are analyzed. Results reveal that wavereflection increases, and transmission decreases as the mound height increases. Moreover, the verticalforce on the mound decreases with an increase in the breakwater’s width and the porous effectparameter.


7 Minisymposia: Modelling and Simulation of Flowand Transport Processes in Porous Media

Organizers:

• Prof . G. P. Raja Sekhar, Department of Mathematics, Indian Institute of Technology Kharag-pur, India. Email: [email protected]


Modeling of Fluid-poroelastic Structure Interaction

Ivan Yotov, Department of Mathematics, University of Pittsburgh, USAemail: [email protected]

ABSTRACT

We study mathematical models and their finite element approximations for solving the coupled prob-lem arising in the interaction between a free fluid and a fluid in a poroelastic material. Applicationsof interest include flows in fractured poroelastic media and arterial flows. The free fluid flow is gov-erned by the Navier-Stokes or Stokes/Brinkman equations, while the poroelastic material is modeledusing the Biot system of poroelasticity. We present several approaches to impose the continuity ofnormal flux, including an interior penalty method and a Lagrange multiplier method. A dimension-ally reduced fracture model based on averaging the equations over the cross-sections will also bepresented. Stability, accuracy, and robustness of the methods will be discussed.



Upscaling of Flow Models in a Fractured Porous Medium

Kundan Kumar, Department of Mathematics, University of Bergen, Bergen, Norway, email:[email protected]

ABSTRACT

We consider a mathematical model for flow in an unsaturated porous medium containing a fracture.In all subdomains (the fracture and the adjacent matrix blocks) the flow is governed by Richards’equation. The submodels are coupled by physical transmission conditions expressing the continuityof the normal fluxes and of the pressures. We start by analyzing the case of a fracture having afixed width-length ratio, called ε. Then we take the limit ε tending to zero and give a rigorousproof for the convergence toward effective models. Our results show that the ratio of porosities andpermeabilities in the fracture to matrix determine, to the leading order of approximation, the appro-priate effective model. In these models the fracture is a lower dimensional object for which differenttransversally averaged models are derived depending on the ratio of the porosities and permeabili-ties of the fracture and respective matrix blocks. We obtain a catalogue of effective models whichare validated by numerical computations. Numerical simulations confirm the theoretical upscalingresults. We extend these results for polymer enhanced oil recovery models and two phase flow models.

This is a joint work with Florin Radu (Bergen), Sorin Pop (Hasselt), Florian List (Sydney/Hasselt),and Martin Dugstad (Bergen).



A Two-scale Adaptive Scheme for Mineral Precipitation and Dissolutionin Porous Media

Carina Bringedal, Institute for Modelling Hydraulic and Environmental Systems, University ofStuttgart, Stuttgart, Germany, email: [email protected]

Manuela Bastidas and Iuliu Sorin Pop, Hasselt University, Diepenbeek, Belgium.

ABSTRACT

Mineral precipitation and dissolution processes in a porous medium can alter the structure of themedium at the pore scale, and consequently affect the Darcy-scale flow and transport through themedium. Such processes are challenging to include in a numerical model, especially as the pore-scalechanges of the geometry are a-priori unknown, but depend on the solute concentration.

In order to deal with such aspects, we here adopt a two-scale model that couples the Darcy-scaletransport with pore-scale cell problems for the effective properties such as permeability and effectivediffusivity. A phase field is applied to account for the evolving fluid-mineral interface at the porescale [1]. We tailor an iterative numerical scheme which couples the two scales and also handles thenon-linear character of the model. Through introducing a stabilization term we can prove that thetwo-scale iterations are convergent with only a mild restriction on the time step size [2, 3].

Two types of adaptivity are introduced to accelerate the scheme: On the pore scale we applyadaptive mesh refinement that follows the diffuse interface of the phase field, while on the Darcyscale we use an indicator for whether updates of the cell problems are needed. We present somenumerical tests showing the efficiency and accuracy of the scheme, also in the case of anisotropy andheterogeneity.

References[1] C. Bringedal, L. von Wolff, and I. S. Pop. “Phase Field Modeling of Precipitation and Dissolution

Processes in Porous Media: Upscaling and Numerical Experiments”. In: Multiscale Modeling &Simulation 18.2, pp. 10761112. doi: 10.1137/19M1239003, 2020.

[2] M. Bastidas, C. Bringedal, and I. S. Pop. “An adaptive multi-scale iterative scheme for a phase-field model for precipitation and dissolution in porous media”. In: arXiv:2007.05413. Submittedto Applied Mathematics and Computation (in review), 2020.

[3] M. Bastidas, C. Bringedal, and I. S. Pop. “Numerical simulation of a phasefield model for reactivetransport in porous media”. In: Numerical Mathematics and Advanced Applications ENUMATH.Vol. 139. Lecture Notes in Computational Science and Engineering, 2019.



Unsteady Flow in a Porous Media Considering Variable Permeability andLocal Thermal Non–equilibrium

Lalrinpuia Tlau, Department of Basic Sciences & HSS, National Institute of Technology Mizoram,Aizawl 796012, Mizoram, India email: [email protected]

Surender Ontela, Department of Basic Sciences & HSS, National Institute of Technology Mizoram,Aizawl 796012, Mizoram, India

ABSTRACT

An unsteady flow in a horizontal channel filled with a saturated porous medium having variablepermeability is considered. The flow is also assumed to be laminar and fully developed. A localthermal non–equilibrium condition is considered such that the temperature of the flowing fluid isnot equal to the temperature of the porous medium. The governing equations are given as [1, 2, 3]:

ρ∂u

∂τ= µf

∂2u

∂y2− µf

u

K(y)−∂p

∂x(5)

(ρCp)f

[∂T ∗f

∂τ+ u

∂T ∗f

∂x

]= κf

∂2T ∗f

∂y2+ ah(T ∗s − T ∗f ) (6)

(ρCp)s∂T ∗s∂τ

= κs∂2T ∗s∂y2

− ah(T ∗s − T ∗f ) (7)

We consider the no slip condition for the flow such that:

u(y = 0) = u(y = L) = 0 (8)

The temperature at the lower wall of the channel is assumed to vary along the horizontal directionwhile the temperature of the upper wall is assumed to be isothermal. Thus, the boundary conditionsfor the temperature are given as below:

T ∗f (y = 0) = T ∗s (y = 0) = Tw = T0 + Γx

L(9)

∂T ∗f

∂y

∣∣∣∣∣y=L

=∂T ∗s∂y

∣∣∣∣y=L

= 0 (10)

The governing partial differential equations along with the boundary conditions are changedto dimensionless form by suitable substitutions. The dimensionless form of the partial differentialequations governing the flow are reduced to a set of coupled ordinary differential equations whichseperates the harmonic and non–harmonic part of the equations [4, 5]. The new set of equations arethen solved using the homotopy analysis method.

The obtained results show that the temperature of the fluid is sightly higher than the porousmedium. The velocity profile shows that the peak velocity is bent towards the upper half of thechannel, which is caused by the variable permeability of the porous medium.

References[1] A. A. Raptis, Unsteady free convective flow through a porous medium, International Journal of

Engineering Science, 21, 345–348 (1983)

[2] Y. J. Kim, Unsteady convetion flow of micropolar fluids past a vertical porous plate embeddedin a porous medium, Acta Mechanica, 148, 105–116 (2001)

[3] Y. Mahmoudi, Constant wall heat flux boundary condition in micro-channels filled with a porousmedium with internal heat generation under local thermal non-equilibrium condition, Interna-tional Journal of Heat and Mass Transfer, 85 524–542 (2015)

[4] C. Israel–Cookey, A. Ogulu, V. B Omubo–Pepple, Influence of viscous dissipation and radiationon unsteady MHD free-convection flow past an infinite heated vertical plate in a porous mediumwith time-dependent suction International Journal of Heat and Mass Transfer 46 2305–2311(2003)


[5] J. C. Umavathi, A. J. CHamkha, A. Mateen, A. Al–Mudhaf, Unsteady oscillatory flow and heattransfer in a horizontal composite porous medium channel, Nonlinear Analysis: Modelling andControl 14 397–415 (2009)


Stability of Miscible Rayleigh-Taylor Fingers in Porous Media withNon-monotonic Density Profiles

Satyajit Pramanik, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar - 382 355,Gujarat, India, email: [email protected]

ABSTRACT

We study miscible Rayleigh-Taylor (RT) fingering instability in two-dimensional homogeneous porousmedia, in which the fluid density varies non-monotonically as a function of the solute concentrationsuch that the maximum density lies in the interior of the domain, thus creating a stable and anunstable density gradients that evolve in time. With the help of linear stability analysis (LSA) as wellas nonlinear simulations the effects of density gradients on the stability of RT fingers are investigated.As diffusion relaxes the concentration gradient, a non-monotonic density profile emerges in time.Our simple mathematical treatment addresses the importance of density gradients on the stabilityof miscible Rayleigh-Taylor fingering. In this process we identify that RT fingering instabilities arebetter understood combining Rayleigh number (Ra) with the density gradients–hence defining a newdimensionless group, gradient Rayleigh number (Rag). We show that controlling the stable andunstable density gradients, miscible Rayleigh-Taylor fingers are controllable.



Two-phase Flow Through Highly Heterogeneous Porous Media:Modelling and Upscaling

Tufan Ghosh, 1,2,∗ Carina Bringedal, 2,3 Rainer Helmig, 2 G. P. Raja Sekhar, 1

∗ email: [email protected] Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

2 Institute for Modelling Hydraulic and Environmental Systems, University of Stuttgart,Pfaffenwaldring 61, Stuttgart 70569, Germany

3 Stuttgart Center for Simulation Science (SimTech), University of Stuttgart, Pfaffenwaldring 5a,Stuttgart 70569, Germany

ABSTRACT

In this work we consider a two-phase flow model in a highly heterogeneous porous column. Theporous column consists of homogeneous blocks, where the porosity and permeability vary from oneblock to the other. The flow direction is perpendicular to the layering of the porous column, andhence can be approximated by one dimensional model equations. The periodic change in porosityand absolute permeability enforce the fluid to be trapped at the interface between the blocks, leadingto a highly varying saturation. In order to capture the effective behavior, upscaled equations for theaverage saturation are derived via homogenization. This technique relies on a notion of periodicityand allows averaging over any number of blocks that may have any internal distributions of the rockparameters. Moreover, the present study also derives effective equations for randomly distributedlayers of different porosity and absolute permeability. Numerical experiments are performed whichshow good agreement between the averaged solutions of the original micro-scale equations and thesolutions of the upscaled equations, also in the case of randomly distributed layers. In particularthese numerical experiments show how the internal distribution of the permeability and porosityaffect the effective behavior of the flow.



Differential Transform Method for Solving One Dimensional HeatConduction Equation With Variable Coefficient

Neelam Gupta, Research Scholar, Jaypee University of Information Technology Waknaghat, Solan,India, email: [email protected]

Neel Kanth, Faculty, Jaypee University of Information Technology Waknaghat, Solan, India.

ABSTRACT

The differential transform method is one of the numerical method which provide solutions in terms ofconvergent series with easily computable components. In this paper, one dimensional heat conductionequation with variable coefficient is solved using differential transform method. The solution obtainedby differential transform method is an infinite power series. Some examples have been solved andthe obtained results reveals the efficiency of the method.


8 Minisymposia: Recent Advances in the Analy-sis and Development of Numerical Methods forNonlinear Problems

Organizers:

• Prof. Vinay Kanwar, University Institute of Engineering and Technology, Panjab University,Chandigarh 160014, India. E-mail: [email protected]

• Prof. Munish Kansal, School of Mathematics, Thapar Institute of Engineering and Technol-ogy, Patiala 147001, India. E-mail: [email protected]


Numerical Solutions of Some Non-Linear Evolutionary Equations byB-Splines

R. C. Mittal

Department of Mathematics, Jaypee Institute of Information Technology, Sector 62, NoidaEmail: [email protected]

ABSTRACT

Finding solutions of non-linear differential equations is a challenging problem. In this talk, numericalsolutions of some non-linear evolutionary equations by B-Splines collocation method are discussed.The method of lines is adopted by using B-Splines collocation method in space direction x to get asystem of ordinary differential equations. The obtained system of ODEs is solved by fourth orderRunge-Kutta method. Method is illustrated by applying on some non-linear equations and resultsare presented.



Iterative Mthods for Solving Equations: Convergence and Dynamics

Angel Alberto Magrenan RuizDepartment of Mathematics and Computation, University of La Rioja, Spain Email:

[email protected]

ABSTRACT

Iterative methods have gain visibility and weight along the last decades due to the improvementsfound. Several researchers in are have focused their efforts in the study of three different types ofconvergence of those methods: semilocal one, in which conditions on the starting point and theoperator involved should be imposed, local convergence, in which conditions on the solution and alsothe operator involved should be imposed and global convergence which is the most restrictive one.Moreover, the dynamical study of this iterative methods is also important to obtain the behaviourof the method for different polynomials or even to other kind of functions. In the dynamical analysisthere are two different fields, real dynamics study or complex dynamics study. Each field has its owntools to study them and can reach conclusions. In this talk, we will focus on the convergence andreal and complex dynamics of some iterative methods in which we will find differences both in theconvergence and the dynamical behaviour.



Basic Iterative Methods for Solving Nonlinear Equations and TheirExtensions for Systems

J.R. Sharma

Department of Mathematics, Sant Longowal Institute of Engineering and Technology,Longowal, Punjab

Email: [email protected]

ABSTRACT

A large number of computational problems are ultimately reduced to the problem of solving anonlinear equation f(x) = 0 or to the problems of solving a system of nonlinear equations F (X) = 0.In general, such equations arise in the cases of Initial Value and Boundary Value Problems. Analyticmethods are almost non-existent to solve such cases and therefore, we seek the help of numericalmethods which are iterative in nature. The aim of this talk is to introduce the basic iterative methods(IM) along with their properties. Especially, the following points will be discussed during the courseof this action:

1. Basic IM

2. Classification of IM

3. Convergence

4. Generalization of IM for Systems



On the Semilocal Convergence of a Computationally EfficientFifth-Order Method in Banach Spaces under Relaxed Condition

Jai Prakash Jaiswal

Department of Pure & Applied Mathematics, Guru Ghasidas Vishwavidyalaya (A CentralUniversity), Bilaspur (Chattisgarh)-495009, India


ABSTRACT

The current paper is concerned with the study of semilocal convergence of a fifth-order method forsolving nonlinear equations in Banach spaces under mild condition. The existence and uniquenesstheorem has been proved followed by the error estimates. The computational efficiency of the con-sidered scheme over the identical order methods is also examined, which endorses the nobility of thepresent scheme from computational point of view. Lastly, application of theoretical development hasbeen made in nonlinear integral equation.



Multipoint Iterative Methods without Memory for Nonlinear Equationand its Applications

Kalyanasundaram Madhu

Department of Mathematics, Khalifa University, Abu Dhabi - 127788, UAEEmail: [email protected]

ABSTRACT

Higher order iterative methods for finding solution of system of nonlinear equation. Two-step fifthorder method requires two function, two first order derivative and only one inverse. The multi-stepversion requires one more function evaluation for each step and each step increase the convergenceorder with three. The performance of this method has been tested with finding solutions to sev-eral test problems. Then applied to solving pseudorange nonlinear equations on Global NavigationSatellite Signal (GNSS). To solve the problem, at least four satellites measurements are needed tofind out the user position and receiver time offset. In this work, the number of satellites from 4 to 8are considered that is number of equations are more than number of unknown variables to calculatethe user position. The conventional Taylor series and Bancroft methods utilized in the Positioncalculation Module of the GNSS receiver typically converge with 6 iterations for finding the userposition whereas the proposed method takes only 3 iterations which really decrease the computationtime which yield quicker position calculation. From the simulation outcomes, it has been noted thatthe new member is more efficient and converges 33% faster than the conventional iterative methodswith good precision of 92%.



Introduction to Numerical Methods and its Applications

Vinay Kanwar

University Institute of Engineering and Technology, Panjab University, Chandigarh-160014,India


ABSTRACT

In this work, we discuss the geometry of iterative methods for finding the roots of nonlinear equationsnumerically. In particular, we have included the parabolic and elliptic versions of classical Newton’smethod. The beauty of the proposed methods is that they have same convergence order as that ofNewton’s method and does not fail even if the derivative is zero in the vicinity of required root unlikeNewton’s method. The numerical examples considered show that in many cases proposed methodsare efficient alternative to Newtons method which may converge slowly or even fail.



Matrix Iterative Methods for Computing Generalized Outer Inverses

Munish KansalSchool of Mathematics, Thapar Institute of Engineering & Technology, Patiala-147004, India


ABSTRACT

Generalized inverses for a given complex matrix play a significant role in various mathematicalmodels arising in different disciplines of science and engineering. In the literature, various type ofapproaches for computing the generalized outer inverse such as Schulz-iterative technique, (T, S)splitting methods, Successive Matrix Square (SMS) algorithms have been discussed. We will dis-cuss Schulz-iterative method for finding the generalized outer inverse. The convergence analysis isestablished under certain necessary conditions, which indicates that the methods possess at leastfourth-order convergence. Some real-world and academic problems are chosen to validate our meth-ods for solving the linear systems arising from statically determinate truss problems, steady-stateanalysis of a system of reactors, and elliptic partial differential equations. We include a wide varietyof large sparse test matrices obtained from the matrix market library. The performance measuresused are the number of iterations, computational order of convergence, residual norm, efficiencyindex, and the computational time.



Convergence of Newton’s Method and its Higher Order Variants inBanach Spaces

Sukhjit Singh

Dr. B.R. Ambedkar National Institute of Technology, Jalandhar, IndiaEmail: [email protected]

ABSTRACT

The content of this presentation briefly explains the recent developments on the convergence ofNewton’s method and its higher order variants in Banach spaces for solving nonlinear equations.These types of equations appear in different applied problems such as electrostatics and radiativeheat transfer problems. Moreover, the semilocal convergence of a third order iterative method ispresented for solving nonlinear Fredholm integral equations under mild conditions. The novelty ofour work lies in the fact that this study involves first order Frchet derivative and mild conditions.The existence and uniqueness theorem is established using recurrence relations. Numerical examplesinvolving nonlinear Fredholm integral equations are worked out to show the applicability of themethod and domains of the existence and uniqueness of the solution are obtained.



A Monotone Domain-Decomposition Technique for Singularly PerturbedProblems

Vivek Sangwan

School of Mathematics, Thapar Institute of Engineering & Technology, Patiala-147004,India


ABSTRACT

Singularly perturbed problems model many real-life phenomena. Traditional numerical schemes failto capture the boundary layers arising in the solutions of these problems as the singular perturbationparameter approaches zero. In the present work, a uniform convergent numerical technique has beenpresented to capture these sharp boundary layers. The domain of interest is divided into overlapingsubdomains, then a monotone iterative scheme has been proposed to solve the problem on thesesubdomains parallelly. Convergence and numerical results of the proposed iterative scheme has beenpresented which depicts the robustness of the scheme.



Numerical Techniques for Solving Systems of Nonlinear Equations andtheir Applications

Dr. Mona Narang, DAV College, Sector :10 Chandigarh, India, [email protected]

ABSTRACT

The root-finding problems are frequently seen in many areas of natural and physical sciences. Due tolack of analytical solution for most type of nonlinear equations or systems of nonlinear equations inclosed form, numerical techniques which are iterative in nature play a fundamental role. This has ledto the development of several high quality iterative algorithms having higher-order of convergence,minimal computational cost, minimum execution time (e-Time), high precision, wider basins ofattraction and simple body structure for solving systems of nonlinear equations, numerically. Thispresentations discusses these aspects about higher-order Newton-type methods and Steffensen-typemethods with and without memory.



The Impact of Aaverage Flow on Forward Sites in Lattice HydrodynamicModel

Nikita Madaan∗, email: [email protected]

Sapna Sharma∗, ∗ Thapar Institute of Engineering and Technology, Patiala, India

ABSTRACT

In this paper, a novel lattice hydrodynamic model is presented to simulate the traffic flow by incor-porating the effect of forward sites flow on current traffic dynamics. The stability condition of thismodel is attained via linear stability analysis. The modified Korteweg de Vries (mKdV) equation isderived and solved near the critical point to describe the evolution properties of traffic density waveswith the help of nonlinear stability analysis. The information of average flow and optimal averageflow on prospective sites can effectively enhance the stabilization of traffic dynamics. In addition, tovalidate the theoretical results, numerical simulation is carried out and it agree with the theoreticalresults.



Method for Solving Nonlinear Multiobjective Integer OptimizationProblem

Prerna, Thapar Institute of Engineering and Technology, Patiala, India,[email protected]

Vikas Sharma , Thapar Institute of Engineering and Technology, Patiala, India,[email protected]

ABSTRACT

In this paper, we have discussed a method for finding the set of all efficient solutions of a multiob-jective integer quadratic programming problem. The proposed method is based on the aspect thatthe efficient solutions of a multiobjective integer quadratic programming problem can be obtainedby systematically scanning the ranked solutions of an integer quadratic programming problem. Forthis purpose, we have constructed a related integer linear programming problem and cutting planetechnique is utilized for enumerating the ranked solutions of integer linear programming problem.Moreover, we have also justified that none of the efficient point is missed by the proposed algorithm,also our proposed algorithm scans only the efficient points. Numerical and computational results arealso presented to analyze the performance of the developed method.



Optimal Iterative Scheme for Finding Multiple Zeros of Scalar NonlinearEquations

Manpreet Kaur, Thapar Institute of Engineering and Technology, Patiala-147001, India,[email protected]

ABSTRACT

The objective is to present the iterative scheme for solving the scalar nonlinear equation havingmultiple zeros. The algorithm is developed with the use of a weight function approach involvingfunctions and is applicable for known multiplicity. The systematic analysis of the convergence showsthat the scheme attains fourth-order under some conditions. Moreover, the algorithm is optimal inthe sense of Kung-Traub conjecture. Several computational tests have been conducted to show theeffectiveness of the proposed algorithm. The numerical outcomes emphasize the scheme’s efficacy interms of its precision and time complexity.



Studying the Effect of Extraneous Fixed Points on Dynamical Behaviorof Iterative Methods via Basins of Attraction

Dr. RajBala, Government PG College, Sector-1, PanchkulaKurukshetra University Kurukshetra, Haryana, India


ABSTRACT

During the last two decades, a lot of research has been done for the comprehensive study of dynamicalbehavior of iterative methods of nonlinear equations in complex plane. The pivotal works of Cayleyand Schroder in 19th Century, with Newtons method applied to quadratic polynomial showed a pathto researchers to begin in this direction and has been in continuous evolution till then. We usuallylocate a zero α of a nonlinear equation f(x) = 0 by means of a fixed point ξ of iterative methods ofthe form

xn+1 = Rf (xn) = xn −f(xn)

f ′(xn)Hf (xn), n = 0, 1, ...,

where Rf is the iteration function under consideration. It is obvious that α is a fixed point of Rf .But, in general, Rf might possess other fixed points ξ 6= α for which Hf (ξ) = 0. Such fixed pointsare called the extraneous fixed points of the iteration function Rf . In 1988, Vrscay and Gilbertdemonstrated for simple zeros via Konig functions and Schroder functions applied to a family offunctions fk(x) = xk − 1, k ≥ 2 and showed that the existence of such extraneous fixed pointswould affect the global iteration dynamics. Further, it is well known that extraneous fixed pointsmay result in attractive, indifferent or repulsive cycles as well as other periodic orbits. The presenceof possible attractive, indifferent, or repulsive, and other periodic or chaotic orbits underlying theextraneous fixed points might influenced the dynamical behavior of Rf . The present work is devotedto investigating the extraneous fixed points of any iterative map and relevant dynamics associatedwith their basins of attraction.



The Role of Information Technology (IT) in Vehicular Dynamics

Daljeet Kaur∗, email:[email protected] Sharma∗, ∗Thapar Institute of Engineering and Technology, Patiala, India.

ABSTRACT

Nowadays, with the increase of vehicles on roads, traffic congestion has become a more and moreserious topic for scientists and researchers. Traffic congestion has a enormous impact on the life ofpeople. Environmental pollution, travel times delay, energy waste, and even traffic accidents arecaused by traffic congestion. Therefore, the main motive of this paper is to develop an optimal roadnetwork with a maximum flow of traffic and minimum traffic congestion.

With the rapid advancement of information technology (IT) and sensor technology, drivers canget an accurate estimation of forward traffic conditions in the next moment. After getting this priorinformation (predictive effect), drivers modify their driving skills and adjust their speed accordingly.As we know, in real traffic, if the downstream flow is greater than the upstream flow, basically, itmeans that the traffic conditions of the downstream road is much better than that of the upstreamroad. Therefore, the upstream drivers could accelerate, so the flow from upstream to downstreambecomes larger, this phenomenon is known as the flow difference effect (FDE). Upstream flow isdependent on the downstream flow conditions. Therefore, to explain the importance of the predictiveeffect and the FDE on traffic dynamics, a new lattice hydrodynamic model is proposed. It is observedthat the predictive effect plays an important role to improve the stability of traffic flow when FDEis considered. The numerical results are consistent with the analytical results.



Generalized Local Projection Stabilization for Galerkin Approximationsof Darcy and Stokes Problems

Deepika Garg, Indian Institute of Science, Bangalore, Indiaemail: [email protected]

Sashikumaar Ganesan, Indian Institute of Science, Bangalore, Indiaemail: [email protected]

ABSTRACT

A priori analysis for a generalized local projection stabilized conforming finite element approximationof Darcy flow and Stokes problems is presented in this paper. A first-order conforming Pc1 finiteelement space is used to approximate both the velocity and the pressure. It is shown that thestabilized discrete bilinear form satisfy the inf-sup condition with respect to a generalized localprojection norm. Moreover, a priori error estimates are derived for both problems. Finally, thevalidation of the proposed stabilization scheme is demonstrated with appropriate numerical examples.



On Complex Dynamics of Some Third Order Methods for ComputingMultiple Roots

Sunil Kumar, Department of Mathematics, Sant Longowal Institute of Engineering andTechnology, Longowal 148106, Punjab, India, email: [email protected]

Janak Raj Sharma, Department of Mathematics, Sant Longowal Institute of Engineering andTechnology, Longowal 148106, Punjab, India.

ABSTRACT

Numerous methods are available in literature for finding the multiple roots of nonlinear equations.These methods are categorized by the their order, informational efficiency and efficiency index.Another important criterion for comparing the methods is to study their complex dynamics usingthe graphical tool, namely basins of attraction. In this paper, we consider several methods of orderthree and characterize their basins of attraction by applying them on different polynomials.

Keywords Nonlinear equations, Basin of attraction, Iterative method, Multiple roots.



The Study of Convergence and Dynamics Analysis of an IterativeMethod Under Weak Conditions

Deepak Kumar, Department of Mathematics, Chandigarh University, Mohali, India, email:[email protected]

Janak Raj Sharma, Department of Mathematics, Sant Longowal Institute of Engineering andTechnology, Longowal, India.

ABSTRACT

We study the local convergence and dynamical analysis of a fourth order method, to approximatea locally-unique solution of a nonlinear equation in Banach space. Our approach establishes com-putable radius of convergence as well as error bounds on the distances involved and estimates onthe uniqueness of the solution based on some functions appearing in these generalized conditions.Dynamical nature of iterative method is presented with basin of attraction. Finally, numerical ex-amples are provided to show that the present results can be applied to solve equations in the caseswhere earlier results cannot be applied.

Keywords Iterative methods · Local convergence · Weak conditions · Banach space · Frechet-derivativeMathematics Subject Classifications (2010) 49M15 · 47H17 · 65H10



A Numerical Approach to Solve Non-Linear Systems With Higher Orderof Convergence

Sonia Bhalla

Chandigarh University, Gharun, Mohali, IndiaEmail: [email protected]

ABSTRACT

Numerical techniques are a powerful tool to solve the non-linear differential equations, and to solvemany non-linear problems in diverse fields like Heat equation, load flow problem in engineering,etc. In this work, we propose a new derivative-free iterative approach to solve systems of nonlinearequations. Moreover, the validity and numerical accuracy of this technique have been proved interms of higher-order convergence, minimum computation cost, time and efficiency over existingtechniques. In addition, proposed approach is applied to some prominent ordinary and partialdifferential equations via numerical simulations.


9 Minisymposia: Stability of Nonlinear DynamicalSystems

Organizer:

• Prof. S. Marshal Anthoni, Department of Mathematics, Anna University Regional Campus,Coimbatore - 641046, India, Email: [email protected]


An Invitation to Control Theory of Stochastic Distributed ParameterSystems

Xu ZhangSchool of Mathematics, Sichuan University, Chengdu 610064, China

email: zhang [email protected]

ABSTRACT

Control theory for ODE systems is now relatively mature. There exist a huge list of works on controltheory for (deterministic) distributed parameter systems though it is still quite active; while thesame can be said for control theory for stochastic systems in finite dimensions. In this talk, I willgive a short introduction to control theory of stochastic distributed parameter systems (governed bystochastic differential equations in infinite dimensions, typically by stochastic PDEs), which is, in myopinion, almost at its very beginning stage. I will mainly explain the new phenomenon and difficultiesin the study of controllability and optimal control problems for these sort of equations. In particular,I will show by some examples that both the formulation of stochastic control problems and thetools to solve them may differ considerably from their deterministic/finite-dimensional counterparts.Interestingly enough, one has to develop new mathematical tools to solve some problems in this field.



Stability and Stabilizability of Fractional Dynamical Systems

K. BalachandranDepartment of Mathematics, Bharathiar University, Coimbatore, India.


ABSTRACT

Many scientific and engineering problems can be modeled by differential equations, integral equationsor integrodifferential equations. Recently many models are reformulated and expressed in terms offractional differential equations so that their physical meaning will be incorporated in the math-ematical models more realistically. In fact, fractional calculus attract many physicists, engineers,biologists and mathe- maticians for its interdisciplinary applications. So it becomes important tostudy the existence of solutions and qualitative behavior of solutions of such equations.

The main aim of this talk is to discuss the stability and stabilizability of solutions of fractionaldynamical systems represented by fractional differential equations. Many examples are presented tounderstand the theory.



Geometrical Methods for Modeling Complex Systems

Sivaguru. S. SritharanRamaiah University of Applied Sciences, Bengaluru, India.


ABSTRACT

In this talk we will discuss some of the benchmark advances in Riemannian and Khlerian geometry toindicate some research opportunities in modern differential geometry to enable applications such asbig data science, machine learning and complex physics. We will discuss some well-known embeddingtheorems, geometric partial differential equations of evolutions and also some aspects of geometricmeasure theory. Some interesting research directions will be indicated. The talk is introductory andself contained to some extent.



Stabilization of Burgers Equation by Boundary Control

S. Marshal AnthoniDepartment of Mathematics, Anna University Regional Campus, Coimbatore, India.


ABSTRACT

In this talk, the problem of Neumann and Dirichlet control for a class of nonlinear partial differentialequations is considered. The aim is to derive appropriate boundary control laws which make thesystem globally, asymptotically stable. Numerical examples are also shown to demonstrate theeffectiveness of the derived controls. The derived boundary control law makes the Burgers equationglobally asymptotically stable. The purpose is to have minimum control effort and avoid derivativesand integrals in control laws and to guarantee the existence of solutions.



Mathematical Modeling COVID-19: Forecasting and Analyzing theDynamics of the Outbreak in India

K. Chinnadurai, SRM Institute of Science and Technology, Kattankulathur - 603 203, India,email:[email protected]

S. Athithan, SRM Institute of Science and Technology, Kattankulathur - 603 203, India.email:[email protected]

ABSTRACT

As the Coronavirus Disease 2019 (COVID-19) pandemic rages around the world , it is importantto correctly model the complexities of it. However, when collection of data and consistency differsignificantly from region to region, it is difficult, if not entirely impossible, to reliably model directlyfrom a global view point.Nevertheless, it is possible to create precise local prediction methods throughlocal data obtained by some countries, which can be combined to build global models. In thisanalysis, a combined system of ordinary differential equations ( ODEs) analyses the dynamics oflocal outbreaks of COVID-19. Using the vast volume of data available as a tested from the ebbingoutbreak in India, we estimate the simple reproductive number, R0 of COVID-19 and predict with ahigh degree of precision the total cases, total deaths, and other characteristics of the India outbreak.We observe the impacts of quarantine, social distancing, and COVID-19 research on the nature ofthe epidemic by computational studies. We apply our model to examine the dynamics of outbreaksin India / Asia, using information obtained from the India outbreak. For various degrees of socialdistancing, quarantine, and COVID-19 monitoring, we provide projections for the duration of theepidemic and the overall number of cases / deaths in India / Asia.



Robust Feedback Control of Nonlinear PDEs by NumericalApproximation of High-dimensional Hamilton-Jacobi-Isaacs Equations

Sudeep Kundu, University of Graz, Austria, email: [email protected]

Dante Kalise, University of Nottingham, Nottingham, UK.Karl Kunisch, University of Graz and RICAM, Linz, Austria.

ABSTRACT

In this talk, we propose an approach for the synthesis of robust and optimal feedback controllers fornonlinear PDEs. Our approach considers the approximation of infinite-dimensional control systemsby a pseudospectral collocation method, leading to high-dimensional nonlinear dynamics. For thereduced-order model, we construct a robust feedback control based on theH∞ control method, whichrequires the solution of an associated high-dimensional Hamilton-Jacobi-Isaacs nonlinear PDE. Thedimensionality of the Isaacs PDE is tackled by means of a separable representation of the controlsystem, and a polynomial approximation ansatz for the corresponding value function. Our methodproves to be effective for the robust stabilization of nonlinear dynamics up to dimension d ≈ 12.We assess the robustness and optimality features of our design over a class of nonlinear parabolicPDEs, including nonlinear advection and reaction terms. The proposed design yields a feedbackcontroller achieving optimal stabilization and disturbance rejection properties, along with providinga modelling framework for the robust control of PDEs under parametric uncertainties.



Impulsive Complex-valued Stochastic BAM Neural Networks withLeakage and Mixed Time Delays: An Exponential Stability Problem

1 R.Raja, Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi 630003,Tamilnadu, India, email: [email protected]

2 C.Maharajan, Department of Mathematis, V.S.B Engineering College, Karur, Tamilnadu, India.

ABSTRACT

In this paper, the stability criteria for exponential sense is concerned for the complex-valued stochas-tic BAM neural networks with leakage, discrete, distributed time delays and impulses. By utilizingthe contracting mapping theorem, the existence and uniqueness of the equilibrium point for theproposed complex-valued neural networks are investigated. Moreover, based on the construction ofLyapunov-Krasovskii functional, matrix inequality techniques and stability theory, some novel time-delayed sufficient conditions are attained in linear matrix inequalities (LMIs) form, which ensure theexponential stability of the trivial solution for the addressed neural networks. Finally, via MATLABLMI control toolbox, few numerical examples with their simulations are provided to illustrate theeffectiveness and merits of our theoretical results.



Adaptive Controllability of Chaos Generated in GeneralizedLotka-Volterra Biological Model using Projective Combination Difference

Synchronization Technique

Harindri Chaudhary, Jamia Millia Islamia, New Delhi, India.email: [email protected]

Taqseer Khan, Jamia Millia Islamia, New Delhi, India.

ABSTRACT

This manuscript puts forward a systematic investigation of projective combination difference syn-chronization among identical generalized Lotka-Volterra biological models of integer order usingadaptive control method. We use Lyapunov stability theory (LST) to construct the desired adaptivecontrollers to ensure the global asymptotic convergence of trajectory following synchronization errors.In addition, some numerical simulations in MATLAB environment are performed to illustrate theaccuracy and effectiveness of the proposed technique. Exceptionally, both experimental and theo-retical results are in excellent agreement. Finally, a comparative analysis between the consideredtechnique and the previously published literature is made. Furthermore, the discussed technique hasapplications in the fields of image encryption and secure communication.



Impulsive Effects on Complex-valued Neural Networks with TimeDelays: An Asymptotic Stability Issue

R. Samidurai, Thiruvalluvar University, Vellore, Tamil Nadu-632 115, Indiaemail:[email protected]

ABSTRACT

A generalization of real-valued neural networks is represented by complex-valued neural networks, forwhich the states and weights are matrices. Inspired by this aspect, the global asymptotic stabilityof continuous-time complex-valued neural networks with fixed time impulsive and discrete timedelays is investigated. Firstly, by means of non-linear Lipschitz measure and some matrix inequalitytechniques, the existence and uniqueness of the network equilibrium point are proved. Secondly,by using Lyapunov-Krasovskii method and linear matrix inequality the global asymptotic stabilitycriteria are established for the considered complex-valued neural networks. Finally, a numericalexample with their simulations is shown to illustrate the essence and merits of our obtained analyticalresults.



Non-fragile Event-triggered Control Design for Fuzzy Systems withActuator Saturation

Palanisamy Selvaraj,School of Electrical Engineering, Chungbuk National University, Cheongju - 28644, South Korea.


ABSTRACT

This paper is concerned with the problem of event-triggered non-fragile dynamic output feedbackcontroller design for Takagi-Sugeno fuzzy systems with actuator saturation. The controller to bedesigned is supposed to include additive gain variations. By using Lyapunov stability theory andlinear matrix inequality technique, a set of sufficient conditions are derived to design the event-triggered parameters and the controller gains, which ensures the robust stabilization of the closed-loop system with an estimation of the domain of attraction. At last, numerical illustrations are givento exhibit the rightness and important features of the acquired theoretical results.



FDI Attacks on Various Dynamical Systems: The Known to be Explored

Maya Joby, KCG College of Technology, Chennai, India. email:[email protected]

ABSTRACT

A prime research area which requires significant attention is the security of control systems when thethe sensor or actuator is under attack, to be specific false data injection (FDI) attack. This type ofattack can severely compromise stability, performance and integrity of the system. This presentationis an outline on some of the existing studies about FDI attack in the areas of power systems, cyberphysical systems and wireless sensor networks. The talk addresses the effects/impacts of the attacksand detection methods. The conclusion reflects the limitations in the structure of the consideredsystems and suggestions to explore further on what we know.



Robust Stability of Fuzzy Genetic Regulatory Networks with MarkovianJumping Parameters

P.Vadivel, Kongu Engineering College, Erode, India, email: [email protected]

ABSTRACT

Genetic regulatory networks have become an important area of research in the biological sciences.Genetic regulatory networks can be described by differential equations with time delays. In thistalk, we address the problem of robust asymptotic stability analysis of fuzzy genetic regulatorynetworks (GRNs) with Markovian jumping and mixed time delays. By employing a novel Lyapunovfunctional approach combined with the delay fractioning technique, a new set of sufficient conditionsis derived for the asymptotic stability of the considered fuzzy GRN model with constant delay.Further, this result is extended to study the robust asymptotic stability criteria for uncertain fuzzyGRN model. In particular, the parametric uncertainties are assumed to be norm bounded. Moreprecisely, we will establish linear matrix inequalities (LMIs) based solvability problems encompassingnominal and uncertain cases. The derived stability conditions are expressed in the form of linearmatrix inequalities, which can be tested efficiently by the available commercial software packagessuch as MATLAB LMI Control Toolbox. Finally, a numerical example is provided to illustrate theeffectiveness of the obtained results.



Synchronization Analysis for a Network of Dynamical Systems ThroughComposite Control Approach

B. Kaviarasan, Chungbuk National University, Cheongju, South Korea, email:[email protected]

ABSTRACT

The issue of robust synchronization for a network of dynamical systems with multiple disturbancesis presented. In particular, the network is affected by two different types of disturbances in whichone type produced by exogenous systems is unknown and the other type is norm-bounded. Inorder to ensure synchronization of the considered network, a disturbance observer is constructed toestimate the unknown disturbance, which is integrated with the common state feedback controller.Meanwhile, the norm-bounded disturbance is handled with the help of the H∞ control method.Using the Lyapunov stability theory and integral inequalities, the sufficient conditions required tosolve the aforementioned issue are obtained in terms of linear matrix inequalities. A simulationexample based on the F-18 aircraft model is provided in order to certify the correctness of thedeveloped theoretical findings. The simulations show that the proposed composite control methodcan provide better performance than the existing traditional H∞ control method.

The author is affiliated with the School of Electrical Engineering, Chungbuk National University,Cheongju 28644, South Korea.



Mathematical Modelling of Effects of COVID-19 in Poverty

K. Chinnadurai, SRM Institute of Science and Technology, Kattankulathur - 603 203, India,email:[email protected]


ABSTRACT

As the Coronavirus Disease 2019 (COVID-19) pandemic rages around the world , it is importantto correctly model the complexities of it. However, when collection of data and consistency differsignificantly from region to region, it is difficult, if not entirely impossible, to reliably model directlyfrom a global view point.Nevertheless, it is possible to create precise local prediction methods throughlocal data obtained by some countries, which can be combined to build global models. In thisanalysis, a combined system of ordinary differential equations ( ODEs) analyses the dynamics oflocal outbreaks of COVID-19. Using the vast volume of data available as a tested from the ebbingoutbreak in India, we estimate the simple reproductive number, R0 of COVID-19 and predict with ahigh degree of precision the total cases, total deaths, and other characteristics of the India outbreak.We observe the impacts of quarantine, social distancing, and COVID-19 research on the nature ofthe epidemic by computational studies. We apply our model to examine the dynamics of outbreaksin India / Asia, using information obtained from the India outbreak. For various degrees of socialdistancing, quarantine, and COVID-19 monitoring, we provide projections for the duration of theepidemic and the overall number of cases / deaths in India / Asia.



Jacobi Stability Analysis of Prey-Predator Model

T. N. Mishra, DST-CIMS, BHU, Varanasi, India, email: [email protected]

B. Tiwari, DST-CIMS, BHU, Varanasi, India.

ABSTRACT

The stability of prey-predator model has been studied by using KCC theory. The KCC theory is basedon the assumption that the second-order dynamical system and geodesics equation, in associatedFinsler space, are topologically equivalent. The stability (Jacobi stability) based on KCC theoryand linear stability of the model has been discussed in details. Further, the effect of parameters onstability of model and presence of chaos in the model has been investigated. The numerical examplesof particular interest have been taken to compare the results of Jacobi stability and linear stabilityand it has been found that Jacobi stability on the basis of KCC theory is global than the linearstability.



Water Resource Sharing and its Effects:A Mathematical ModellingApproach

K. Siva, SRM Institute of Science and Technology, Kattankulathur - 603 203, India,email:[email protected]


ABSTRACT

We formulated and studied a predatorprey system has five compartments. The model has six equi-librium points, and these are analyzed both analytically and numerically through stability analysis.Further, the model is enhanced by introducing stochastic and analyzed the effect of stochastic modelby comparing it with deterministic model. To validate our analytical findings, numerical simulationof both the deterministic and stochastic models is shown better results.



Modelling Dynamics of Violence Against Women

G.Divya, SRM Institute of Science and Technology, Kattankulathur - 603 203, India,email:[email protected]


ABSTRACT

Violence against women is a social infectious epidemic disease. This means that violence is a conta-gious disease, it can spread from one person to another person. In this paper, we build a mathematicalmodel and analyzed stability of two equilibria namely violence free equilibrium point and non-trivialequilibrium point. Further this ODE model extended to stochastic approach and the results arecompared with deterministic model through numerical simulation. Also we tried to increase the rateof reported cases and reduce the rate of abusement



Sliding Mode Boundary Control Design for Parabolic Systems

K. MathiyalaganDepartment of Mathematics, Bharathiar University, Coimbatore, India.


ABSTRACT

This paper deals with the boundary control problem for reaction-diffusion-advection model withspatially dependent parabolic systems. The sliding mode control (SMC) approach and the linearquadratic regulator technique are used in designing the desired boundary controller. An integralswitching surface with average system state and possible error deviation from the reference value isintroduced to obtain the desired characteristics for the proposed model. The sufficient conditions forthe stability are analyzed by using the Lyapunov approach together with linear matrix inequalitytechnique. Furthermore, the SMC law forces the state trajectories into the switching surface withinthe finite-time. Finally, a numerical example is presented to illustrate the significance and efficiencyof the proposed results.

References[1] C. Edwards and S.K. Spurgeon, Sliding mode control: Theory and Applications, Taylors and

Francis Ltd, London, (1998).

[2] N.I.C. Molina, J. Paulo and V.S. Cunha, Non-collocated sliding mode control of partial differen-tial equations, Journal of Process Control, 73, 1-8, (2019).

[3] A. Seuret, M. Prathyush and C. Edwards, LQR performance for multi-agent systems: benefitsof introducing delayed inter-agent measurements, IEEE Conference on Decision and Control,Florence, Italy, 5150-5155, (2013).

[4] H. Xing, D. Li, C. Gao and Y. Kao, Delay independent sliding mode control for a class of quasi-linear parabolic distributed parameter systems with time-varying delay, Journal of the FranklinInstitute, 350, 397-418, (2013).



Stochastic Model for the Co-existence of Diabetes and COVID-19

S. Anusha, SRM Institute of Science and Technology, Kattankulathur - 603 203, India,email:[email protected]


ABSTRACT

This article discusses diabetes major threat to human life in several parts of the World. We formulateand analyze a deterministic model for diabetes. Existence of equilibria and the stability of the modelare discussed in-detail. Basic reproduction number R0 of the proposed model is also computed andthis helps in determining the impact of different key parameters on the transmission dynamics ofdisease. Additionally, the proposed model is extended to stochastic model and simulation results ofboth deterministic and stochastic models are compared and analyzed.


10 Minisymposia: The Use of Block Methods forSolving Differential Problems

Organizer:

• Prof. Higinio Ramos, Department of Applied Mathematics, University of Salamanca, Sala-manca, Spain. Email: [email protected]


On the Use of Block Methods for Solving Singular Boundary ValueProblems

Higinio Ramos, Department of Applied Mathematics, Higher Polytechnic School, University ofSalamanca, Ada. Requejo-33 49033, Zamora, Spain


ABSTRACT

Block methods have been used extensively to solve different types of differential problems. One ofthose types is the singular boundary value problems, which are characterized by the presence of asingularity in the coefficients of the differential equation. There are many models that use this typeof problem in applied science, for example in thermal explosions, in oxygen diffusion, in a non-linearheat conduction model of the human head, in human physiology, etc. There are also many approachesfor solving these problems, as the variational iteration method, the homotopy perturbation method,different method based on spline approximations, methods that use Pad or Taylor approximations,methods using different types of wavelets, the Adomian decomposition method, and many more.

This talk is devoted to the application of block methods for solving singular boundary valueproblems. The adopted approach consists in the development of appropriate block formulas to dealwith the singularity. These formulas are combined with other formulas used on the remaining intervalto obtain a global method that produces the approximations at the nodal points at once.



Extended Use of One-step Hybrid Block Method on Partial DifferentialEquations

Anurag Kaur, Panjab University, Chandigarh, India, email:[email protected]

ABSTRACT

A new optimized one-step hybrid block method is generated by considering two intra-steps nodes.In deriving this method, a power series approximate function is interpolated at xn and collocated at xn, xn+r, xn+s, xn+1 in the given interval. The characteristics of the hybrid block method, thatis, consistency, convergence and stability are verified. This method is applied on partial differentialequation along with the method of collocation using B-splines for spatial grid. This collaboration istested on Buckmasters equation and has given better results than the existing results.



Adapted Hybrid Second Derivative Block Method for Initial ValueProblems with Oscillating Solutions

Ridwanulahi I. Abdulganiy, Distance Learning institute, University of Lagos, Lagos, Nigeria.,[email protected]:

Ganiyu O. Inakoju, Department of Mathematics, University of Lagos, Lagos, Nigeria.Muslihat A. Gaffari, Department of Mathematics, University of Lagos, Lagos, Nigeria.

Solomon A. Okunuga, Department of Mathematics, University of Lagos, Lagos, Nigeria.

ABSTRACT

In this talk, we will start with general overview of block methods, their advantages, their rolein circumventing the problem associated with the traditional hybrid method and their adaptationto solving Oscillatory Initial Value Problems. Specifically, a sixth-order convergence one step blockhybrid formula for solving first-order differential systems whose results present an apparent oscillatorybehaviour will be proposed. The formula is built such that it skirts the complications connected withthe conformist hybrid formulas. In order to obtain better accuracies, the formula combines secondderivative in its development and is constructed on the basis that it provides no errors when the truesolution is a linear combination of polynomial and trigonometric functions containing a parameter.The performance of the proposed formula will be established through some standard numericalexamples whose solutions show apparent oscillatory behaviour.



An Efficient Optimized Hybrid Block Method for Integrating InitialValue Ordinary Differential Systems

Gurjinder Singh, Department of Mathematical Sciences, I. K. Gujral Punjab Technical UniversityJalandhar, Main Campus, Kapurthala-144603, Punjab, India


ABSTRACT

This work deals with an optimized hybrid block method for integrating numerically initial valueordinary differential systems. The hybrid nature of the one-step scheme allows us to bypass thefirst Dahlquist’s barrier on linear multi-step methods. The development of the method is based onthe theory of interpolation and collocation. Further, an optimized version of the method has beenobtained by following an optimization strategy. In this way, the resulting method is of order at leastfive having the characteristic of A-stability. The method is then formulated in a variable step-sizemode by using an embedded-type procedure. This variable step-size code has been tested on somewell-known first order initial value systems.



Block Hybrid Method for the Numerical solution of Fourth orderBoundary Value Problems

Mark I. Modebeia, Samuel N. Jatorb, Higinio Ramosc,d,aDepartment of Mathematics Programme, National Mathematical Centre, Abuja, Nigeria,

bDepartment of Mathematics and StatisticsAustin Peay State University

cDepartment of Applied Mathematics, University of Salamanca, Salamanca, SpaindEscuela Politcnica Superior de Zamora, Campus Viriato, 49022 Zamora, Spain


ABSTRACT

A linear Multistep Block Hybrid Method with four off-grid points is presented for the direct ap-proximation of the solution of fourth order Boundary Value Problems. Multiple Finite Differenceformulas are derived and grouped into a unique block to form a numerical integrator to solve directlythe fourth order problem, without the need to reduce it previously to a first-order system. The con-vergence of the proposed method is discussed. The superiority of this method over existing methodsis established numerically considering different problems that have appeared in the literature.



An Adaptive Fourth-Derivative Hybrid Block Strategy for IntegratingThird-order IVPs with Applications to Problems in Fluid Dynamics and

Engineering

M.A. RufaiDepartment of Mathematics, University of Bari Aldo Moro, 70125 Bari, Italy., email:

[email protected]

Higinio RamosDepartment of Applied Mathematics, University of Salamanca, Plaza de la Merced , 37008

Salamanca, Spain., email: [email protected]

ABSTRACT

In this talk, I will present an efficient fourth-derivative two-step hybrid block strategy (FDTHBS)for solving third-order IVPs with applications to problems in Fluid Dynamics and Engineering.The mathematical derivation of the proposed FDTHBS depends on the procedure of interpolationand collocation of a reliable function and its derivatives at the selected equidistant grid and off-grid points. The essential characteristics of the proposed method are analysed. An embedding-likeprocedure is considered and executed in variable step-size mode to acquire better performance of thenewly developed strategy. Some test problems are integrated numerically to ascertain the accuracyof the FDTHBS, showing that the obtained outcomes support the analytical results.



A High Order Block Hybrid Method for Integrating Oscillatory GeneralSecond Order Initial Value Problems

S. JatorDepartment of Mathematics and Statistics, Austin Peay State University


ABSTRACT

High order methods are known to provide greater accuracy with larger step-sizes than low orderones. Thus, we present a Block Hybrid Method (BHM) of order 11 for solving systems of generalsecond-order initial value problems (IVPs), including Hamiltonian systems and partial differentialequations (PDEs). The BHM is constructed from a continuous scheme based on a hybrid methodof a linear multistep type and then implemented in a block-by-block fashion. The properties of theBHM are discussed and the performance of the method is demonstrated on some numerical examples.The superiority of the BHM over the Generalized Adams Method (GAM) of order 11 is establishednumerically.



Implicit Three-Point Block Numerical Algorithm for Solving ThirdOrder Initial Value Problem Directly with Applications

Reem Allogmany, Department of Mathematics, Faculty of Science, Taibah University, P.O. Box30097,Al-Madinah Al-Munawarah, Saudi Arabia, email: [email protected]

Fudziah Ismail, Department of Mathematics, Faculty of Science, Universiti Putra Malaysia,Serdang 43400 UPM, Malaysia, email: [email protected]

ABSTRACT

Recently, direct methods that involve higher derivatives to numerically approximate higher orderinitial value problems (IVPs) are being explored, which aim to construct numerical methods withhigher order and very high precision of the solutions. This article aims to construct a fourth andfifth derivatives, three-point implicit block method to tackle general third-order ordinary differen-tial equations directly. As a consequence to the increased in order acquired via the implicit blockmethod of higher derivative, a significant improvement in efficiency has been observed. The newmethod derived in a block mode to simultaneously evaluate the approximations at three points. Thederivation of the new method can be easily implemented. We established the proposed method’scharacteristics, including order, zero-stability, and convergence. Numerical experiments are used toconfirm the superiority of the method. Applications to problems in Physics and engineering are givento assess the significance of the method.


Multistep Block Method for Solving Neutral Delay Differential Equations

Z. A. Majid Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400UPM Serdang, Selangor Darul Ehsan, Malaysia, email: am [email protected]

ABSTRACT

The initial-value problem for neutral delay differential equations (NDDEs) of constant and pan-tograph delay types have been solved by using a multistep block method. The method has beenderived by applying Taylor series interpolation. Both types of NDDEs will be solved at two-pointsimultaneously with constant step-size. In order to find the solution for NDDEs, the delay solutionsof the unknown function will be interpolated using Lagrange interpolation polynomial. The order,consistency and implementation of the block method will be discussed. Numerical results presentedhave concluded that the proposed method is comparable with the existing method and is assumedto be reliable for solving NDDEs with constant and pantograph delays.


Solving Differential Systems Y′ = F(x,Y) by Using a Novel TwoParameter Class of Optimized Hybrid Block Methods

Rajat Singla, Department of Mathematics, Akal University, Talwandi Sabo, Bathinda-151302,Punjab, India. email: [email protected]

ABSTRACT

This work presents a two parameter family of hybrid block methods for integrating first order initialvalue ordinary differential systems. The method exhibits hybrid nature which helps in bypassingthe first Dahlquist barrier existing for linear multi-step methods. The method is developed intoa block form which produces approximate numerical solution at several points simultaneously. Apurely interpolation and collocation approach has been used in development of the class of methods.An optimized version of this class of methods is obtained by using an optimization strategy. Theproposed method has atleast sixth order of accuracy. Further, the method is formulated into anadaptive step-size algorithm using an embeded type procedure. The proposed class of methods hasbeen tested on well-known first order differential systems.



Variable Step-Size Formulation of a Three-Point Fourth-DerivativeHybrid Block Method for Solving Third-Order IVPs

M.A. Rufai, Department of Mathematics, University of Bari Aldo Moro, 70125 Bari, Italy.email: [email protected]

Higinio Ramos, Department of Applied Mathematics, University of Salamanca, Plaza de la Merced,37008 Salamanca, Spain.email: [email protected]

ABSTRACT

In this talk, we will introduce an efficient two-step fourth-derivative hybrid block method (2FDHBM)formulated in variable step-size for solving third-order IVPs directly. The approach of interpolationand collocation of power series polynomial will be adopted in the mathematical formulation of the2FDHBM. The fundamental properties of the proposed strategy will be theoretically analysed. Anembedding-type procedure will be used to get better performance of the proposed method. Some testproblems, including the Blasius equation and non-linear thin-film flow problems, will be numericallysolved to determine the efficiency and utility of the 2FDHBM in solving real-world problems inEngineering and Applied Sciences.


11 Minisymposia: Young Researchers in Numericsfor Evolutionary Problems

Organizers:

• Dr. Stefano Di Giovacchino, Department of Information Engineering and Computer Scienceand Mathematics, University of L’Aquila, Italy. Email: [email protected]

• Dr. Carmela Scalone, Department of Information Engineering and Computer Science andMathematics, University of L’Aquila, Italy. Email: [email protected]


Geometric Analysis of a Phantom Bursting Model

Iulia Martina Bulai, Department of Mathematics, Informatics and Economics, University ofBasilicata, Viale dell’Ateneo Lucano, 10, 85100 Potenza, Italy, email:[email protected]

Richard Bertram, Department of Mathematics, Florida State University, Tallahassee.Morten Gram Pedersen, Department of Information Engineering, University of Padova, Italy.

Theodor Vo, School of Mathematics, Monash University, Australia.

ABSTRACT

The phantom bursting model was introduced to describe the episodic bursting of the pancreaticβ-cells, where active phases are interspersed by silent ones. The model is characterised by two slowand two fast variables with the two slow variables having very different time scales [1]. Consideringthe different time scales of the four variables of ordinary differential equations, Mixed-Mode BurstingOscillations (MMBOs) solutions can be found. MMBOs are characterized by both small-amplitudeoscillations (SAOs) and bursts consisting of one or multiple large-amplitude oscillations (LAOs) [2].

Here we focus our attention on the mechanism that generate the MMBOs due to both canardsand delayed-Hopf-bifurcation [3]. Canards are central to the dynamics of MMBOs and we studythem starting from the folded singularities, that are equilibria of the desingularized system of thephantom burster model. The canard phenomenon explains the very fast transition upon variationof a parameter from a small amplitude limit cycle via canard cycles to a large amplitude relaxationcycle. Furthermore the presence of the subcritical Hopf bifurcation via fast/slow analysis of the fastsubsystem is found. A detailed geometric explanation of MMBOs is done using numerical simulationsand the slow attracting manifold is obtained using numerical continuation technics.

References

[1] Richard Bertram, Joseph Previte, Arthur Sherman, Tracie A. Kinard, Leslie S. Satin,The Phantom Burster Model for Pancreatic β-Cells, Biophysical Journal, 79(6) (2000)2880-2892.

[2] Mathieu Desroches, Tasso J. Kaper, Martin Krupa, Mixed-mode bursting oscillations:Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster, Chaos: An Interdisciplinary Journal of Nonlinear Science, 23(4) (2013)1054-1500.

[3] Richard Bertram, Joseph Rhoads, Wendy P. Cimbora, A Phantom Bursting Mechanismfor Episodic Bursting, Bull. Math. Biol. 70(1979) (2008) 1989-2017.



Numerical Stability Analysis of Linear Periodic Delay Equations viaPseudospectral Methods

Davide Liessi, University of Udine, Italy, email: [email protected]

Dimitri Breda, University of Udine, Italy.

ABSTRACT

Realistic models for various phenomena, from control theory to mathematical biology, areoften based on delay equations. Due to the complexity of such models, their dynamicscannot usually be studied analitically and must be approximated numerically. I will presenta method based on pseudospectral collocation for approximating the spectra of evolutionoperators of linear delay equations, focusing on renewal equations [1] with some hints tothe cases of delay differential equations [2, 3] and coupled renewal and delay differentialequations [4]. The method allows to study the stability of periodic orbits via the principleof linearised stability and Floquet theory.

References

[1] D. Breda and D. Liessi, Approximation of eigenvalues of evolution operators forlinear renewal equations, SIAM J. Numer. Anal., 56 (2018), pp. 1456–1481, DOI:10.1137/17M1140534.

[2] D. Breda, S. Maset, and R. Vermiglio, Approximation of eigenvalues of evolution op-erators for linear retarded functional differential equations, SIAM J. Numer. Anal., 50(2012), pp. 1456–1483, DOI: 10.1137/100815505.

[3] D. Breda, S. Maset, and R. Vermiglio, Stability of Linear Delay Differential Equations:A Numerical Approach with MATLAB, SpringerBriefs Control Autom. Robot., Springer,New York, 2015, DOI: 10.1007/978-1-4939-2107-2.

[4] D. Breda and D. Liessi, Approximation of eigenvalues of evolution operators for linearcoupled renewal and retarded functional differential equations, Ric. Mat. (2020), DOI:10.1007/s11587-020-00513-9.



Convergence of the Piecewise Orthogonal Collocation for PeriodicSolutions of Retarded Functional Differential Equations

Alessia And, University of Udine, Udine, Italyemail: [email protected]

Dimitri Breda, University of Udine, Udine, Italy.

ABSTRACT

I will present an analysis of the convergence of (a variant of) the piecewise orthogonal collo-cation for periodic solutions of retarded functional differential equations defined by a genericright-hand side [1]. Such analysis is highly based on [2] where a general framework for solv-ing a certain class of boundary value problems (BVPs) is presented and accompanied by arigorous proof of convergence of the corresponding iterative method. The novel contributionsconsist in the proofs of the validity of the assumptions required to apply the abstract ap-proach of [2] in the case of periodic BVPs. Indeed, although the general BVP in [2] considersthe presence of unknown parameters, it does not explicitly deal with the periodic case. Inthe presentation I will highlight the role of the period as the (main) unknown parameterof the problem, which leads to some effort in validating the required assumptions, being itdirectly linked to the course of time. I will conclude the presentation with some commentson the possibility of extending the convergence proof to the case of renewal equations.

References

[1] A. And and D. Breda, Convergence analysis of collocation methods for computing periodicsolutions of retarded functional differential equations, SIAM J. Numer. Anal., (2020) toappear.

[2] S. Maset, An abstract framework in the numerical solution of boundary value problemsfor neutral functional differential equations, Numer. Math., 133(3):525 555 (2016).



Highly Stable Multivalue Almost Collocation Methods with StructuredCoefficient Matrix

Maria Pia D’Arienzo, University of Salerno, Fisciano, Italy, email:[email protected] Conte, University of Salerno, Fisciano, Italy.

Raffaele D’Ambrosio, University of L’Aquila, L’Aquila, Italy.Beatrice Paternoster, University of Salerno, Fisciano, Italy.

ABSTRACT

Our aim is to solve systems of ordinary differential equations potentially candidate to be stiff,so developed methods must have high stability properties and uniform order of convergence.We focus our attention on multivalue methods [1], which are a generalization of classicalmethods, such as multistep and Runge-Kutta methods, and we extend the solution smoothlyby approximating them though a collocation polynomial. These methods require at eachtime-step the solution of a non linear system of internal stages, so the computational effortis strictly connected to the nature of this system. We are interested in the construction ofmethods that allow a reduction of this computational cost, so we propose methods with fullmatrix [6] and structured matrix (triangular [2], singly triangular [3, 4], diagonal [5]). In thecase of structured matrix, we perform almost collocation since it is not possible to imposeall the collocation conditions. We compare these methods calculating the error in the finalstep point and the experimental order of convergence.

References

[1] J.C. Butcher, General linear methods. Computers & Mathematics with Applications.31 (4-5): 105-112. doi:10.1016/0898-1221(95)00222-7 (1996).

[2] D. Conte, R. D’Amborsio, M.P. D’Arienzo, B. Paternoster, Highly stable multivaluecollocation methods, J. Phys.: Conf. Ser. 1564, 012012 (2020).

[3] D. Conte, R. D’Ambrosio, M.P. D’Arienzo, B. Paternoster, Singly diagonally im-plicit multivalue collocation methods, in 2020 International Conference on Mathe-matics and Computers in Science and Engineering (MACISE), Madrid, Spain, DOI:10.1109/MACISE49704.2020.00018, pp. 65-68 (2020).

[4] D. Conte, R. D’Amborsio, M.P. D’Arienzo, B. Paternoster, One-point spectrum Nord-sieck almost collocation methods, International Journal of Circuits, Systems and SignalProcessing, vol. 14, pag. 266-275, DOI: 10.46300/9106.2020.14.38 (2020).

[5] D. Conte, R. D’Amborsio, M.P. D’Arienzo, B. Paternoster, Multivalue almost colloca-tion methods with diagonal coefficient matrix, Lecture Notes in Comput. Sci., in press.

[6] R. D’Ambrosio, B. Paternoster, Multivalue collocation methods free from order reduc-tion, J. Comput. Appl. Math. DOI: 10.1016/j.cam.2019.112515 (2019).



Nonreflecting Boundary Conditions for a CSF model of the FourthVentricle - Spinal SAS Dynamics

Licia Romagnoli, Universita Cattolica del Sacro Cuore, Brescia, Italy,email: [email protected]

Donatella Donatelli, University of L’Aquila, L’Aquila, Italy.

ABSTRACT

In this talk we will introduce a one-dimensional model for analyzing the cerebrospinal fluid(CSF) dynamics within the fourth ventricle and the spinal subarachnoid space (SSAS) ([2]).The model has been derived starting from an original model of Linninger et al. ([3]) andfrom the detailed mathematical analysis of two different reformulations developed in [1]. Wewill show the steps of the modelization and the rigorous analysis of the first-order non-linearhyperbolic system of equations which rules the new CSF model, whose conservative-law formand characteristic form are required for the boundary conditions treatment. By assumingsub-critical flows, for the particular dynamics we are dealing with, the most desirable optionis to employ the nonreflecting boundary conditions ([4], [5]), that allow the simple wave as-sociated to the outgoing characteristic to exit the computational domain with no reflections.Finally, we will show some numerical simulations related to different cerebral physiologicalconditions.

References

[1] D. Donatelli, P. Marcati and L. Romagnoli, A comparison of two mathematical models ofthe cerebrospinal fluid dynamics, Mathematical Biosciences and Engineering, 16: 2811– 2851, 2019.

[2] D. Donatelli and L. Romagnoli, Nonreflecting boundary conditions for a CSF model ofthe fourth ventricle - spinal SAS dynamics, Bulletin of Mathematical Biology (BMAB),82 (2020).

[3] A. A. Linninger, C. Tsakiris, D. C. Zhu et al. , Pulsatile cerebrospinal fluid dynamics inthe human brain, IEEE T. Bio-Med. Eng., 52: 557 – 565, 2005.

[4] G. W. Hedstrom, Nonreflecting boundary conditions for nonlinear hyperbolic systems, J.Comput. Phys., 30: 222 – 237, 1979.

[5] K. W. Thompson, Time dependent boundary conditions for hyperbolic systems, J. Com-put. Phys.,



Virtual Element Method for Bulk-surface Reaction-diffusion Systemswith Electrochemical Applications

Massimo Frittelli, Department of Innovation Engineering, University of SalentoVia per Arnesano, 73100 Lecce, Italy


ABSTRACT

We present a Bulk-Surface Virtual Element Method (BSVEM) for the spatial discretisationof bulk-surface reaction-diffusion systems (BSRDSs) in two space dimensions. The methodis based on coupling the Virtual Element Method (VEM) [1] in the bulk domain to a sur-face finite element method [2] on the surface. To the best of the authors’ knowledge, theproposed method is the first application of the VEM to bulk-surface PDEs. The methodexhibits second-order convergence in space, provided the exact solution is H2+1/4 in the bulkand H2 on the surface, where the additional 1

4is required only in the simultaneous presence

of surface curvature and non-triangular elements. Two novel techniques introduced in ouranalysis are (i) an L2-preserving inverse trace operator for the analysis of boundary condi-tions and (ii) the Sobolev extension as a replacement of the lifting operator [3] for sufficientlysmooth exact solutions. The generality of the polygonal mesh can be exploited to optimizethe computational time of matrix assembly.We present a novel bulk-surface model for electrodeposition based on the reaction-diffusionmodel considered in [4]. The proposed model couples a linear diffusion system in the bulkwith a RDS on the boundary through nonlinear boundary conditions. Numerical examplesillustrate (i) pattern formation in the proposed BSRDS for electrodeposition, (ii) the com-putational advantages of BSVEM and (iii) the optimal convergence rate.Joint work with Anotida Madzvamuse (University of Sussex, UK) and Ivonne Sgura (Uni-versity of Salento, Italy).

References

1. L Beirao da Veiga, F Brezzi, A Cangiani, G Manzini, L D Marini, and A Russo. Basicprinciples of virtual element methods. Math Mod Meth Appl Sci, 23(01):199–214, 2013.

2. G Dziuk and C M Elliott. Finite element methods for surface PDEs. Acta Numerica,22:289-396, 2013.

3. C M Elliott and T Ranner. Finite element analysis for a coupled bulk–surface partialdifferential equation. IMA J Num Anal, 33(2):377–402, 2013.

4. D Lacitignola, B Bozzini, and I Sgura. Spatio-temporal organization in a morpho-chemical electrodeposition model: Hopf and turing instabilities and their interplay.Europ J Appl Math, 26(2):143–173, 2015.



Numerical Analysis of 1D Blood Flow Coupled with a New PressureArea Relation

Mahesh Udupa, Vellore Institute of Technology, Vellore, India, email:[email protected]

Sunanda Saha, Vellore Institute of Technology, Vellore, India.

ABSTRACT

In the present work, one dimensional blood flow through an arterial vessel has been con-sidered and coupled with a generalized pressure-area relation, which thus far has not beenconsidered as per author’s knowledge. A theoretical study on the derivation of the character-istic variables including the generalized pressure-area relation has been carried out. Furtherthe results available in the literature has been verified with Lax Wendroff scheme and thesubsequent results have been compared with a Finite difference scheme and two other FiniteVolume Methods, using rms-error analysis.



Perturbative Analysis of the Discretization to Stochastic HamiltonianProblems

Giuseppe Giordano, University of Salerno, Fisciano (SA), Italy, email:[email protected]

Raffaele D’Ambrosio, University of L’Aquila, L’Aquila (AQ), Italy.Beatrice Paternoster, University of Salerno, Fisciano (SA), Italy.

ABSTRACT

Stochastic differential equations (SDEs) are used to describe several real-life phenomenawhose underlying dynamics depends on random fluctuations. This is the case, for example,of weather forecasts, turbulent diffusion or investment finance. In fact, SDEs provide a keytool for a mesoscopic approach to describe the effects of external environments to a physicalmodel. In this talk we specifically focus on the numerical discretization of stochastic Hamil-tonian problem with additive noise, that are the most suitable candidate to conciliate theclassic Hamiltonian mechanics with the non-differentiable Wiener process which describesthe continuous innovative character of stochastic diffusion. Specifically, our analysis in-volves stochastic Runge-Kutta methods obtained as a stochastic perturbation of symplecticRunge-Kutta methods, in order to understand their role in retaining invariance laws of theunderlying dynamical system. In particular, we are interested in maintaining the linear driftvisible in the expected Hamiltonian of the system. We give explanation to the preservationof this linear drift by means of a perturbative analysis, in terms of ε-expansions, being ε theamplitude of the stochastic part of the right-hand side. The presence of spurious terms grow-ing in time and with ε is also visible and explained. Numerical tests confirm the theoreticalanalysis.

References

[1] A. Bazzani, Hamiltonian systems and Stochastic processes, Lecture Notes, Universityof Bologna (2018).

[2] P.M. Burrage, K. Burrage, Low rankRunge-Kutta methods, symplecticity and stochas-tic Hamiltonian problems with additive noise, Journal of Computational and AppliedMathematics, 236, 3920-3930 (2012).

[3] P.M. Burrage, K. Burrage, Structure-preserving Runge-Kutta methods for stochasticHamiltonian equations with additive noise, Numer. Algorithms 65, 519-532 (2014).

[4] C. Chen, D. Cohen, R. D’Ambrosio, A. Lang, Drift-preserving numerical integrators forstochastic Hamiltonian systems, Adv. Comput. Math 46, 27 (2020).

[5] R. D’Ambrosio, G. Giordano, B. Paternoster, Numerical conservation issues for stochas-tic Hamiltonian problems, submitted.



Nonlinear Stability Issues in Stochastic Discretizations

Stefano Di Giovacchino, University of L’Aquila, L’aquila, Italy, email:[email protected]

Raffaele D’Ambrosio, University of L’Aquila, Italy, Italy.

ABSTRACT

We analyze conservation properties of numerical methods for nonlinear stochastic differentialequations (SDEs), hidden behind proper conditional stability issues. We study the numericalapproximation of nonlinear SDEs of Ito type with an exponential mean-square contractivebehaviour by stochastic teta-methods, in order to provide stepsize restrictions ensuring sim-ilar exponential mean-square properties also numerically, without adding further constraintson the numerical method itself. The analysis is also provided for stochastic Runge-Kuttamethods. A selection of numerical experiments confirms the sharpness of the estimates.



Numerical Preservation Issues for Nonlinear Stochastic Oscillators

Carmela Scalone, University of L’Aquila, L’Aquila, Italy,email: [email protected]

Raffaele D’Ambrosio, University of L’Aquila, L’Aquila, Italy.

ABSTRACT

Long-term preservation of meaningful features is a dominant topic of the numerical analysisof differential problems. This talk is focused on analyzing the conservation issues of stochasticθ-methods when applied to nonlinear damped stochastic oscillators. We deal with a secondorder stochastic differential equation of the form

x = f(x)− ηx+ εξ(t) (11)

where ξ(t) satisfy E|ξ(t)ξ(t′)| = δ(t− t′) and η is the damping parameter. The motion of aparticle described by (12), is characterized by a deterministic force f(x), which derives from apotential V (x), i.e., f(x) = −V ′(x). The random forcing ξ(t) has amplitude ε, satisfying therelation ε2 = 2ηKT , where η is the amplitude of the damping term and T is the temperature.We are interested in reproducing the long-term properties of the continuous problem overits discretization through stochastic θ-methods, by preserving the correlation matrix. Thisevidence is equivalent to accurately maintaining the stationary density of the position and thevelocity of a particle driven by a nonlinear deterministic forcing term and an additive noiseas stochastic forcing term. The provided analysis relies on a linearization of the nonlinearproblem, whose effectiveness is proved theoretically and numerically confirmed.



Convergence of the Piecewise Orthogonal Collocation for PeriodicSolutions of Retarded Functional Differential Equations

Alessia And, University of Udine, Udine, Italyemail: [email protected]

Dimitri Breda, University of Udine, Udine, Italy.

ABSTRACT

I will present an analysis of the convergence of (a variant of) the piecewise orthogonal collo-cation for periodic solutions of retarded functional differential equations defined by a genericright-hand side [1]. Such analysis is highly based on [2] where a general framework for solv-ing a certain class of boundary value problems (BVPs) is presented and accompanied by arigorous proof of convergence of the corresponding iterative method. The novel contributionsconsist in the proofs of the validity of the assumptions required to apply the abstract ap-proach of [2] in the case of periodic BVPs. Indeed, although the general BVP in [2] considersthe presence of unknown parameters, it does not explicitly deal with the periodic case. Inthe presentation I will highlight the role of the period as the (main) unknown parameterof the problem, which leads to some effort in validating the required assumptions, being itdirectly linked to the course of time. I will conclude the presentation with some commentson the possibility of extending the convergence proof to the case of renewal equations.

References

[1] A. And and D. Breda, Convergence analysis of collocation methods for computing periodicsolutions of retarded functional differential equations, SIAM J. Numer. Anal., (2020) toappear.

[2] S. Maset, An abstract framework in the numerical solution of boundary value problemsfor neutral functional differential equations, Numer. Math., 133(3):525 555 (2016).


Numerical Stability Analysis of Linear Periodic Delay Equations viaPseudospectral Methods

Davide Liessi, University of Udine, Italy, email: [email protected]

Dimitri Breda, University of Udine, Italy.

ABSTRACT

Realistic models for various phenomena, from control theory to mathematical biology, areoften based on delay equations. Due to the complexity of such models, their dynamicscannot usually be studied analitically and must be approximated numerically. I will presenta method based on pseudospectral collocation for approximating the spectra of evolutionoperators of linear delay equations, focusing on renewal equations [1] with some hints tothe cases of delay differential equations [2, 3] and coupled renewal and delay differentialequations [4]. The method allows to study the stability of periodic orbits via the principleof linearised stability and Floquet theory.

References

[1] D. Breda and D. Liessi, Approximation of eigenvalues of evolution operators forlinear renewal equations, SIAM J. Numer. Anal., 56 (2018), pp. 1456–1481, DOI:10.1137/17M1140534.

[2] D. Breda, S. Maset, and R. Vermiglio, Approximation of eigenvalues of evolution op-erators for linear retarded functional differential equations, SIAM J. Numer. Anal., 50(2012), pp. 1456–1483, DOI: 10.1137/100815505.

[3] D. Breda, S. Maset, and R. Vermiglio, Stability of Linear Delay Differential Equations:A Numerical Approach with MATLAB, SpringerBriefs Control Autom. Robot., Springer,New York, 2015, DOI: 10.1007/978-1-4939-2107-2.

[4] D. Breda and D. Liessi, Approximation of eigenvalues of evolution operators for linearcoupled renewal and retarded functional differential equations, Ric. Mat. (2020), DOI:10.1007/s11587-020-00513-9.

Numerical Preservation Issues for Nonlinear Stochastic Oscillators

Carmela Scalone, University of L’Aquila, L’Aquila, Italy,email: [email protected]

Raffaele D’Ambrosio, University of L’Aquila, L’Aquila, Italy.

ABSTRACT

Long-term preservation of meaningful features is a dominant topic of the numerical analysisof differential problems. This talk is focused on analyzing the conservation issues of stochasticθ-methods when applied to nonlinear damped stochastic oscillators. We deal with a secondorder stochastic differential equation of the form

x = f(x)− ηx+ εξ(t) (12)

where ξ(t) satisfy E|ξ(t)ξ(t′)| = δ(t− t′) and η is the damping parameter. The motion of aparticle described by (12), is characterized by a deterministic force f(x), which derives from apotential V (x), i.e., f(x) = −V ′(x). The random forcing ξ(t) has amplitude ε, satisfying therelation ε2 = 2ηKT , where η is the amplitude of the damping term and T is the temperature.We are interested in reproducing the long-term properties of the continuous problem overits discretization through stochastic θ-methods, by preserving the correlation matrix. Thisevidence is equivalent to accurately maintaining the stationary density of the position and thevelocity of a particle driven by a nonlinear deterministic forcing term and an additive noiseas stochastic forcing term. The provided analysis relies on a linearization of the nonlinearproblem, whose effectiveness is proved theoretically and numerically confirmed.

Nonreflecting Boundary Conditions for a CSF Model of the FourthVentricle - Spinal SAS Dynamics

Licia Romagnoli, Universita Cattolica del Sacro Cuore, Brescia, Italy,email: [email protected]

Donatella Donatelli, University of L’Aquila, L’Aquila, Italy.

ABSTRACT

In this talk we will introduce a one-dimensional model for analyzing the cerebrospinal fluid(CSF) dynamics within the fourth ventricle and the spinal subarachnoid space (SSAS) ([2]).The model has been derived starting from an original model of Linninger et al. ([3]) andfrom the detailed mathematical analysis of two different reformulations developed in [1]. Wewill show the steps of the modelization and the rigorous analysis of the first-order non-linearhyperbolic system of equations which rules the new CSF model, whose conservative-law formand characteristic form are required for the boundary conditions treatment. By assumingsub-critical flows, for the particular dynamics we are dealing with, the most desirable optionis to employ the nonreflecting boundary conditions ([4], [5]), that allow the simple wave as-sociated to the outgoing characteristic to exit the computational domain with no reflections.Finally, we will show some numerical simulations related to different cerebral physiologicalconditions.

References

[1] D. Donatelli, P. Marcati and L. Romagnoli, A comparison of two mathematical models ofthe cerebrospinal fluid dynamics, Mathematical Biosciences and Engineering, 16: 2811– 2851, 2019.

[2] D. Donatelli and L. Romagnoli, Nonreflecting boundary conditions for a CSF model ofthe fourth ventricle - spinal SAS dynamics, Bulletin of Mathematical Biology (BMAB),82 (2020).

[3] A. A. Linninger, C. Tsakiris, D. C. Zhu et al. , Pulsatile cerebrospinal fluid dynamics inthe human brain, IEEE T. Bio-Med. Eng., 52: 557 – 565, 2005.

[4] G. W. Hedstrom, Nonreflecting boundary conditions for nonlinear hyperbolic systems, J.Comput. Phys., 30: 222 – 237, 1979.

[5] K. W. Thompson, Time dependent boundary conditions for hyperbolic systems, J. Com-put. Phys.,


Highly Stable Multivalue Almost Collocation Methods with StructuredCoefficient Matrix

Maria Pia D’Arienzo, University of Salerno, Fisciano, Italy, email:[email protected] Conte, University of Salerno, Fisciano, Italy.

Raffaele D’Ambrosio, University of L’Aquila, L’Aquila, Italy.Beatrice Paternoster, University of Salerno, Fisciano, Italy.

ABSTRACT

Our aim is to solve systems of ordinary differential equations potentially candidate to be stiff,so developed methods must have high stability properties and uniform order of convergence.We focus our attention on multivalue methods [1], which are a generalization of classicalmethods, such as multistep and Runge-Kutta methods, and we extend the solution smoothlyby approximating them though a collocation polynomial. These methods require at eachtime-step the solution of a non linear system of internal stages, so the computational effortis strictly connected to the nature of this system. We are interested in the construction ofmethods that allow a reduction of this computational cost, so we propose methods with fullmatrix [6] and structured matrix (triangular [2], singly triangular [3, 4], diagonal [5]). In thecase of structured matrix, we perform almost collocation since it is not possible to imposeall the collocation conditions. We compare these methods calculating the error in the finalstep point and the experimental order of convergence.

References

[1] J.C. Butcher, General linear methods. Computers & Mathematics with Applications.31 (4-5): 105-112. doi:10.1016/0898-1221(95)00222-7 (1996).

[2] D. Conte, R. D’Amborsio, M.P. D’Arienzo, B. Paternoster, Highly stable multivaluecollocation methods, J. Phys.: Conf. Ser. 1564, 012012 (2020).

[3] D. Conte, R. D’Ambrosio, M.P. D’Arienzo, B. Paternoster, Singly diagonally im-plicit multivalue collocation methods, in 2020 International Conference on Mathe-matics and Computers in Science and Engineering (MACISE), Madrid, Spain, DOI:10.1109/MACISE49704.2020.00018, pp. 65-68 (2020).

[4] D. Conte, R. D’Amborsio, M.P. D’Arienzo, B. Paternoster, One-point spectrum Nord-sieck almost collocation methods, International Journal of Circuits, Systems and SignalProcessing, vol. 14, pag. 266-275, DOI: 10.46300/9106.2020.14.38 (2020).

[5] D. Conte, R. D’Amborsio, M.P. D’Arienzo, B. Paternoster, Multivalue almost colloca-tion methods with diagonal coefficient matrix, Lecture Notes in Comput. Sci., in press.

[6] R. D’Ambrosio, B. Paternoster, Multivalue collocation methods free from order reduc-tion, J. Comput. Appl. Math. DOI: 10.1016/j.cam.2019.112515 (2019).



Perturbative Analysis of the Discretization to Stochastic HamiltonianProblems

Giuseppe Giordano, University of Salerno, Fisciano (SA), Italy, email:[email protected]

Raffaele D’Ambrosio, University of L’Aquila, L’Aquila (AQ), Italy.Beatrice Paternoster, University of Salerno, Fisciano (SA), Italy.

ABSTRACT

Stochastic differential equations (SDEs) are used to describe several real-life phenomenawhose underlying dynamics depends on random fluctuations. This is the case, for example,of weather forecasts, turbulent diffusion or investment finance. In fact, SDEs provide a keytool for a mesoscopic approach to describe the effects of external environments to a physicalmodel. In this talk we specifically focus on the numerical discretization of stochastic Hamil-tonian problem with additive noise, that are the most suitable candidate to conciliate theclassic Hamiltonian mechanics with the non-differentiable Wiener process which describesthe continuous innovative character of stochastic diffusion. Specifically, our analysis in-volves stochastic Runge-Kutta methods obtained as a stochastic perturbation of symplecticRunge-Kutta methods, in order to understand their role in retaining invariance laws of theunderlying dynamical system. In particular, we are interested in maintaining the linear driftvisible in the expected Hamiltonian of the system. We give explanation to the preservationof this linear drift by means of a perturbative analysis, in terms of ε-expansions, being ε theamplitude of the stochastic part of the right-hand side. The presence of spurious terms grow-ing in time and with ε is also visible and explained. Numerical tests confirm the theoreticalanalysis.

References

[1] A. Bazzani, Hamiltonian systems and Stochastic processes, Lecture Notes, Universityof Bologna (2018).

[2] P.M. Burrage, K. Burrage, Low rankRunge-Kutta methods, symplecticity and stochas-tic Hamiltonian problems with additive noise, Journal of Computational and AppliedMathematics, 236, 3920-3930 (2012).

[3] P.M. Burrage, K. Burrage, Structure-preserving Runge-Kutta methods for stochasticHamiltonian equations with additive noise, Numer. Algorithms 65, 519-532 (2014).

[4] C. Chen, D. Cohen, R. D’Ambrosio, A. Lang, Drift-preserving numerical integrators forstochastic Hamiltonian systems, Adv. Comput. Math 46, 27 (2020).

[5] R. D’Ambrosio, G. Giordano, B. Paternoster, Numerical conservation issues for stochas-tic Hamiltonian problems, submitted.



Virtual Element Method for Bulk-Surface Reaction-Diffusion Systemswith Electrochemical Applications

Massimo Frittelli, Department of Innovation Engineering, University of SalentoVia per Arnesano, 73100 Lecce, Italy


ABSTRACT

We present a Bulk-Surface Virtual Element Method (BSVEM) for the spatial discretisationof bulk-surface reaction-diffusion systems (BSRDSs) in two space dimensions. The methodis based on coupling the Virtual Element Method (VEM) [1] in the bulk domain to a sur-face finite element method [2] on the surface. To the best of the authors’ knowledge, theproposed method is the first application of the VEM to bulk-surface PDEs. The methodexhibits second-order convergence in space, provided the exact solution is H2+1/4 in the bulkand H2 on the surface, where the additional 1

4is required only in the simultaneous presence

of surface curvature and non-triangular elements. Two novel techniques introduced in ouranalysis are (i) an L2-preserving inverse trace operator for the analysis of boundary condi-tions and (ii) the Sobolev extension as a replacement of the lifting operator [3] for sufficientlysmooth exact solutions. The generality of the polygonal mesh can be exploited to optimizethe computational time of matrix assembly.We present a novel bulk-surface model for electrodeposition based on the reaction-diffusionmodel considered in [4]. The proposed model couples a linear diffusion system in the bulkwith a RDS on the boundary through nonlinear boundary conditions. Numerical examplesillustrate (i) pattern formation in the proposed BSRDS for electrodeposition, (ii) the com-putational advantages of BSVEM and (iii) the optimal convergence rate.Joint work with Anotida Madzvamuse (University of Sussex, UK) and Ivonne Sgura (Uni-versity of Salento, Italy).

References

1. L Beirao da Veiga, F Brezzi, A Cangiani, G Manzini, L D Marini, and A Russo. Basicprinciples of virtual element methods. Math Mod Meth Appl Sci, 23(01):199–214, 2013.

2. G Dziuk and C M Elliott. Finite element methods for surface PDEs. Acta Numerica,22:289-396, 2013.

3. C M Elliott and T Ranner. Finite element analysis for a coupled bulk–surface partialdifferential equation. IMA J Num Anal, 33(2):377–402, 2013.

4. D Lacitignola, B Bozzini, and I Sgura. Spatio-temporal organization in a morpho-chemical electrodeposition model: Hopf and turing instabilities and their interplay.Europ J Appl Math, 26(2):143–173, 2015.



Geometric Analysis of a Phantom Bursting Model

Iulia Martina Bulai, Department of Mathematics, Informatics and Economics, Universityof Basilicata, Viale dell’Ateneo Lucano, 10, 85100 Potenza, Italy,


Richard Bertram, Department of Mathematics, Florida State University, Tallahassee.Morten Gram Pedersen, Department of Information Engineering, University of Padova,

Italy.Theodor Vo, School of Mathematics, Monash University, Australia.

ABSTRACT

The phantom bursting model was introduced to describe the episodic bursting of the pancre-atic β-cells, where active phases are interspersed by silent ones. The model is characterisedby two slow and two fast variables with the two slow variables having very different timescales [1]. Considering the different time scales of the four variables of ordinary differentialequations, Mixed-Mode Bursting Oscillations (MMBOs) solutions can be found. MMBOsare characterized by both small-amplitude oscillations (SAOs) and bursts consisting of oneor multiple large-amplitude oscillations (LAOs) [2].

Here we focus our attention on the mechanism that generate the MMBOs due to bothcanards and delayed-Hopf-bifurcation [3]. Canards are central to the dynamics of MMBOsand we study them starting from the folded singularities, that are equilibria of the desin-gularized system of the phantom burster model. The canard phenomenon explains the veryfast transition upon variation of a parameter from a small amplitude limit cycle via canardcycles to a large amplitude relaxation cycle. Furthermore the presence of the subcriticalHopf bifurcation via fast/slow analysis of the fast subsystem is found. A detailed geometricexplanation of MMBOs is done using numerical simulations and the slow attracting manifoldis obtained using numerical continuation technics.

References

[1] Richard Bertram, Joseph Previte, Arthur Sherman, Tracie A. Kinard, Leslie S. Satin,The Phantom Burster Model for Pancreatic β-Cells, Biophysical Journal, 79(6) (2000)2880-2892.

[2] Mathieu Desroches, Tasso J. Kaper, Martin Krupa, Mixed-mode bursting oscillations:Dynamics created by a slow passage through spike-adding canard explosion in a square-wave burster, Chaos: An Interdisciplinary Journal of Nonlinear Science, 23(4) (2013)1054-1500.

[3] Richard Bertram, Joseph Rhoads, Wendy P. Cimbora, A Phantom Bursting Mechanismfor Episodic Bursting, Bull. Math. Biol. 70(1979) (2008) 1989-2017.


12 General Contributed Talk


Pointwise Error Estimation for Nonhomogeneous Dirichlet BoundaryValue Problems

Swapnil V Kale , Defence Institute of Advanced Technology, Pune, India,email:[email protected]

Debasish Pradhan, Defence Institute of Advanced Technology, Pune, India.

ABSTRACT

We give a pointwise error estimate for the Laplace’s problem −∆u = f in Ω with non-homogeneous Dirichlet boundary condition u = Ψ on ∂Ω, by approximating it into theRobin BVP with a penalty factor ε, and u is the solution of original problem and uε is thesolution of Robin BVP, respectively. For finding the convergence of uε → u, the L2-errorestimate has been done for the continuous problem [2]. Here, the domain Ω is not necessarilyconvex, so we decompose the doamin Ω into the Base-2 annulus regions known as a dyadicdecomposition of the domain and we use the idea of regularised Green’s functions combinedwith the local H1 and L2 estimates over these dyadic regions [1]. Also as the computationaldomain Ωh 6= Ω we take care of boundary skin layers known as the perturbation terms.Using linear finite element method, we find an approximate solution uhε to the new RobinBVP, and L∞ and W 1,∞ error estimates are derived between the solutions uε and uhε , whichare of O(h2| log h|) and O(h), respectively.

Keywords: Pointwise Error Estimate, Linear Finite Element Method, Penalty factor,Dyadic Decomposition

[1] T. Kashiwabara, T. Kemmochi: Pointwise error estimates of linear finite element methodfor Neumann boundary value problems in a smooth domain,Numerische Mathematik,(2020)[2] J. Barrett and C. Elliott: Finite element approximation of the dirichlet problem using theboundary penalty method. Numerische Mathematik, 49:343366, 1986.[3] Krasovski, J.P.: Isolation of singularities of the Greens function. Math. USSR Izvest. 1,935966 (1967).[4] Wahlbin, L.B.: Maximum norm error estimates in the finite element method with isopara-metric quadratic elements and numerical integration. R.A.I.R.O. Numer. Anal. 12, 173202(1978).



Fourier Integral Representation of the R-Function

Ankit Pal, S. V. National Institute of Technology, Surat, Gujarat, India. email:

[email protected]

R. K. Jana, S. V. National Institute of Technology, Surat, Gujarat, India.A. K. Shukla, S. V. National Institute of Technology, Surat, Gujarat, India.

ABSTRACT

In this paper, we propose a Fourier transform representation of theR-function which leads tothe distributional representation. Further we use this representation to obtain the integralsof products of two R-functions by using the Parseval’s identity of the Fourier transform interms of Kampe de Feriet function. An application related to Mittag-Leffler functions havealso been discussed.



A New Approximation for Conformable Time Fractional NonlinearDelayed Differential Equations via Two Efficient Methods

Saloni AgrawalDepartment of Mathematics, School of Physical and Decision Sciences,

Babasaheb Bhimrao Ambedkar University, Lucknow-226025 (UP), India.email: [email protected]

Brajesh Kumar SinghDepartment of Mathematics, School of Physical and Decision Sciences,

Babasaheb Bhimrao Ambedkar University, Lucknow-226025 (UP), India.

ABSTRACT

This article shows a comparative study for conformable time fractional nonlinear partialdifferential equations with proportional delay by adopting two efficient techniques: Newintegral decomposition transform method (NIDTM) and optimal homotopy analysis new in-tegral transform method (OHANITM). Some properties of new integral transform basedupon conformable derivatives are analysed. Three test examples of conformable time frac-tional generalized Burger equation with proportional delay (CTGBEPD) are taken for theeffectiveness and validity of both methods. The findings demonstrate that for given orderof approximation NIDTM requires less computational time in comparison to OHANITM. Inaddition, OHANITM is more general and converges faster than NIDTM at optimal value of~ while OHANITM results coincide with NIDTM for controlling parameter ~ = −1.



Variable Order Nonlocal Choquard Problem with Variable Exponents

Reshmi Biswas, Indian Institute of Technology Guwahati, Guwahati, India,email:[email protected]

Sweta Tiwari, Indian Institute of Technology Guwahati, Guwahati, India.

ABSTRACT

In this talk, our main focus is to discuss the study of a class of nonlocal Choquard type of equationsinvolving nonlocal non-linearity as well as some non-homogeneous nonlocal operator which is havingvariable order-exponents feature.

To investigate the existence/multiplicity results of the solutions to such problem, first we requireto ensure the desired smoothness of the associated energy functional, which is itself a challengingjob due to the presence of the nonlocal non-linearity involving variable exponents.

Therefore, in this work first we introduced some new fractional Sobolev spaces with variable orderand variable exponents associated to the variable-order fractional p(·)-Laplace operator, studied theproperties of these spaces, and proved some continuous and compact type embedding results.Next, we established a Hardy-Littlewood-Sobolev type inequality in variable exponents set-up forthe functions belonging to fractional Sobolev spaces with variable order and variable exponents,which is the most crucial tool to study the Choquard type of problem. Finally, using this result weguaranteed the existence of solution to our problem involving variable-order fractional p(·)-Laplaceoperator and generalized Choquard type non-linearity in some appropriate fractional Sobolev spaceswith variable order and variable exponents.



Approximate Controllability of Semilinear Hilfer Fractional DifferentialEquation with Control in the Nonlinear Term

Bandita Roy, Indian Institute of Technology Guwahati, Assam, India, email:[email protected]

Swaroop Nandan Bora, Indian Institute of Technology Guwahati, Assam, India.

ABSTRACT

Fractional differential equations have emerged as an important area of investigation as it can beapplied to a variety of areas such as biological, physical, and engineering sciences. Further, controltheory is a branch of engineering and mathematics that deals with influencing the behavior of dy-namical systems. Its objective is to make systems perform specific tasks by using suitable controlactions. Controllability is an important concept in mathematical control theory. It is concerned withthe question of the existence of a control function which steers the solution of the system from aninitial state to a final state. This work studies the approximate controllability for a class of fractionalcontrol systems with analytic semigroup governed by differential equations with Hilfer derivatives inan abstract space. The existence and uniqueness of the mild solutions are established with the helpof semigroup theory, fractional power of operators, and a generalized contraction type fixed pointtheorem. A set of sufficient conditions is formulated to ensure the approximate controllability of oursystem.

References[1] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. Theory and Applications of Fractional Differ-

ential Equations. Elsevier, London, 2006.

[2] Hilfer, R. Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000.

[3] Mahmudov, N.I. and Zorlu, S. On the approximate controllability of fractional evolution equa-tions with compact analytic semigroup. J. Comput. Appl. Math., 259 (2014), 194–204.



Unsteady Flow of Thin Liquid Film Over a Heated Stretching Sheet inPresence of Uniform Transverse Magnetic Field and Thermal Radiation

Palky Handique, NIT Arunachal Pradesh, Yupia, India,email: [email protected]

ABSTRACT

The unsteady two dimensional flow of thin Casson liquid film over a porous heated horizontal stretch-ing surface are discussed considering the stretching velocity and temperature distribution in theirgeneral forms in presence of thermal radiation. The effects of uniform transverse magnetic field andsection/injection are also considered for investigation. The nonlinear governing set of equations willbe solved numerically. An evolution equation for the film thickness, that retains the convective heattransport effect; are derived using long wave theory of an a thin liquid film are solved numericallyfor some values of non dimensional parameters. It is found that the film thickness decreases withthe increasing value of Casson Parameter. The thermocapillary effects are also observed.



Numerical Study of Errors Obtained by Combining the Shooting Methodwith the FOM Method in Solving Boundary Value Problems

Mahendra Kumar Jena, Veer Surendra Sai University of Technology, Odisha, India, email:mkjena [email protected]

Kshama Sagar Sahu, Veer Surendra Sai University of Technology, Odisha, India.

ABSTRACT

Most of the fundamental processes of the nature is converted to boundary value problems. Solvinga boundary value problem (BVP) is not easy and there exist few methods to solve BVPs. Oneway is to convert a BVP into an initial value problem (IVP) and then solve the IVP. One methodwhich converts a BVP into an IVP is the shooting method. In this paper, we combine the shootingmethod with the frame operational matrix (FOM) method to solve a BVP with Dirichlet boundarycondition, approximately. The error of this approximation is not much satisfactory. But, we foundthat if we subdivide the interval of consideration and take boundary conditions for this subdivideinterval appropriately then error can be made smaller and smaller. This is an advantage since wecan subdivide the interval of consideration as many times we want.



New Fourth-Order Efficient Numerical Solutions of Heat-Like Equationand Klein-Gordon Equation

MukeshDepartment of Mathematics, Babasaheb Bhimrao Ambedkar University Lucknow-226025 (UP)

India,email: [email protected]

Brajesh Kumar SinghDepartment of Mathematics, Babasaheb Bhimrao Ambedkar University Lucknow-226025 (UP)

India.

ABSTRACT

In the present paper an efficient fourth-order collocation scheme with use of cubic B-spline (CBCM4)as base functions is adopted for the evaluation of numerical solutions of heat-like equations and Klein-Gordon, which occurs in various physical phenomenon like - temperature-changes of the composite-materials, nonlinear optics, solid state physics, and quantum field theory. First, the given timedependent partial differential equations is discretized via Crank-Nicolson scheme, in which afterlinearization of each nonlinear term in the equation, the spatial derivatives of fourth order accuracyevaluated from cubic b-spline collocation and Taylor formula is utilized. In this way the obtainedsystem of linear recurrence equations in time with suitable initial and boundary conditions is solved byGauss elimination method. The effectiveness, accuracy and validity of the adopted scheme CBCM4is verified by considering two test problems of each Klein-Gordon and heat-like equations, in termsof order of convergence, L2 and L∞ errors. The findings shows that the proposed solutions havefourth-order spatial accuracy.



PDE Optimal Control Problems: An Automatic Differentiation Approach

Renu Saraswat, School of Natural Science, Shiv Nadar University, Gautam Buddh Nagar(U.P),India email:[email protected]

Ajit Kumar, School of Natural Science, Shiv Nadar University, Gautam Buddh Nagar (U.P),India.

ABSTRACT

Optimal Control problems with PDE constraints find applications in many fields such as Aeronau-tics, Mechanical engineering, Medicines, etc. The hardest part of solving such problems is findinggradients of cost functionals with respect to control parameters. The standard approach is to writea Lagrangian functional, and take its derivative with respect to the control variables and form aso-called adjoint equation. This approach is simple in theory but is extremely complicated in prac-tice especially when the constraints are PDEs. We demonstrate here that one can avoid learningthe theory of Lagrangian and adjoint variables if we use Automatic Differentiation (AD). AD isa trending tool used primarily to find derivatives of functions which are described by a computerprogram. This is applicable in PDE-Optimal control problems because cost functions are typicallyintegration of solutions of PDEs which are generally solved computationally.

We test and report our findings of combining a Matlab PDE solver and a AD Matlab code:ADiMat.In this seminar, we demonstrate how a Matlab PDE solver code can be restructured and interfacedwith ADiMat. A wrapper function is created which solves the constraint PDE in each execution ofa cost-function call and integrates the result to evaluate the cost function. The cost-function needsto follow a strict pattern for interfacing with ADiMat. The ideas discussed here are demonstratedon simple textbook PDE-constrained optimization problems: Bang bang control and the Distributedcontrol with Neumann boundary conditions. Finite difference techniques are used for the discretiza-tion of the PDEs. The gradients are calculated with ADiMat. The optimization steps are simplegradient descent method. For improved convergence, step lengths are calculated with a backtrackingmethod. It is observed that the Automatic differentiation approach is much easier and it makes PDEoptimal control problems more accessible.

Keywords: ADiMat;PDE optimal control problems; Automatic Differentiation ;Bang bang control,Distributed control



The Lumped Mass Finite Element Method for Fractional ParabolicIntegro-Differential Equations

Shantiram Mahata, Indian Institute of Technology Guwahati, Assam - 781039, India, email:[email protected]

Rajen Kumar Sinha, Indian Institute of Technology Guwahati, Assam - 781039, India, email:[email protected]

ABSTRACT

We study lumped mass finite element method for a time-fractional parabolic integro-differential equa-tion of fractional order α ∈ (0, 1) in a bounded convex polygonal domain Ω. The spatially discretefinite element method for both smooth and nonsmooth data cases are analyzed using piecewise linearand continuous finite elements. We prove optimal order error bounds in the L2 and H1 norms whenthe initial function u0 ∈ H2(Ω) ∩H1

0 (Ω). Moreover, almost optimal order error bounds are shownto hold for nonsmooth initial data (u0 ∈ L2(Ω)) in the H1-norm, whereas L2-norm error relies onan additional assumption on the mesh.



An Improvement of Third Order WENO Scheme for Convergence Rateat Critical Points with New Non-linear Weights

Anurag Kumar , Department of Mathematics, University of Delhi, Delhi 110007, India,email:[email protected]

Bhavneet Kaur, Department of Mathematics, Lady Shri Ram College for Women, University ofDelhi, Delhi 110024, India.

ABSTRACT

In this paper, we construct and implement a new improvement of third order weighted essentiallynon-oscillatory (WENO) scheme in the finite difference framework for hyperbolic conservation laws.In our approach, a modification in the global smoothness measurement is reported by applying allthree points on global stencil (i − 1, i, i + 1) which is used for convergence of non-linear weightstowards the optimal weights at critical points and achieves the desired order of accuracy for thirdorder WENO scheme. We use the third order accurate total variation diminishing (TVD) Runge-Kutta time stepping method. The major advantage of the proposed scheme is its better numericalaccuracy in smooth regions. The computational performance of the proposed WENO scheme withthis global smoothness measurement is verified in several benchmark one- and two-dimensional testcases for scalar and vector hyperbolic equations. Extensive computational results confirm that thenew proposed scheme achieves better performance as compared with WENO-JS3, WENO-Z3 andWENO-F3 schemes.



Port-Hamiltonian Control Approach for CRTBP with Non-idealSolar-sail and Albedo Effect

Arun Kumar Yadav, Indian Institute of Technology(ISM), Dhanbad, India,email:[email protected]

Badam Singh Kushvah, Indian Institute of Technology(ISM), Dhanbad, India.

ABSTRACT

Libration point orbits around collinear Lagrange points present many properties that are advanta-geous for space missions. These Lagrange point orbits are exponentially unstable. Station keepingstrategies targets periodic orbits around the unstable Lagrangian points. These Control strategiesare based on the circular restricted three body problem (CRTBP) linearized around an equilibriumpoint and canot endure global stability. The problem of stabilization of Lagrangian point usingNon-ideal solar sail and albedo effect in the Sun-Jupiter system is investigated in this paper. A port-Hamiltonian approach is use to reformulate the CRTBP with input, which preserves the originalnon-linear dynamics.

Keyword : Port-Hamiltonian Method; Non-Ideal Solar Sail; Energy Shaping; Dissipation Injec-tion; Albedo effect.



An Exponentially Fitted Numerical Algorithm for a Boundary ValueProblem of Singularly Perturbed Delay Differential Equation

N. Sathya Kumar, Vellore Institute of Technology, Vellore, Tamil Nadu–632014, India,email:[email protected]

R. Nageshwar Rao, Vellore Institute of Technology, Vellore, Tamil Nadu–632014, India.

ABSTRACT

In this paper, a numerical algorithm is presented for solving a linear singularly perturbed delaydifferential equation. Central differences for the derivatives are re-approximated and exponentialfitting is used. Convergence analysis of the numerical scheme is carried out. Two test problems aresolved to demonstrate the efficiency of the method.



Exponential Spline Method for Two-Point Singularly PerturbedDifferential-Difference Equations

E. Siva Prasad1∗ and K. Phaneendra2

Department of Mathematics, University College of Engineering, Osmania University, Hyderabad,Telangana, 500004, India.emineni.yahoo.co.in

[email protected]

ABSTRACT

In this paper, we proposed a second order finite difference method via exponential splines in con-nection with numerical treatment for a class of two-point differential-difference equations havingnegative shift in the differentiated term with boundary layer at left (or right) end of the domain.A fitting factor is introduced in the difference scheme and evaluated it with the help of theory ofsingular perturbations. Convergence of the method is analyzed. Also we have focused on the effectof shift on the boundary layer behaviour or oscillatory behaviour of the solutions using exponentialspline method with a special type of mesh. The maximum absolute errors in the solution are tabu-lated to illustrate the efficiency of the proposed numerical method compare with the existing methods.

Key words: Exponential spline, Differential- difference equation, Delay, Fitting parameter, Dif-ference approximation.



Wave Scattering by a Submerged Circular Porous Membrane

R. Gayathri, Department of Mathematics, College of Engineering and Technology, SRM Institute ofScience and Technology, SRM Nagar, Kattankulathur, 603203, Kanchipuram, Chennai, TN, India,


Siluvai Antony Selvan, Department of Mathematics, College of Engineering and Technology, SRMInstitute of Science and Technology, SRM Nagar, Kattankulathur, 603203, Kanchipuram, Chennai,

TN, India.Dr. Harekrushna Behera, Department of Mathematics, College of Engineering and Technology,SRM Institute of Science and Technology, SRM Nagar, Kattankulathur, 603203, Kanchipuram,

Chennai, TN, India.

ABSTRACT

Using the plane wave integration technique of Bessel and Hankel functions, wave scattering bya submerged circular permeable membrane is investigated to examine the impact of porosity indampening the far-field wave energy. The effect of three different edge conditions are employed suchas (i) free-edge, (ii) clamped-edge and (iii) mooring-edge. The unknown potential for the free surfaceand the membrane covered regions are obtained by coupling the respective boundary conditions andthe Darcy’s law for permeable structure. Further, the Bessel series solution is obtained by using thematched eigenfunction expansion method. To understand the efficiency of the present study, waveforce excitation on the circular membrane, far field wave reflection and transmission amplitude,energy dissipation, flow distribution and deflection of the membrane are computed and examined.The study shows that the wave amplitude on the membrane’s leeward side is reduced drastically. Inaddition, the submergence depth plays a vital role in the wave energy dissipation. This model willbe useful in the development of a mechanism to reduce the wave load on coastal/sea facilities forvarious marine development activities. In addition, this structure has minimal environmental effectson different coastal processes and can therefore be used in a variety of coastal/ocean applications.



A New Hybrid Cubic B-spline Differential Quadrature Method for aClass of Convection–diffusion Equations in Extended Domains

Jai Prakash ShuklaDepartment of Mathematics, School of Physical and Decision Sciences,Babasaheb Bhimrao Ambedkar University Lucknow-226025 (UP) India,


Brajesh Kumar SinghDepartment of Mathematics, School of Physical and Decision Sciences,Babasaheb Bhimrao Ambedkar University Lucknow-226025 (UP) India.

ABSTRACT

In the present paper, a new hybrid cubic B–spline differential quadrature method (in short nHCB–DQM) has been developed to solve convection–diffusion equations. The nHCB–DQM is base ondifferential quadrature method (DQM) with new hybrid cubic B–splines as base functions. We haveapplied the proposed method for convection-diffusion equations, and so, the convection–diffusionequations is converted into a system of first order ordinary differential equations (ODEs) which issolved by using SSP–RK43 scheme. The efficiency and effectiveness of the proposed method is testedfor three different problems of convection–diffusion equation in terms of L2 and L∞ error norms.The computed results are compared with the exact results. Also, we compare the computed resultswith existing recently published results. It is seen that the computed results are agreed well withthe exact solutions. Less data complexity, straightforwardness and easy to implementation are themain advantages of nHCB–DQM.



Numerical Methods for Modelling an Oscillating Water Column on aPorous Sloping Ocean Bed

Mohamin B M Khan, Department of Mathematics, College of Engineering and Technology, SRMInstitute of Science and Technology, SRM Nagar, Kattankulathur, 603203, Kanchipuram, Chennai,

TN, Indiaemail: [email protected]

Harekrushna Behera, Department of Mathematics, College of Engineering and Technology, SRMInstitute of Science and Technology, SRM Nagar, Kattankulathur, 603203, Kanchipuram, Chennai,

TN, India

ABSTRACT

As world energy scenario is shifting towards green and safe renewable energy resources, wave energyderived from the oceans has the potential of providing a substantial amount of this new energy tothe world. Oscillating Water Column (OWC) devices are of considerable interest in this regard. Themaximum power absorption by the OWC is attained when the resonance frequency of the watercolumn inside the chamber is equal to the frequency of the incoming waves. The advantages ofthe OWC devices with attached air turbine stem from their reliability, efficiency and low cost ofmaintenance.

This work examines the wave energy conversion efficiency of an OWC over a porous slopingocean floor, using the analytical eigenfunction expansion method in conjunction with numericalmulti-domain Boundary Element Method (BEM) using constant elements. Additionally, the roles ofintroducing stepped bottom and undulated slope in front of the OWC or a reflecting wall etc. arealso studied with the aim of increasing performance of the system. Various proposed structural andhydrodynamic conditions are analyzed to maximize energy conversion efficiency, wherein radiationsusceptance, radiation conductance and maximum efficiency parameters are optimized.



Boundary Control Design for Parabolic Systems Using BacksteppingMethod

T. RenugadeviDepartment of Mathematics, Bharathiar University, Coimbatore, India


ABSTRACT

This paper focuses on the boundary control problem for a linear parabolic system using backsteppingmethod. The system consists of mixed boundary conditions and a Dirichlet local term which leadsinstability in the systems. A Volterra integral transformation is used to eliminate the destabilizingterm and subsequently an equivalent stable target system is derived. The explicit solution of kernelfunction is found using Laplace transforms and subsequently the boundary control is designed. Thestability of the target system is analysed by the Lyapunov approach. Finally, the effectiveness of theachieved results are validated through an example.

References[1] D.M. Boskovic and M. Krstic, “Stabilization of a solid propellant rocket instability by state

feedback”, International Journal of Robust and Nonlinear Control, vol. 13, pg. 483-495, 2003.

[2] M. Krstic and A. Smyshlyaev, Boundary control of PDEs: A course on backstepping designs,volume 16, Siam, 2008.

[3] A. Alessandri, P. Bagnerini, R. Cianci, S. Donnarumma and A. Taddeo, “Stabilization of diffusivesytems using backstepping and the circle criterion”, International Journal of Heat and MassTransfer, vol. 149, pg. 119132, 2020.



Design and Analysis of a Numerical Method for Fractional NeutronDiffusion Equation with Delayed Neutrons

Vikas Rohil, Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur,440010, India, email: [email protected]

Pradip Roul, Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur,440010, India.

ABSTRACT

The main purpose of this work is to construct and analyze an efficient numerical scheme for solvingthe fractional neutron diffusion equation with delayed neutrons, which describes neutron transportin a nuclear reactor, given by:

ταΣa∂αΦ(x, t)

∂tα=D

∂2Φ(x, t)

∂x2+[(1− β)γΣf − Σa

]Φ(x, t) + λC(x, t) ,

0 < α <1

2, (x, t) ∈ (0, X)× (0, T ) ,

(13)

τα−1 ∂αC(x, t)

∂tα= βγΣfΦ(x, t)− λC(x, t) , (14)

with initial condition

Φ(x, 0) = Φ0(x) (15)

and boundary conditions

Φ(0, t) = 0 , Φ(X, t) = 0 . (16)

The L1 approximation is used for discretization of time derivative and finite difference methodis used for discretization of space derivative. The stability and convergence analysis of the proposedmethod are studied. The method is shown to be second-order convergent in space and (2 − 2α)-thorder convergent in time, where is the order of fractional derivative. Numerical experiments arecarried out to demonstrate the performance of the method and theoretical analysis. The effectsof fractional order derivative, relaxation time and radioactive decay constant on the neutron fluxbehaviour are investigated.



Natural Cubic Spline for Singular Boundary Value Problems

M.Santoshi Kumari, B.M.S College of Engineering, Bangalore, India,email:[email protected] and [email protected]

B. Mallikarjuna, BMS College of engineering, Bangalore, India.Shrivalli, H.Y, BMS College of engineering, Bangalore, India.

ABSTRACT

In this paper, we developed and presented a natural cubic spline for solving singular boundary valueproblems (BVP). The BVPs are solved at various boundary conditions. The solution is approximatedby cubic spline functions and converted BVP’s into tridiagonal form. Thomas algorithm is used tosolve the tridiagonal form more accurately. To check the accuracy of the present method we comparedresults with another numerical method. The obtained results show that the cubic spline method isgiving more accurate results compare to other techniques for singular boundary value problems.Some examples are presented and compared with the exact solutions numerically and graphically.Residual of the differential equations are also presented.



Mathematical Modeling of COVID-19 Transmission: The Roles ofIntervention Strategies and Tockdown

Sarita Bugalia, Department of Mathematics, Central University of Rajasthan, Bandar Sindri,Kishangarh-305817, Ajmer, Rajasthan, India [email protected]

Vijay Pal Bajiya, Department of Mathematics, Central University of Rajasthan, Bandar Sindri,Kishangarh-305817, Ajmer, Rajasthan, India

Jai Prakash Tripathi, Department of Mathematics, Central University of Rajasthan, Bandar Sindri,Kishangarh-305817, Ajmer, Rajasthan, India

ABSTRACT

An outbreak of rapidly spreading coronavirus established human-to-human transmission and nowbecame a pandemic across the world. The new confirmed cases of infected individuals of COVID-19are increasing day by day. Therefore, the prediction of infected individuals has become of utmostimportant for health care arrangements and to control the spread of COVID-19. In this study, wepropose a compartmental epidemic model with intervention strategies such as lockdown, quarantine,and hospitalization. We compute the basic reproduction number (R0), which plays a vital role inmathematical epidemiology. Based on R0, it is revealed that the system has two equilibrium, namelydisease-free and endemic. We also demonstrate the non-negativity and boundedness of the solutions,local and global stability of equilibria, transcritical bifurcation to analyze its epidemiological rele-vance. Furthermore, to validate our system, we fit the cumulative and new daily cases in India.We estimate the model parameters and predict the near future scenario of the disease. The globalsensitivity analysis has also been performed to observe the impact of different parameters on R0. Wealso investigate the dynamics of disease in respect of different situations of lockdown, e.g., completelockdown, partial lockdown, and no lockdown. Our analysis concludes that if there is partial orno lockdown case, then endemic level would be high. Along with this, the high transmission rateensures higher level of endemicity. From the short time prediction, we predict that India may face acrucial phase (approx 6000000 infected individuals within 140 days) in near future due to COVID-19.Finally, numerical results show that COVID-19 may be controllable by reducing the contacts andincreasing the efficacy of lockdown.



Output Feedback Boundary Controller for an ODE Coupled to a FirstOrder Hyperbolic PDE

A. Shree NidhiDepartment of Mathematics, Bharathiar University, Coimbatore, India


ABSTRACT

This work focuses on the output feedback boundary controller design for a first order ODE-transportPDE coupled at the boundary point. An anti-collocated observer is designed for the chosen coupledsystem and the output feedback boundary control law is formulated using the backstepping method.The novelty of the work is that the linear matrix inequality approach has been implemented todesign the gains. The exponential stability of the error system is discussed using Lyapunov stabilitytheory. The effectiveness of the output feedback boundary controller is illustrated through numericalexamples.

References[1] L. Baudouin, A. Seuret, M. Safi. (2016) Stability analysis of a system coupled to a transport

equation using integral inequalities. 2nd IFAC Workshop on Control of Systems Governed byPartial Differential Equations. Vol.49. 092-097.

[2] M. Krstic, A. Smyshlyaev. (2008). Boundary control of PDEs. A course on backstepping designs.Philadelphia, PA.

[3] S. Tang, C. Xie. (2011). State and output feedback boundary control for a coupled PDE-ODEsystem. Systems and control letters. Vol.60. 540-545.



A New Approach for Solving Third-order Partial Differential Equationsvia Discrete Spline Technique

Talat Sultana, Lakshmibai College,University of Delhi, New Delhi, India,email:[email protected]

Pooja Khandelwal, Department of Mathematics, M.L.V. Textile & Engineering College,Bhilwara-311001, INDIA.

D.N. Vyas, Department of Mathematics, M.L.V. Textile & Engineering College, Bhilwara-311001,INDIA.

ABSTRACT

In this paper, we propose a new approach to solve third-order homogeneous and non-homogeneouspartial differential equations (PDEs) which occurs in the field of science and engineering. Ourapproach depends on discrete cubic spline technique based on central differences. The proposedalgorithm is stable and convergent. Three numerical examples are given to validate the efficiencyand accuracy of the proposed method. Comparisons are made to confirm the reliability and accuracyof the proposed technique.



Hybrid Approach to Find Approximate Analytical Solution of Burger’sEquation

Archana C. Varsoliwala, Applied Mathematics and Humanities Department, Sardar VallabhbhaiNational Institute of Technology, Surat-395 007 (Gujarat), India, [email protected]. Twinkle R. Singh, Applied Mathematics and Humanities Department, Sardar Vallabhbhai

National Institute of Technology, Surat-395 007 (Gujarat), India.

ABSTRACT

In this article, the effectualness of the Elzaki Adomain Decomposition Method has been discussed.Elzaki Adomain decomposition method is a fusion of Elzaki transform and Adomian decompositionmethodology. Adomian polynomials handle the non-linear terms. Burger’s equation is well-knownmodel in applied mathematics and engineering field. Here the precise and approximate analyticalsolution of Burger’s equation is defined by this approach. The technique has been connected to(2+1)-dimensional, (3+1)-dimensional and (n+1)-dimensional Burger’s equation.



Numerical Technique for Singularly Perturbed Delay DifferentialEquations Using Gaussion Quadrature

M. Lalu, University College of Engineering, Osmania University, Hyderabad, India,email: [email protected]

K. Phaneendra,University College of Engineering, Osmania University, Hyderabad, India.

ABSTRACT

A quadrature method is suggested for the solution of singularly perturbed delay differential equa-tion. Initially, a first-order delay differential equation is achieved, which is asymptotically equivalentto the given singularly perturbed delay differential equation. Then Gaussian quadrature two-pointformula is implemented on the first order equation to get a tridiagonal system. Thomas algorithmis used to solve this system. The proposed method is implemented on model examples, for differentvalue of delay parameter and perturbation parameter. Maximum absolute errors are tabulated witha comparison to authorize the method. Theoretical convergence of the method is discussed. Thelayer behaviour is discussed using the graphical representation.

Keywards: singularly perturbed delay differential equations, Gaussian quadrature two-pointformula, layer behaviour, linear interpolation



Surfactant-laden Newtonian Falling Film Down a Wavy Channel: Linearand Nonlinear Stability Analysis

Md. Mouzakkir Hossain, Siluvai Antony Selvan and Dr. Harekrushna BeheraDepartment of Mathematics, SRM Institute of Science and Technoloy, Kattankulathur, India.

email:mouzakkir123gmail.com

ABSTRACT

An instability of surfactant-laden falling film down the wavy plane is analyzed using the methodof small aspect ratio and multiple scales. The falling film and insoluble surfactant are governedby the Navier–Stokes equation and surfactant-transport equation, respectively. Using the techniqueof small-aspect ratio, the non-linear evolution equation corresponding to the physical problem isobtained. Further, the method of multiple scales are employed directly in the evolution equationfor determining the nonlinear instability in the neighborhood of criticality. Moreover, the criticalconditions of the primary instability are determined, which depend on the surfactant properties. Thenumerical results carried out for different physical variables suggest that the behaviour of surfaceand surfactant modes on the primary instability are solely depend on the properties of an insolublesurfactant and wavy bottom. The amplitude of non-linear disturbance varies and shows anomalousbehaviour in the sub-critical and supercritical regime depending on the surfactant and amplitude ofwavy bottom. This work may find application in the coating and biomedical industries, where thefilm thickness is sufficiently small and the method of small aspect ratio holds good for estimatingthe primary instability.



Long Wave Stability Analysis of a Film Flow in the Presence of anInsoluble Surfactant

Muhammad Sani, Department of Mathematics College of Engineering and Technology, SRMInstitute of Science and Technology,

Kattankulathur-603 203, Tamil Nadu, [email protected]

Dr. Harekrushna Behera, Department of Mathematics College of Engineering and Technology,SRM Institute of Science and Technology,

Kattankulathur-603 203, Tamil Nadu, India.

ABSTRACT

Long wave stability analysis can serve as a powerful tool for determining stability/instability of fluidflowing horizontally with surfactant at the free surface which has an important effects on appliedMathematics, Physics, Ocean Engineering, etc.. This study investigates the long wave stability of ahorizontal film flow in the presence of an insoluble surfactant at the free surface. The motion of thefluid is described using the Navier-Stokes equation and the normal mode approach is employed forderiving the corresponding Orr-Sommerfeld system of equations. The effects of insoluble surfactantand other physical parameters are analyzed through the growth rate results. The contours of thebandwidth for varying Reynolds numbers against the Marongoni and Capillary numbers are elabo-rated in this work. It is observed that the insoluble surfactant plays a vital role to suppress the freesurface instability.



Wave Scattering by a Pair of Floating Pontoons Having a SubmergedMesh Cage

V. Venkateswarlu, Department of Civil Engineering, Bapatla Engineering College,Mahatmajipuram, Bapatla, Andhra Pradesh-522102, India

C.S. Nishad, Department of Mathematics, School of Technology, Pandit Deendayal PetroleumUniversity, Gandhinagar, Gujarat-382007, India

K.G. Vijay, Oceaneering International Services Limited, Chandigarh-160101, IndiaT. Sahoo, Department of Ocean Engineering and Naval Architecture, IIT Kharagpur, West

Bengal-721302, India

ABSTRACT

In recent decades, recurrence of storm surges is on the rise due to global warming which has resultedin a significant threat to coastal facilities and infrastructure. Often temporary floating structuresprovide a cost-effective and feasible solution which is because these structures are independent ofwater depth, reusable and easy to install and decommission. In the present study, scattering of gravitywaves by a pair of floating pontoons having a mesh cage is studied under the assumptions of smallamplitude wave theory. The waves past the mesh cage are assumed to follow nonlinear pressure dropboundary condition to capture the effect of non-breaking wave height on the energy dissipation. Tosolve the boundary value problem, a numerical model based on the dual boundary element method isdeveloped. The code is initially validated with known results in the literature. For different wave andstructural parameters such as relative water depth, relative spacing and different barrier porositiesvarious physical quantities of interests like scattering coefficients, horizontal and vertical forces arecomputed and analyzed. The study will be useful in the design of effective breakwaters as waveabsorption systems.

Keywords: Floating pontoons, mesh cage, non-linear pressure drop, dual boundary elementmethod.



Sonofragmentation of Rectangular Plate-like Crystals: BivariatePopulation Balance Modeling and Experimental Validation

Ashok Das, IIT Kharagpur, Kharagpur, West Bengal, 721302, India, email:[email protected]

Jitendra Kumar, IIT Kharagpur, Kharagpur, West Bengal, 721302, India

ABSTRACT

The use of multidimensional population balance equations in predicting ultrasound assisted sonocrys-talization processes are curbed due to the unavailability of physically motivated macroscopic mathe-matical kernels. This study develops a comprehensive population balance modeling of the ultrasoundassisted sonofragmentation of thin rectangular plate-type pyrazinamide crystals. A new bivariatebreakage selection function is presented, which takes care of the time dependency in particle selec-tions along with the size dependency. Further, a new mathematical formulation of the bivariatebreakage distribution function is introduced to predict the outcomes of sonofragmentation experi-ments of Bhoi et al. (Chem. Eng. Sci. 2019, 203, 12-27). Finally, the results obtained from solvingthe developed population balance model is successfully validated against the experimental results.The developed population balance model accurately captures the effects of variations in ultrasonicamplitude and sonication time.



Lie Group Transformation Analysis of Fractional Partial DifferentialEquations with Variable Coefficients

Baljinder Kour, Central University of Punjab, Bathinda, India, email:[email protected]

ABSTRACT

The present study is devoted for the Lie group transformation analysis of space time fractional systemwith variable coefficients involving the Riemann-Liouville derivative. Feasible vector field of thesystem is obtained by applying the invariance attribute of one-parameter Lie group of transformation.The reduction of the number of independent variables by this method gives the reduction of fractionalsystems with variable coefficients into a system of fractional ordinary differential equations involvingErdelyi-Kober operators. The application of Lie symmetry on a fractional system is basically forreduction of order and very helpful to obtain exact solution of fractional partial differential.



Analysis of Reflection of Wave at the Free Boundary of a FlexoelectricMicrostructured Half-space

Geetika Gupta, Panjab University, Chandigarh, India, email: [email protected]

Baljeet Singh, Post Graduate Government College, Sector 11, Chandigarh, India .

ABSTRACT

In the present work, isotropic dielectrics with centrosymmetric microstructures specialized in a planeis considered. The effects of micro-inertia, flexoelectricity and non-uniform strain is studied. Byconsidering the plane wave solution of field equations, it is confirmed that two quasi plane wavespropagate in a flexoelectric medium. Reflection phenomenon of the plane harmonic waves in adielectric halfspace model is studied by using stress free boundary conditions. The coefficient ofreflection of reflected waves are analytically obtained for both incident plane waves. An experimentaldata of strontium titanate is chosen for computations of the speeds, the coefficient of reflectionof reflected waves. These wave characteristics are illustrated graphically to show the influencesof incident angle, wavenumber, length parameters and flexocoupling coefficients. The results arevalidated using sum of energy ratios.



Numerical Approach for Solutions of Delay Differential Equations

Giriraj Methi , Department of Mathematics & Statistics,Manipal University Jaipur, Rajasthan,India, email::[email protected]

Anil Kumar, Department of Mathematics & Statistics,Manipal University Jaipur, Rajasthan, India

ABSTRACT

Aim of the talk is to obtain numerical solutions of some nonlinear delay differential equations usingCoded 2 Method (CDTM). The CDTM is developed and applied to some nonlinear delay problemsto show the efficiency of the proposed method. The coded differential transform method is a com-bination of the differential transform method and Mathematica software. The numerical solutionobtained by CDTM is compared with an exact solution. Numerical results, convergence and erroranalysis are presented for delay differential equations to show that the proposed method is suitablefor solving delay differential equations. The coded differential transform method reduces complexcalculations, avoids discretization, linearization, and saves calculation time.



Mathematical Study of Reflection of Plane Waves From theStress-free/rigid Surface of a Micro-mechanically Modeled Piezoelectric

Fiber-Reinforced Composite Half-space

Sayantan Guha, Indian Institute of Technology (Indian School of Mines), Dhanbad, India,email:[email protected]

ABSTRACT

The present work has two primary objectives: The first one is to present the micro-mechanics modelof Piezoelectric Fiber-Reinforced Composite (PFRC) and demonstrate some of its advantages overmonolithic piezoelectric materials. The second one is to analytically investigate wave reflectionphenomenon at the stress-free/rigid surface of a PFRC. With the emergence of more sophisticatedtechnological advancements, the development of composite structures such as PFRCs is ongoing at amassive pace due to their enormous significance owing to substantial assets like strength, light-weight,low thermal effects, etc. They offer superior performance in comparison to monolithic piezoelectricmaterials, as they can be optimized to enhance the properties which are desired in respective scientificand engineering applications. Ergo, continuous and determined attempts are being made to developand improvise such materials focusing on their effective commercial utilization in countless areaslike aeronautics, sports, constructions, remote explorations, medical services, ultrasonic imaging,applications of surface acoustic wave (SAW) and bulk acoustic wave (BAW) devices, etc. amongothers. The prediction of the electro-mechanical properties of PFRCs has been a dynamic researcharea for several years. Consequently, several important works on developing the micro-mechanicsof composite materials by analytical and numerical methods using techniques such as AsymptoticHomogenization Method, Continuum Mechanics, Method of Cells, Finite Element Method, etc. canbe found in extant literature. In this work, the PFRC structure is modeled using the Strength ofMaterials technique with the Rule of Mixtures approach. Due to the incidence of a quasi-longitudinalwave, three reflected waves viz. quasi-longitudinal (qP), quasi-transverse (qSV), and electroacoustic(EA) waves are generated in the half-space. The angles of propagation of the reflected waves aregraphically represented as functions of the incident angle. The closed-form expressions of amplituderatios of all reflected waves are derived utilizing appropriate electro-mechanical boundary conditionsat both the stress-free, as well as the rigid surface. However, the amplitude ratios of reflected wavescannot be used exclusively to validate the numerical results. Hence, the expressions of energy ratiosof all reflected waves and interaction energy are derived using the expressions of amplitude ratios,and the Law of Conservation of Energy is established. The influences of the incident angle onthe energy ratios are illustrated graphically. Some special cases exclusive to this study are shown.Despite the apparently endless advantages of PFRC, no mathematical studies have been performedyet on the wave reflection phenomenon at the stress-free/rigid surface of a PFRC half-space. Thepresent work is framed to explore the same for contemplating the phenomenon in constructed smartstructures. Therefore, this work presents a novel effort to develop a connection between derivationof the composite’s micro-mechanics model and analysis of the wave reflection phenomenon in it.



Mathematical Analysis of Surface Wave Velocity in a Rotating BeddedStructure

Nidhi Dewangan, Department of Mathematics, Govt. Pt. Shayamacharan Shukla College,Dharsiwa, Raipur (CG), India.


S. A. Sahu, Department of Mathematics & Computing, Indian Institute of Technology (ISM)Dhanbad (JH), India.


ABSTRACT

This paper aims to analyse the velocity profile of Rayleigh waves (a surface seismic wave) prop-agating in a two-layered anisotropic bedded structure comprises of orthotropic and a generalizedpiezo-thermoelastic substrate. Moreover, we aim to investigate the influence of rotation, stiffnessof orthotropic medium, pre-stress and piezo-thermoelastic coupling constant on the phase velocityof considered wave. Mathematical methods have been employed to obtain the required frequencyequation, under certain boundary conditions. The frequency equation of Rayleigh waves for chargefree as well as electrically shorted cases is obtained. Numerical illustrations are shown throughself-explanatory graphs.



Numerical Approximation of Time Fractional Telegraph Equation by anRBF Based Meshless Method

Akanksha Bhardwaj, Rajiv Gandhi Institute of Petroleum Technology, Amethi, India, email:[email protected]

Alpesh Kumar, Rajiv Gandhi Institute of Petroleum Technology, Amethi, India.Shruti Dubey, Indian Institute of Technology Madras, Chennai, India.

ABSTRACT

In the present manuscript, we solved the time-fractional telegraph equation by a local meshlessmethod. The fractional-order derivative is defined in the Caputos sense. The theoretical convergenceanalysis and unconditional stability of the proposed numerical scheme are also proved. The timesemi-discretization was carried out using finite difference method and for spatial discretization, weproposed radial basis function-based local collocation method. Several test problems with regular andirregular domains with uniform and non-uniform points are considered to demonstrate the accuracyand efficiency of the proposed method.




Mathematical Modeling of Elastic Plastic Transitional Stresses in HumanTooth Enamel and Dentine under Pressure using Seths Transition Theory

Shivdev Shahi, Department of Mathematics, Punjabi University, Patiala, India,email:[email protected]

Satya Bir Singh, Department of Mathematics, Punjabi University, Patiala, India.

ABSTRACT

In this paper, elastic-plastic transitional stresses in human tooth enamel and dentine are calculatedanalytically. The tooth is modelled in the form of a shell which exhibits transversely isotropicmacrostructural symmetry. Transition theory of B.R. Seth(1962) has been used to model the elastic-plastic state of stresses. The shell so modelled is subjected to external pressure to analyse the stateof stress. The results for enamel and dentine are compared with hydroxyapatite (HAP). It is anaturally occurring mineral form of calcium and the enamel is composed 95 percent by weight of thismineral. The elastic stiffness constants for these are taken from the available literatures which havebeen obtained using resonance spectroscopy, a non-destructive technique of obtaining the elasticityconstants. The radial and circumferential stresses are obtained for radius ratios which can handleany type of dataset for thicknesses of enamel and dentine. The findings allow us to conclude thatenamel and dentin under uniaxial compression behaves as a functionally graded strong hard tissuewith necessary elastic and plastic limit, which demonstrates considerable ability to suppress a crackgrowth. Varying values of pressure required for initial yielding and fully plastic state were calculatedfor various radius ratios depending on the geometry of the sample. Trends of the graphs were similarfor enamel and hydroxyapatite due to enamels composition. Significant difference between stressbuildup at crown and root dentin is observed by varying the radii ratios in the modeled sphericalshell.

Mr. Shivdev Shahi, Research Scholar (Ph.D.), Department of Mathematics, Punjabi University,Patiala, IndiaDr. Satya Bir Singh, Professor, Department of Mathematics, Punjabi University, Patiala, India

c© Abstracts of ADENA2020c© Abstracts of ADENA2020


Impact of Inclined Magnetic Field on Mixed Convection of Nanofluid ina Lid-driven Square Enclosure with Different Heater Locations Using

Buingirno’s Two-phase Model

Subhasree Dutta, Department of Mathematics, Indian Institute of Technology, Kharagpur - 721302,India


Somnath Bhattacharyya, Department of Mathematics, Indian Institute of Technology, Kharagpur -721302, India

ABSTRACT

A numerical study to analyze the MHD mixed convective heat transfer in presence of discrete heatersis performed to evaluate the impact of inclined magnetic field on heat transport and energy consump-tion. The heat sources are considered in three different locations of the left wall, whereas the rightwall is maintained at a lower temperature. The aim of our study is to investigate numerically theeffect of Lorentz force generated by the applied inclined magnetic field on the flow and thermal fieldinside the enclosure filled with Al2O3-water nanofluid. Buingirno’s two-phase model [1] is adoptedto study the nonhomogeneous distribution of nanoparticles due to the slip velocity between thebase fluid and nanoparticles generated by the two slip mechanisms namely Brownian diffusion andthermophoresis. A control volume method over a staggered grid arrangement is used to discretizethe governing equations. The two-dimensional continuity, momentum, energy and volume fractionare solved through a pressure-correction based SIMPLE algorithm [2]. A comprehensive parametricstudy is employed to investigate the effect of governing parameters including Richardson number(Ri), Hartman number (Ha), nanoparticle volume fraction (ϕb), Reynolds number (Re) and incli-nation angle (λ) of the imposed magnetic field on flow field, heat transfer and entropy generation.Our results show that the Lorentz force has an adverse effect on the flow as well as thermal field,consequently the heat transfer and entropy generation generation attenuates with the increment ofHartman number. The position of the heater has a significant influence to obtain the thermodynam-ically optimal situation where the heat transfer is enhanced but the entropy generation is minimized.This work has a remarkable contribution to analyze the thermal performance depending upon thelocations of the heater and the imposed magnetic field in micro-electromechanical systems, MHDgenerators, plasma studies and in several engineering applications.

Keywords: Nanofluid; Two-phase model; Magnetic field; Entropy generation.

References[1] J. Buongiorno, Convective transport in nanofluids, Journal of Heat Transfer 128(3) (2006) 240-

250.

[2] C. A. Fletcher, Computational techniques for fluid dynamics 2: Specific techniques for differentflow categories, Springer Science and Business Media (2012).



Radial Miscible Viscous Fingering Induced by an Infinitely FastChemical Reaction

Priya Verma (email:[email protected])Vandita Sharma, Manoranjan Mishra

Indian Institute of Technology Ropar, Rupnagar, Punjab, 140001 India.

ABSTRACT

Viscous fingering (VF) or Saffman-Taylor instability occurs at the fluid-fluid interface when a lessviscous fluid displaces a more viscous fluid in a porous medium [1]. This instability can be observedin various physical phenomena such as enhanced oil recovery, contamination in aquifers, chromatog-raphy seperation, to name a few. The effect of chemical reaction on miscible VF has been extensivelystudied in literature [2, 3]. Recently, Sharma et.al. [3] reported that the non-uniform radial velocityis also a factor contributing to VF due to reaction. We consider miscible, viscosity-matched reactantswith one radially displacing the other undergoing an infinitely fast reaction to generate a producthaving different viscosity. We solve a system of reaction-diffusion-convection (RDC) equations cou-pled with the Darcy’s law. Numerical simulations are performed using a hybridization of compactfinite difference method and the pseudo-spectral method [3, 4]. The chemo hydrodynamic instabilitywith an infinitely fast reaction is explored by suitable modifying the governing RDC equations toincorporate infinite reaction term. The onset of instability is found to be a function of Pe numberwith reaction producing more viscous product being more unstable.

References[1] C. T. Tan and G. M. Homsy. Stability of miscible displacements in porous media: Rectilinear

flow. Physics of Fluids, 29(11):3549-3556, 1986.

[2] Yuichiro Nagatsu, Kenji Matsuda, Yoshihito Kato, and Yutaka Tada. Experimental study onmiscible viscous fingering involving viscosity changes induced by variations in chemical speciesconcentrations due to chemical reactions. Journal of Fluid Mechanics, 571:475-493, 2007.

[3] Vandita Sharma, Satyajit Pramanik, Ching-Yao Chen, and Manoranjan Mishra. A numericalstudy on reaction-induced radial fingering instability. Journal of Fluid Mechanics, 862:624-638,2019.

[4] Ching-Yao Chen, C-W Huang, Hermes Gadelha, and Jose A Miranda. Radial viscous fingeringin miscible Hele-Shaw flows: A numerical study. Physical Review E, 78(1):016306, 2008.



Heterogeneous Multiscale Method in Drug Delivery

Kuldeep Singh Yadav, Indian Institute of Technology Guwahati, Guwahati, India, email:[email protected]

D. C. Dalal, Indian Institute of Technology Guwahati, Guwahati, India

ABSTRACT

In this study, we develop a multiscale method for drug delivery in the biological tissues. Themultiscale method is developed in the heterogeneous multiscale method [1] framework, which is atop-down approach that includes models at two scales, microscale and macroscale.

The multiscale method accounts for the cell scale features to the tissue scale to include thetransport processes occurring there. We study the drug diffusion that occurs at the cellular scale inthree steps: diffusion in the extracellular space, across the cell membrane, and inside the intracellularspace. We simplify the biological cells to be elliptical. The simulation results reveal that the cellulargeometry has a significant impact on the tissue scale effective diffusion even though the porosity ofthe tissue (extracellular volume) does not change. The resulted model can be used to identify theoptimal strategies in drug delivery.

References[1] E Weinan, and Bjorn Engquist, The heterognous multiscale methods. Communications in Math-

ematical Sciences, 1(1):87–132, 2003.



A Particular Soliton-type Analytic Solution and ComputationalModeling of Non-linear Regularized Long Wave Model

Sanjay kumar, Indian Institute of Technology Roorkee, Roorkee, India, email:[email protected]

Ram Jiwari, Indian Institute of Technology Roorkee, Roorkee, India.

ABSTRACT

In this article, the authors study soliton-type analytic solutions and numerical simulation of 1D and2D regularized long wave (RLW) models [1,2]. The model occurred in various fields such as shallowwater waves, plasma drift waves, rotating flow down a tube, longitudinal dispersive waves in elas-tic rods and the anharmonic lattice and pressure waves in liquid-gas bubble mixtures. First of all,tanh-coth method [3] is applied to obtained soliton-type analytic solution of RLW equations. Forcomputational modeling of the problems, a mesh-free method based on local radial basis functionsand differential quadrature method [4,5] is developed. The mesh-free method converts the problemsinto a system of non-linear ordinary differential equations (ODEs) and then the attained system ofODEs is simulated by Runge-Kutta methods (RK4). Further, the stability of the proposed mesh-freemethod is discussed using matrix scheme [4,5]. Finally, in numerical experiments some problems areconsidered to check the competence and chastity of the developed method.References[1] D. H. Peregrine, Calculations of the development of an undular bore, Journal of Fluid Mechanics,vol. 25, no. 2, pp. 321–330, 1966.[2] D. H. Peregrine, Long waves on a beach, Journal of fluid mechanics, vol. 27, no. 4, pp. 815–827,1967.[3] A. Bekir and A. C. Cevikel, The tanh-coth method combined with the riccati equation for solvingnonlinear coupled equation in mathematical physics, Journal of King Saud University-Science, vol.23, no. 2, pp. 127–132, 2011.[4] S. Kumar, R. Jiwari, and R. Mittal, Numerical simulation for computational modelling of reaction-diffusion brusselator model arising in chemical processes, Journal of Mathematical Chemistry, vol.57, no. 1, pp. 149–179, 2019.[5] R. Jiwari, S. Kumar, and R. Mittal, Meshfree algorithms based on radial basis functions for numer-ical simulation and to capture shocks behavior of burgers type problems, Engineering Computations,2019.



Water Wave Scattering and Energy Dissipation by Interface PiercingPorous Plates

Najnin Islam, IIT Kharagpur,West Bengal, India,email: [email protected]

R Gayen, IIT Kharagpur,West Bengal, India.

ABSTRACT

An integral equation method is developed to study the wave interaction with two symmetric per-meable plates submerged in a two-layer fluid. The plates are inclined and penetrate the commoninterface between the layers. The existence of two different wave modes for the incident wave givesrise to two problems. Both of these are tackled by reducing them to a set of coupled hypersin-gular integral equations of the second kind. Unknown functions of the integral equations are thediscontinuities in the potential functions across portions of the plates. These are computed nu-merically by employing an expansion collocation method. New results for the reflection coefficientsand the amount of energy loss are presented by varying several parameters like porosity, angle ofinclination, plate-length, separation between the plates, interface position and density ratio. Knownresults for two symmetric vertical permeable and impermeable plates, single vertical impermeableand horizontal permeable plates are recovered from the present analysis.



Substructuring Waveform Relaxation Methods with Time-dependentRelaxation Parameter

Soura Sana, IIT Bhubaneswar, Bhubaneswar, India, email: [email protected]

Bankim C. Mandal, IIT Bhubaneswar, Bhubaneswar, India, email: [email protected]

ABSTRACT

We present in this paper a modified variant of Dirichlet–Neumann (DNWR) and Neumann–NeumannWaveform Relaxation (NNWR) methods for solving time-dependent Partial Differential Equations(PDEs). These two domain decomposition methods are based on non-overlapping splitting of do-mains. Unlike in the case of classical version, we introduce time-dependent relaxation parameter inthe update stage of the iterative process before moving to next iteration. This makes the analysiscomplex, but leads to an understanding of the effect of the relaxation parameter on the iteration.We present the details of the modified algorithms for two non-overlapping subdomains, and showconditional convergence properties in few special cases. We illustrate our findings with numericalresults.



Role of CD8-cells in a HIV-immune Cell System Through DynamicalAnalysis

Sudipa Chauhan, Amity University, Sector-125, Noida, INDIA email:[email protected]

Payal Rana, Amity University, Sector-125, Noida, INDIA.Kuldeep Chaudhary, Amity University, Sector-125, Noida, INDIA.

ABSTRACT

CD4(T cells) and CD8(Z cells) cells have a major impact on HIV and the cure of HIV is still an unmetchallenge worldwide. However, the activation of CD8 cell has a major role in fighting against thisdisease as it not only kills the infected T-cells but also reduces the virus load indirectly. Therefore,this paper focuses on the formulation of a model by involving both T cells and Z cells. The basicreproduction number has been calculated followed by existence of disease free equilibrium, endemicequilibrium with-out and with immune response and the dynamical analysis for the model. Thenumerical discussion and sensitivity analysis has also been carried for basic reproduction numberand endemic equilibrium point with immune response to recognise the sensitive parameters whichmay help in controlling the disease.



Dynamical Analysis of A Mathematical Model with Incubation TimeDelay for COVID-19

Sumit Kaur Bhatia, Amity Institute of Technology/Amity University Uttar pradesh, India, email:[email protected]

Yashika Bahri, Amity Institute of Technology/Amity University Uttar pradesh, India,Riya Jain, Amity Institute of Technology/Amity University Uttar pradesh, India,

ABSTRACT

The coronavirus disease started spreading around December 2019, and it still poses a major threatto the survival of population. There have been 33.1 Million reported cases of the disease and morethan 9 lakh deaths worldwide. Therefore it is imperative to understand the factors that control thespread of the disease. In this paper, we develop and study the dynamics of the COVID-19 modelincorporating incubation delay. We also study effect of several other parameters on the spread of thedisease. We have derived reproduction number R0, and established the stability of disease-free equi-librium for R0 < 1. We have obtained endemic equilibrium point and studied its stability as well. Inorder to control the spread of the disease several strategies like social distancing and wearing maskshave been incorporated. Later, we have done the numerical analysis of the model, using MATLAB.

Keywords: COVID-19, Epidemics, Basic reproduction number, Sensitivity analysis, IncubationDelay.



A New Approach Based on Exponential B-spline Collocation Method forSolving a Class of Nonlinear Singular Boundary Value Problems with

Neumann and Robin Boundary Conditions

Trishna Kumari, VNIT, Nagpur, India, email: [email protected]

Pradip Roul, VNIT, Nagpur, India.

ABSTRACT

In this paper, we develop a numerical scheme to approximate the solution of general class of nonlinearsingular boundary value problems (SBVPs) subject to Neumann and Robin boundary conditions(BCs) of the following form:

(p(x)y′(x))′ = p(x)f(x, y(x)), 0 < x ≤ 1, (17)

y′(0) = 0, ay(1) + by′(1) = e, (18)

with p(x) = xαg(x), α > 0 and g(x) is a nonnegative function and a > 0, b ≥ 0 and e are finiteconstants. The original differential equation has a singularity at the point x = 0, which is removedvia LHospitals law with an assumption about the derivative of the solution at the point x = 0. Anexponential B-spline collocation approach is then constructed to solve the resulting boundary valueproblem. Convergence analysis of the method is discussed. Numerical examples are provided toillustrate the applicability and efficiency of the method. Our results are compared with the resultsobtained by three other existing numerical methods such as uniform mesh cubic B-spline collocation(UCS) method [1], non standard finite difference method [2] and finite difference method based onChawlas identity [3,4]

References[1] Roul P, Prasad Goura V M K, B-spline collocation methods and their convergence for a class

of nonlinear derivative dependent singular boundary value problems, Appl. Math. Comput. 3412019; 341; 428-450.

[2] Verma A K, Kayenat S, On the convergence of Mickens type nonstandard finite difference schemeson Lane-Emden type equations, J. Math. Chem. 2018; 56; 1667-1706.

[3] Pandey R K, Singh A K, On the convergence of finite difference methods for weakly regularsingular boundary value problems, J. Comput. Appl. Math. 2007; 205; 469-478.

[4] Pandey R K , Singh A K, On the convergence of finite difference methods for general singularboundary value problems, Int. J. Comput. Math. 2003; 80; 1323-1331.



Mathematical Modelling of COVID-19 with the Effects of Quarantineand Detection

M. Aakash, Department of Mathematics, Faculty of Engineering and Technology,SRM Institute of Science and Technology,

Kattankulathur-603203.E-mail: [email protected]

S. Athithan, Department of Mathematics, Faculty of Engineering and Technology,SRM Institute of Science and Technology,

Kattankulathur-603203.E-mail: [email protected]

ABSTRACT

In this paper, a mathematical model for the COVID-19 dynamics with the effects of quarantine anddetection/diagnose is studied and analyzed in detailed manner to provide some suggestions for therecovery from the pandemic situation occurred in many countries, particularly in India. Here we arefocusing on the study of deterministic mathematical model. In this way, we approach the method offinding equilibrium and analyzing their stability. Also we enhance our analytic results by numericalsimulations.Keywords:- COVID-19, Mathematical Model, Stability, Equilibrium points.



MHD Natural Convective Flow of a Polar Fluid with Newtonian HeatTransfer in Vertical Concentric Annuli

Lipika Panigrahi, Department of Mathematics, Veer Surendra Sai University of Technology, Burla,Odisha -768018, India. email: [email protected]

J.P.Panda, Department of Mathematics, Veer Surendra Sai University of Technology, Burla,Odisha -768018, India.

ABSTRACT

A fully developed laminar free convective flow of polar fluid between two concentric vertical cylindershas been considered. The outer surface of the inner cylinders is subjected to Newtons law of heattransfer in the presence of an adjustable transverse magnetic field. The magnetic lines of forceinteract with conducting flowing fluid to generate a force act-at-a-distance. The analytic solutions ofthe governing equations are obtained in the form of modified Bessel function. The important findingsare: Newtonian heating/cooling as well as vortex viscosity reduces the velocity distribution; micro-rotation and linear velocity having opposite signs has an experimental bearing; Magnetic force, act-at-a-distance, reduces the primary flow which may have a therapeutic application; Reynolds analogyholds good with respect to the annular spacing. Most importantly, high Newtonian heating of thebounding surface may lead to back flow as well as instability to flow and heat transfer processes.



A Fourth-order Robust Numerical Method for Fuzzy Integro-differentialEquations under Generalized Differentiability

S Indrakumar∗, Department of Mathematics, Kongu Engineering College. Tamilnadu India.email: [email protected]

S Vengataasalam, Department of Mathematics, Kongu Engineering College. Tamilnadu India.

ABSTRACT

In this paper linear fuzzy Volterra integro-differential equations (FVIDE) under generalized differ-entiability concepts are discussed. Examples of these questions have been solved numerically usingvarious methods for Ordinary Differential Equation (ODE) and using the quadrature rules for in-tegral parts. Finally, a new fourth order routine is used for the numerical solution of the fuzzyintegro-differential equation (FIDE) and numerical example to illustrate the theory.



Bianchi type I Cosmological Model with Ghost Dark Energy in LyraGeometry

Jumi Bharali, Gauhati University, Guwahati, Assam, India.email: [email protected]

Kanika Das, Gauhati University, Guwahati, Assam, India.

ABSTRACT

The main purpose of this paper is to explore the solutions of anisotropic Bianchi type I space timewith ghost dark energy (GDE) in the framework of Lyra Geometry. To find the exact solutions ofEinsteins field equations we use Hybrid Expansion Law (HEL) which is a combination of power lawand exponential law. The study of anisotropy parameter demonstrates that our model approachesisotropy at late times. The Equation of state (EOS) parameter tends to −1 which indicates that ourmodel behaves like a cosmological constant. The matter energy density diminishes whereas GDEdensity approaches to a small value as time evolves. The physical and geometrical aspects of thecosmological model have been discussed and it is found that the results are in good agreement withthe present-day observational facts.



Fractional Order Modeling for Dynamics of Crime Transmission

Komal1, Kocherlakota Satya Pritam2, Trilok Mathur3, Shivi Agarwal4

[email protected]

ABSTRACT

Due to the alarming rise in types of crime committed and the number of criminal activities acrossthe world, there is a great need to upgrade the existing policies and models adopted by jurisdictionalinstitutes. Most of the existing crime transmission mathematical models did not include the past/history of the crime committed by the criminal, which is vital in the eradication of crime. Further, dueto various external factors and policies, a considerable number of criminals have not been imprisoned.To address the aforementioned issues prevailing in the society, this research proposes a novel 4-dimensional fractional crime transmission model by categorizing the existing population into fourclusters. These clusters include law-abiding citizens, criminally active individuals who have not beenimprisoned, prisoners and prisoners who completed the prison tenure. The proposed fractional modelis not only well-posed but also uniformly asymptotically stable. Further, this model is extended to themodel with a delay in catching criminals and due to delay in catching criminals endemic equilibriumis locally asymptotically stable up to a certain point and beyond this bifurcation occurs.

Keywords: Fractional differential equation, Crime transmission, Delay model, Mathematicalmodeling.

References

• McMillon, David, Carl P. Simon, and Jeffrey Morenoff. ”Modeling the underlying dynamicsof the spread of crime.” PloS one 9.4 (2014): e88923.

• Srivastav, Akhil Kumar, Mini Ghosh, and Peeyush Chandra. ”Modeling dynamics of thespread of crime in a society.” Stochastic Analysis and Applications 37.6 (2019): 991-1011.

• Suryanto, Agus, et al. ”A fractional-order predatorprey model with ratio-dependent functionalresponse and linear harvesting.” Mathematics 7.11 (2019): 1100.

• Gambo, Y. Y., et al. ”Existence and uniqueness of solutions to fractional differential equationsin the frame of generalized Caputo fractional derivatives.” Advances in Difference Equations2018.1 (2018): 1-13.



Numerical Solution of Non-linear DPL Model for Analyzing HeatTransfer in Tissue During Thermal Therapy

Neha Sharma, Dept. of Mathematics, Akal college of basic science, Eternal University, Baru Sahib- 173101, H.P., India

Email:[email protected]

Surjan Singh, Dept. of Mathematics, Akal college of basic science, Eternal University, Baru Sahib -173101, India.

ABSTRACT

In this paper, the mathematical modeling and simulation of heat transfer in tissue under Dirich-let boundary conditions has been studied. The non-linear dual-phase-lag (DPL) model is used foranalyzing the temperature base thermal therapy treatment of infected cells. The component ofvolumetric heat source such as blood perfusion and metabolism are assumed in non-linear DPLmodel. A hybrid numerical method is used for solution of non-linear problem which is based onfinite difference scheme and Runge-Kutta (4,5) scheme. In a particular case we obtained the exactsolution and compared with the numerical solution, and found that those are in good agreement.When variation of metabolism and blood perfusion rate is realistic function of temperature, threemodels which are DPL bioheat transfer model, single-phase-lag bioheat transfer model and Pennesbioheat transfer model are compared and found that dual phase lag bioheat transfer model is closestto the experimental data. We also analyzed the effect of different parameters which are perfusionrate, dimensionless heating source, relaxation and thermalisation time on dimensionless temperaturedistribution in treatment process.

Keywords: Non-linear DPL model; tissues, Temperature dependent blood perfusion; Temperaturedependent metabolism; Heat transfer; Thermal therapy.

References[1] H.H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forearm,

J. Appl. Physiol. 1 (1948) 93-122.

[2] D. Kumar, P. Kumar, K.N. Rai, Numerical solution of non-linear dual-phase-lag bioheat transferequation within skin tissues, Math. Biosci. 293 (2017) 56-63.

[3] D. Kumar, Surjan Singh, Neha Sharma, K.N. Rai, Verified non-linear DPL model with exper-imental data for analyzing heat transfer in tissue during thermal therapy, Int. J. Therm. Sci.,133 (2018) 320-329.



Bi-variate Extension of an Operator Based on Multivariate q-LagrangePolynomials

Rahul Shukla, Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee,India, email: [email protected]

Purshottam Narain Agrawal, Department of Mathematics, Indian Institute of Technology Roorkee,Roorkee, India, email: [email protected]

Behar Baxhaku, Department of Mathematics, University of Prishtina, Prishtina, Kosovo, email:[email protected]

ABSTRACT

Let (C(I2), ||.||) be a Banach space endowed with the sup norm. For f ∈ C([0, 1]2), we propose thefollowing linear positive operator as;

Sn2,qn2

,ζ(1),ζ(2)··· ζ(r2)

n1,qn1,η(1),η(2)··· η(r1) (f ;x, y) =

r1∏k1=1

r2∏k2=1

(1− xqn1

n1η(k1)n1

)n1(

1− yqn2n2ζ(k2)n2

)n2

∞∑p1=0

∞∑p2=0

∑l1+l2+···+lr1=p1

∑l∗1+l

∗2+···+l∗r2=p2

r1∏s1=1

r2∏s2=1

(qn1n1, qn1

)ls1

(qn2n2, qn2

)l∗s2

(η(s1)n1 )ls1

(qn1 , qn1 )ls1

(ζ(s2)n2 )

l∗s2

(qn2 , qn2 )l∗s2

·f(

[lr1 ]qn1

[n1 + lr1 − 1]qn1

,[l∗r2 ]qn2

[n2 + l∗r2 − 1]qn2

)xp1yp2 , (19)

where

η(i) =η(i)n1

n1∈N

, (i = 1, 2, · · · r1)

ζ(j) =ζ(j)n2

n2∈N

, (j = 1, 2, · · · r2)

are sequences of real numbers such that η(i)n1 , ζ

(j)n2 ∈ (0, 1) and [n]q denotes the q-integers. In the

present work, we propose the above extension of a linear positive operator constructed by means ofthe multivariable q-Lagrange polynomials (Appl. Math. Comput., 219(2013), pp. 6911-6918). Weestablished the order of convergence in terms of the complete and partial modulus of continuity. Inorder to discuss the approximation behavior in Bogel spaces of continuous and differentiable func-tions, we define the GBS (Generalized Boolean Sum) extension of the proposed bi-variate operatorand study the rate of convergence by means of the mixed modulus of smoothness. Lastly, usingTaylor’s polynomials, we define and study the degree of approximation of rth order generalization ofoperators given by (19).



The Behaviour of Creep in Cylinder and Disc

Savita Bansal, Punjabi University, Patiala, Punjab, India.email: savita [email protected]

Satya Bir Singh, Punjabi University, Patiala, Punjab, India.

ABSTRACT

Engineering and mechanical components are manufactured keeping in mind the need to reduce itswear and tear over a period of time. A lot of research over past eight decades has taken place inorder to make better components which are in scientific terms creep resistant. This paper presentsthe reviews from discs and cylinders made of monolithic material till the discs and cylinders madeof functionally graded material. With the help of functionally graded material, one can reduce thethermal stress and are ideal for extreme temperature and pressure conditions. The review has beendivided into two parts. First part is devoted to the analysis of creep in cylinder and second part isfocused on creep analysis in disc done by various researchers. The paper also gives an overview ofthe classical theory, composites, stress-strain relations, yield criteria, constitutive equations, someof the generalized assumptions made by different researchers for analysis, stages of creep and creeplaws. The theoretical investigations of creep stresses and strains have so far proved to be reliable todesign models where in the theoretical results are empirically at par with the experimental results.This is highly significant from the point of view of observation and experimentation because ofthe fact that the results thus obtained bypass the need to perform strenuous experiments involvingsetting up of apparatuses, modeling samples of various materials, and performing the experiments ina controlled environment by applying pressure/heat etc. to obtain the values of stresses, strains andyield strength. Researchers have constructed constitutive equations based on isotropic/anisotropicyield criterion and evaluated the creep parameters on the basis of available experimental data.



A Cubic B-Spline Quasi-interpolation Method for Solving HyperbolicPartial Differential Equations

Sudhir Kumar, Indian Institute of Technology Roorkee, Roorkee-247667, India,[email protected]

R.C. Mittal, Jaypee Institute of Information Technology Sector 62 Noida, India.Ram Jiwari, Indian Institute of Technology Roorkee, Roorkee-247667, India.

ABSTRACT

This work presents a numerical algorithm based on the Cubic B-spline quasi-interpolation (CB-SQI) method for the simulation of one and two-dimensional hyperbolic partial differential equations(PDEs). In this method, CBSQI is used to approximate the spatial derivatives of the dependentvariable. This produces a second-order ordinary differential equation (ODE). Further, this ODEis reduced into a system of first-order ODEs, and then forward difference approximation for timederivative is used to get final solutions. The idea of Kronecker product is used first time with CB-SQI method for 2D problems. Some well-known problems from the literature such as Klein-Gordon,Sine-Gordon, and Dissipative non-linear wave equations are considered to check the accuracy andefficiency of the proposed method.



Toxicity Effect upon the Phytoplankton and Zooplankton Model

Nossaiba Baba, Hassan II university, Casablanca, Morocco, email:[email protected] Agmour, Hassan II university, Casablanca, Morocco.

Youssef El Foutayeni, Hassan II university, Casablanca, Morocco.Naceur Achtaich, Hassan II university, Casablanca, Morocco.

ABSTRACT

In this paper, we consider the phytoplankton and zooplankton populations that are subject toharvest. Their growth follows the logistic growth function. The zooplanktons accumulate the toxicitysubstance, by predation of phytoplankton, and become toxic. The positivity, boundedness, equilibria,stability, bionomic equilibrium studied. By using generalized Nash equilibrium, we have shown thatthe dynamic results of the interacting species will be affected by the parameters of the system andby their initial population volumes. The optimal harvesting policy is discussed using the MaximumPrinciple of Pontryagin. Finally, some numerical examples are cited to illustrate the effect of toxicityupon both the species.

References1. Youssef EL FOUTAYENI, Mohamed KHALADI, Abdelmounaim ZEGZOUTI, A general-

ized Nash equilibrium for a bioeconomicporblem of fishing, Studia Informatica Universalis-HERMANN, 10 (2012) 186-204.

2. Nossaiba Baba, Imane Agmour, Naceur Achtaich, Youssef El Foutayeni, The mathematicalstudy for mortality coefficients of small pelagic species, Communications in MathematicalBiology and Neuroscience, Vol 2019(2019), Article ID 20.

3. Partha Sarathi Mandal a, Linda J.S. Allen b, Malay Banerjee, Stochastic modeling of phyto-plankton allelopathy, Applied Mathematical Modelling, 38 (2014) 1583-1596.

4. Kanza CHOUAYAKH, Chakib EL BEKKALI, Youssef EL FOUTAYENI, Mohamed KHAL-ADI, Mostafa RACHIK,Maximization of the Fishermen’s Profits Exploiting a Fish Populationin Several Fishery Zones, International Journal of Science and Research, 4 (2015) 1141-1147.

5. L. S. Pontryagin, V. G. Boltyansk, K. V. Gamkrelidre and E. F Mishchenko, The MathematicalTheory of Optimal Processes, Pergamon Press, London, 1964.



Numerical Simulation of 1D and 2D Multi-term Fractional Wave Modelwith Non-linear Source Term

Rahul Kumar Maurya, Government Tilak PG College, Katni, India, email:[email protected]

Vineet Kumar Singh, Indian Institute of Technology, Varanasi, India.

ABSTRACT

In this work, multistep finite difference method based on Caputo fractional derivative approximationpresented and analyzed for solving 1D and 2D nonlinear multi-term fractional wave model (MTFWM)which contain both initial and Dirichlet boundary conditions. We approximate Caputo fractionalderivatives in time with a multistep scheme of order O(τ3−α) & O(τ3−β), 1 < β < α < 2, spatialLaplacian operator with a central difference scheme, and nonlinear source term g(B) by using Taylorseries. The proposed multistep schemes transform the MTFWM into the tridiagonal system for 1Dcase and five-diagonal system for 2D case, respectively. The unique solvability and unconditionalstability are derived for both cases. The convergence of schemes is established with the help ofoptimal error bounds. For 1D MTFWM, accuracy of order O(τ3−α + τ3−β + h2) and for 2DMTFWM, accuracy of order O(τ3−α + τ3−β + h21 + h22) are investigated, where 1 < β < α < 2.Finally, some test functions are investigated to verify our theoretical findings.



A Posteriori Error Analysis of Finite Element Method for ParabolicBoundary Control Problems: A Reconstruction Approach

Ram Manohar, Indian Institute of Technology Guwahati, Guwahati-781039, India, email:[email protected]

Rajen Kumar Sinha, Indian Institute of Technology Guwahati, Guwahati-781039, India, email:[email protected]

ABSTRACT

We present space-time a posteriori error estimates of finite element method for linear boundarycontrol problems governed by parabolic partial differential equations. To discretize the control prob-lems, we use piecewise linear and continuous finite elements for the approximations of the state andcostate variables whereas piecewise constant functions are employed for the control variable. Thetemporal discretization is based on the backward Euler implicit scheme. An elliptic reconstructiontechnique in conjunction with energy argument is used to derive a posteriori error estimates for thestate and the costate variables in the L∞(0, T ;H1(Ω))-norm. Further, an a posteriori error boundfor the control variable is also established in the L∞(0, T ;L2(∂Ω))-norm. The main characteristic ofthis approach is that we can virtually use any available a posteriori estimates for the elliptic problemto control the main part of the spatial error. The derived results are optimal order a posteriori errorestimates.



A Finite Element Method for an Elliptic Optimal Control Problem withIntegral State Constraints

Pratibha Shakya, Department of Mathematics, Indian Institute of Technology Delhi, New Delhi-110016, India. email:[email protected]

Kamana Porwal, Department of Mathematics, Indian Institute of Technology Delhi, New Delhi-110016, India

ABSTRACT

In this work, we discuss a priori error estimates of a bubble enriched nonconforming finite elementmethod for a linear quadratic elliptic distributed optimal control problem with integral state con-straints. Therein, using the state equation we reduce the state-control constrained minimizationproblem to a pure state constrained minimization problem. We have obtained the optimal order(with respect to regularity) error estimate of finite element approximation in two cases: the opti-mal control problem with integral state constraints together with (i) integral control constraints (ii)pointwise control constraints. Numerical results are presented to confirm the theoretical findings.



A Stabilized Finite Element Formulation for Numerical Simulation ofConvection-dominated Reactive Models

Suleyman Cengizci, Middle East Technical University, Ankara, Turkey,Antalya Bilim University, Antalya, Turkey, email:[email protected]

Omur Ugur, Middle East Technical University, Ankara, Turkey.Srinivasan Natesan, Indian Institute of Technology Guwahati, Guwahati, India.

ABSTRACT

In this talk, we are interested in the numerical solution of convection-dominated models with nonlin-ear reaction mechanisms. The existence of the advection term(s) in the relevant models causes thenumerical solutions obtained by standard methods to exhibit nonphysical oscillations. Therefore, theGalerkin finite element method (GFEM) is stabilized using the Streamline-Upwind/Petrov-Galerkin(SUPG) technique to prevent spurious oscillations. Moreover, the stabilized formulation is enhancedby the YZβ shock-capturing technique. Finally, the proposed numerical scheme is tested on variousreaction models. All the numerical computations are performed in the FEniCS environment.

REFERENCES

1. Yucel, H., Stoll, M., Benner, P. Discontinuous Galerkin finite element methods with shock-capturing for nonlinear convection dominated models. Computers & Chemical Engineering.58, 278287 (2013).

2. Uzunca, M., Karaszen, B., Manguolu, M. Adaptive discontinuous Galerkin methods for non-linear diffusionconvectionreaction equations. Computers & Chemical Engineering, 68, 24-37(2014).

3. Weng, Z., Yang, J. Z., Lu, X. Two-grid variational multiscale method with bubble stabilizationfor convection diffusion equation. Applied Mathematical Modelling, 40(2), 1097-1109 (2016).

4. Tezduyar, T.E., Senga, M. Stabilization and shock-capturing parameters in SUPG formula-tion of compressible flows. Computer Methods in Applied Mechanics and Engineering. 195,16211632 (2006).

5. Tezduyar, T.E., Senga, M. SUPG finite element computation of inviscid supersonic flows withYZβ shock-Capturing. Computers & Fluids. 36, 147159 (2007).

6. Tezduyar, T.E., Senga, M., Vicker, D. Computation of inviscid supersonic flows around cylin-ders and spheres with the SUPG formulation and YZβ shock-capturing. Comput Mech. 38,469481 (2006).



Multi-layer Perceptron Artificial Neural Network Approach for SolvingSixth-Order Two-Point Boundary Value Problems

Akanksha Verma, Motilal Nehru National Institute of Technology Allahabad,Prayagraj-211004, (U.P.) India,email: [email protected]

Manoj Kumar, Motilal Nehru National Institute of Technology Allahabad,Prayagraj-211004, (U.P.) India.

ABSTRACT

In this article, we have presented a multilayer perceptron artificial neural network method to solvethe sixth-order boundary value problem. The acquired solutions of these boundary value problemsby our strategy are optimal with less computational efforts as compared to other approximationmethods. Further, a few models have been tested in order to decide the strength of the proposedtechnique. The acquired outcomes proved that the proposed strategy is very effective for higher-orderboundary value problems with low memory space and computational time.



Acoustic-gravity Wave (AGW) Propagation Under Sea-ice in an OceanHaving Elastic Floor

Santu Das, Institute of Advanced Study in Science and technology, Guwahati - 781035, Indiaemail: [email protected] \[email protected]

Michael H. Meylan, School of Mathematical and Physical Sciences, University of Newcastle, NSW2308, Australia

ABSTRACT

Acoustic-gravity waves (AGW), the results of slight compressibility of water, have a vertically oscil-latory profile in contrast with the gravity waves, which have exponential vertical profiles. First, fewevanescent modes of gravity waves become real-valued to generate AGW. These waves spontaneouslyoccur in the ocean due to an abrupt movement inside the water, such as a submarine earthquake.These waves travel much faster than the Tsunami waves and can be treated as an early warningfor Tsunami. The physical problem considered here will include an ice-cover at the ocean surface,and the ocean bed consists of an elastic material of infinite depth. A boundary value problem willbe formulated with the help of linearised water wave theory along with the boundary conditionsprescribed at the ice-cover surface and the ocean floor surface. This work aims to obtain the velocitypotential function and the dispersion relation the wave modes follow to analyze the generation ofAGW and its phase speed. The phase speed of different AGW modes will be graphically illustratedjuxtaposing with their counterparts when the ocean bottom is rigid. The impact of ocean bottomelasticity and that of the ice-cover on the AGW will be discussed.



The Performance of Squeeze Film Conical Bearing in Presence of PorousWall with Viscosity Variation:Rabinowitsch Fluid Model

Amit Kumar Rahul, IIT(ISM), Dhanbad, India, email: [email protected]

Pentyala Srinivasa Rao, IIT(ISM), Dhanbad, India.

ABSTRACT

In this study, the combined effect of viscosity variation of non-Newtonian fluid and porous wallon squeeze film lubrication between conical bearings is analyzed. The modified Reynolds equationis derived ob basis of the modified Darcy’s law and viscosityfilm thickness relationship for porousand viscosity variation. A small perturbation techniques and five-point Gauss quadrature integralformula is used to obtained a closed form expressions for the squeeze film pressure, load-carryingcapacity, and squeeze response time. The results are presented for different operating parameters. Itis observed that the effect of applied porous wall on the squeeze film lubrication conical bearings withviscosity variation is to decrease the load carrying capacity significantly and the time of approach ascompared to the corresponding non-porous case.



Numerical Solution of DPL Bio-heat Transfer Model AmidstHyperthermia Treatment

Tejaswini Kumari, National institute of technology,Patna,Indiaemail: [email protected]

S.K. Singh, National institute of technology, Patna,IndiaDinesh Kumar, Government Polytechnic, Nawada, India

K.N. Rai, Indian Institute of Technology (BHU) Varanasi, Varanasi, India

ABSTRACT

Treatment of hyperthermia is a procedure in which a spatial heat source is subjected to tumor posi-tion to destroy the tumor cells without affecting the neighboring healthy tissues. Accurate predictionof temperature and control of temperature withing the living biological tissue are very important fac-tor during the hyperthermia treatment. The main motive of this paper is to solve highly non-linearDPL (dual-phase-lag) bio-heat transfer model by using a method which uses less computational ef-fort to achieve accurate result. Also, the paper focuses on showing the variation of results causedby changing the values of various parameters, which will help in clinical field to improve the hyper-thermia treatment. To achieve the results, a mathematical model of heat transfer together with thevarious parameters under different boundary conditions inside the living biological tissues has beenstudied. Finite difference method is applied to convert the boundary value problem supervising theprocess of bio-heat transfer into a system of ordinary differential equations of initial value problem.Next, Runge-Kutta (4,5) method is used to find out the temperature profile within the living bio-logical tissues. The whole analysis is presented in non-dimensional form. Distinct effects of variousparameters mentioned in the problem like metabolic heat generation, blood perfusion rate, thermalrelaxation times, Gaussian distribution external heat source term are also discussed. Results arealso obtained for different types of general boundary conditions. Study of these effects during thetreatment of hyperthermia makes this study useful for the clinical applications especially for theoncologists.

KEY WORDS: Dual-phase-lag (DPL) model, RungeKutta (4, 5) method, hyperthermia, heattransfer, Gaussian distribution heating source.

Reference[1] Gupta P.K., Singh J., and Rai K.N.,2010. Numerical simulation for heat transfer in tissues duringthermal therapy, J.Therm.Biol ,35(6),295-301.

[2] Jiang S.C., Ma N., Li H.J., and Zhang X.X.,2002. Effects of thermal properties and geometricaldimensions on skin burn injuries, Burns,28,713-717.



A Lobatto Mixed Quadrature of Precision Eleven for NumericalIntegration of Analytic Functions

Name of presenting author: Sanjit Kumar Mohanty,Department of Mathematics B.S Degree College, Nuahat, Jajpur-754296, Odisha, India ,


Co-author: Rajani Ballav Dash, Department of Mathematics Ravenshaw University, Cuttack,Odisha, India

ABSTRACT

A mixed quadrature rule of precision eleven for approximate evaluation of line integral of an analyticfunction has been constructed by using Lobatto six point transformed quadrature rule and Kronrodextension of Lobatto four point quadrature rule. The mixed quadrature rule so formed, has beentested both theoretically through error analysis and numerically using some test integrals. It is foundthat the constructed rule is more effective than that of the constituent rules. It can also be applicablein adaptive environment.

MSC: 60D30, 60D32



Numerical Solution of a Time Fractional Mixed Reaction-convectionDiffusion Problem Involving Weak Singularity

Sudarshan Santra, National Institute of Technology Rourkela, Odisha, India, email:[email protected]

Jugal Mohapatra, National Institute of Technology Rourkela, Odisha, India.

ABSTRACT

A time fractional mixed reaction-convection diffusion problem is considered and the fractional deriva-tive is defined in Caputo sense. The solution of such model exhibits a weak singularity at the initialtime t = 0 and an initial layer occurs at t = 0. The classical L1 scheme is introduced to approximatethe temporal derivative and a second order central difference scheme is used to approximate the spa-tial derivatives. The error analysis is carried out in the entire domain. The numerical experimentsare in support of the desired theoretical findings.


Weak Galerkin Finite Element Methods for Parabolic Problems with L2

Initial Data

Naresh Kumar, Indian Institute of Technology Guwahati, Guwahati, India.Email: [email protected]

Bhupen Deka, Indian Institute of Technology Guwahati, Guwahati, India.

ABSTRACT

In this paper, we consider the weak Galerkin finite element approximations of second order linearparabolic problems in two dimensional convex polygonal domains with non-smooth initial data. Forhomogeneous equation, we have established error estimates in L2 and H1-norms of order O(h2/t)and O(h/t), respectively for t > 0, when the given initial data is in L2. The error analysis has beencarried out on polygonal meshes for discontinuous piecewise polynomials on finite element partitions.In the analysis, we have used only elementary energy techniques. Numerical experiments are reportedfor several test cases to justify our theoretical convergence results.

Radiative MHD Casson Nanofluid Flow and Heat and Mass Transferpast on Nonlinear Stretching Surface considering Viscous Dissipation,

Chemical Reaction and Heat Source

G. Narender, Research Scholar, Jawaharlal Nehru Technological University Hyderabad,Kukatpally, Hyderabad, Telangana State, India.

CVR College of Engineering, Hyderabad, Telangana State, India, email:[email protected]. Govardhan, GITAM University, Hyderabad, Telangana State, India.

ABSTRACT

The magnetohydrodynamics (MHD) stagnation point Casson nanofluid flow towards stretching sur-face with velocity slip and convective boundary condition has been investigated in this article. Effectsof thermal radiation, viscous dissipation, heat source and chemical reaction have also been incorpo-rated. Using appropriate similarity transformation Partial Differential Equations (PDEs) are con-verted into Ordinary Differential Equations (ODEs) and shooting technique along with AdamsMoul-ton Method of order four has been used to obtain the numerical results. Different physical parameterseffects on velocity, temperature and concentration of nanofluid flow have been presented graphicallyand discussed in detail. Numerical values of the skin friction coefficient, Nusselt number and Sher-wood number are also and discussed.

Haar Wavelets Collocation Method for a Class of System of Lane-EmdenEquations with Four Point Boundary Conditions

Narendra Kumar, Indian Institute of Technology Patna, India, email: [email protected]

Amit K. Verma, Indian Institute of Technology Patna, India.

ABSTRACT

The present work consists of the system of Lane-Emden equations that arises in several branchesof science and engineering such as dusty fluid models, stellar structure, thermal explosions, andisothermal gas sphere. In this work, we consider the following class of system of Lane-Emdenequations

−(tk1y′(t))′ = t−ω1f1(t, y(t), z(t)), t ∈ (0, 1),

−(tk2z′(t))′ = t−ω2f2(t, y(t), z(t)), t ∈ (0, 1),

subject to the following four-point boundary conditions:

y(0) = 0, y(1) = n1z(v1), z(0) = 0, z(1) = n2y(v2),

where n1, n2, v1, v2 ∈ (0, 1) and k1 ≥ 0, k2 ≥ 0, ω1 < 1, ω2 < 1 are real constants. We proposean efficient numerical technique based on the Haar wavelets collocation method together with theNewton-Raphson approach to solve the above differential equation. In this technique, we use theHaar wavelets collocation method and get the system of nonlinear equations. Then, we solve thesystem of nonlinear equations using the Newton Raphson method to get the solution of the systemof Lane-Emden equations. We discuss some test problems based on it. We compare our results withthe other existing methods such as the Taylor series method, successive iteration technique and checkthe accuracy and efficiency of the proposed method.

Water Wave Interaction with a Cylindrical Storage Tank in Finite OceanDepth

Abhijit Sarkar, Indian Institute of Technology Guwahati, Guwahati, India, email:[email protected]

Swaroop Nandan Bora, Indian Institute of Technology Guwahati, Guwahati, India.

ABSTRACT

Here we discuss diffraction of linear waves by a cylindrical storage tank, the inner part of whichcontains a cylindrical pile and the outer part contains a coaxial thin hollow porous cylinder, infinite depth. The main aim is to construct a storage tank surrounding the cylindrical pile of theaerogenerators for aquaculture. There are some current efforts the aim of which includes a plan tointegrate aquaculture around wind farm. Here we consider an impermeable solid circular cylindersurrounded by an outer coaxial thin hollow porous cylinder of same height extending from free surfaceto sea-bed. The sea-bed is assumed to be flat and impermeable. The fluid region is divided intotwo sub-domains, and the familiar separation of variables technique is used to obtain the analyticalexpressions for the corresponding potential in each region. Appropriate matching conditions areapplied at the interface of the fluid sub-domains to get a system of linear equations for the unknowncoefficients and solve it. By considering different sets of values of the radius and the porosity of thecylinder, the hydrodynamic force and wave run-up are evaluated. It clearly indicates that any changein values of these parameters can fetch significant alteration in the values of the hydrodynamic loadsand wave run-up. A certain combination of the parameters of the cylinder may help to bring downthe values of the wave forces to a large extent.

References1. Sollitt, C.K., Cross, R.H., 1972. Wave transmission through permeable breakwaters, In: Pro-

ceedings 13th Conf. Coastal Engineering, Vancouver, Canada. Spring,

2. Chwang, A.T., 1983. A porous wavemaker theory. J. Fluid Mech. 132, 395-406.

Water Wave Scattering by a Pair of Submerged Vertical Porous Barriersover Porous Sea-bed

Ayan Chanda, Indian Institute of Technology Guwahati, Guwahati, India,email:[email protected]

Swaroop Nandan Bora, Indian Institute of Technology Guwahati, Guwahati, India.

ABSTRACT

A hydrodynamic model, with incorporation of porosity, is considered to investigate oblique waterwaves scattering by two fully submerged parallel porous barriers with the wave propagating overa porous bed in a homogenous fluid flow with upper surface exposed to atmosphere. The porousbarriers are assumed to follow the theory of thin plates and the wave propagation through the porousstructure follows porous wave-maker theory ([1]). The behaviour and properties of the roots of thedispersion relation are analyzed by adopting counting argument and contour plot. Time-harmonicpropagating waves propagate with exactly one wave number along the free surface for any givenfrequency. Methods of eigenfunction expansion and least square are employed to acquire the completeanalytical solution for interaction of water waves with submerged porous barriers. Subsequentlythe reflection and transmission coefficients as well as the energy loss are computed. Then thoseare examined corresponding to various values of parameters such as porous-effect parameter, thesubmergence depth of barriers from free surface, angle of incidence, porosity of the sea-bed. Presentinvestigation clearly demonstrates that the wave reflection is of oscillatory nature. It further showsthat the occurrence of minima in wave reflection is due to an increase in the inertial effect of theporous barriers which dissipate a significant portion of the wave energy. The effect of the porous bedunder consideration on surface gravity waves is carried out by introducing various numerical valuesto the hydrodynamic wave characteristics and it is noticed that a reasonable change in porosity ofthe bed has a significant impact when the propagating wave encounters the submerged structure(barrier). The present approach is expected to be of great significance in designing and constructionof different types of effective wave absorbers utilized in sea for studying reflection as well as dissipationof wave energy in coastal regions and hence for the purpose of coastal as well as offshore engineering.

References1. Chwang AT. A porous wavemaker theory, Journal of Fluid Mechanics. 132 (1983) 395–406.

Mathematical Studies on Ureter Smooth Muscle: Modeling Ion Channelsand Their Role in Generating Electrical Activity

Chitaranjan Mahapatra, IIT Bombay, Mumbai, India, email: [email protected]

Rohit Manchanda, IIT Bombay, Mumbai, India.

ABSTRACT

Abnormal peristaltic contraction of the ureter smooth muscle (USM) causes the pathophysiologicalcondition to the urinary system. It is demonstrated that the USM action potential (AP) is a dy-namic parameter to investigate the abnormal USM contractions. In the interest of figuring out theinternal membrane ionic currents responsible for USM AP origination, this study aims at developinga mathematical model of the USM cell AP. In line with the recent experimental evidence, adapt-ing the Hodgkin- Huxley formulation in the NEURON platform, we construct mathematical modelsfor five ionic currents of USM cell. The magnitudes and kinetics of each ionic current system in acylinder-shaped single cell with a specified surface area are described by first-order ordinary differen-tial equations, in terms of the maximal conductances, electrochemical gradients, voltage-dependentactivation/inactivation gating variables and temporal changes in intracellular Ca2+ computed fromknown Ca2+ fluxes. Then, all ion channels were integrated to generate the AP after introducinga current stimulus to a single cell model. This virtual electrophysiological workbench reproducedUSM AP successfully that replicates the experimental AP. This model also allows analyzing the ionchannel implications at different phases of the AP. In the future, this primary model can be furtherextended to explore new intracellular insights for abnormal USM contraction.


On Solutions of The Fractional Integral Equation

Bhuban Chandra Deuri and Anupam Das∗

Department of Mathematics, Rajiv Gandhi University, Rono Hills,Doimukh-791112, Arunachal Pradesh, India

Email: [email protected] and [email protected]

ABSTRACT

The goal of this article is to prove some fixed theorems in Banach space E via the measure of non-compactness. Also, we study the existence of solutions to an implicit functional equation involvinga fractional integral with respect to a certain function, which generalizes the Riemann-Liouvillefractional integral and the Hadamard fractional integral. Finally, we give an illustrate to show thatour abstract results are easy to verify.

Key Words: Measure of noncompactness ; Fixed point theorem; Riemann-Liouville fractionalintegral ; Hadamard fractional integral.



Monotone Iterative Technique for Nonlinear Four Point Neumann BVPswith Non Well Ordered Upper and Lower Solutions

Nazia Urus, Indian Institute of Technology Patna, Bihar, India.email: [email protected]

Nazia Urus 1, Indian Institute of Technology Patna 801103, Bihar, India.Amit K. Verma 2, Indian Institute of Technology Patna 801103, Bihar, India.

ABSTRACT

In this article, we develop the Monotone Iterative (MI) technique with non well ordered upper andlower solutions for a class of nonlinear Neumann 4-point BVPs, defined as,

−z′′(y) = x(y, z, z′), 0 < y < 1,

where the nonlinear term x(y, z, z′) : Ω → R, where Ω = [0, 1] × R2, is dependent on derivative ofsolution z. To study the existence of solution we construct iterative sequences for correspondinglinear problem with the help of lower and upper solutions l(y) and u(y) respectively. We prove theanti maximum principle and establish monotonicity of sequences of upper and lower solutions. Thenunder certain assumptions we prove that these sequences converges uniformly to the solution z(y) inthe specific region, dx

dz> 0. Here x(y, z, z′) is Lipschitz in z′(y) and one-sided Lipschitz in z(y). The

motivation for this work originates from many recent investigations on this technique for multipointboundary value problems in a well-ordered case. The novelty of this article is, we have discussednon-well-ordered cases, and to demonstrate that the proposed technique is effective, we compute thesolution of the nonlinear multi-point BVPs which may not be computed easily.

Nazia Urus, Department of Mathematics, Indian Institute of Technology Patna, Bihta, Patna801103, (BR) India, Email: [email protected].

Amit K. Verma, Department of Mathematics, Indian Institute of Technology Patna, Bihta,Patna 801103, (BR) India, Email: [email protected].



On the Stability of Mickens’ Type Nonstandard Finite DifferenceSchemes for the Advection Diffusion Reaction Equation

Sheerin Kayenat, Indian Institute of Technology Patna, Bihta, India,email:[email protected]

Amit K. Verma, Indian Institute of Technology Patna, Bihta, India.

ABSTRACT

We consider a class of advection diffusion reaction (ADR) equation subject to certain initial andboundary conditions. With the help of solitary wave solution of ADR equation, we develop theexact finite difference (EFD) scheme for it. Furthermore a non-standard finite difference (NSFD)scheme is proposed. The properties like positivity and boundedness is proved to be preserved by theproposed NSFD scheme. The scheme is shown to be stable, consistent and first-order accurate inboth space and time. Approximate solutions of the ADR equation under given initial and boundaryconditions are obtained using NSFD scheme and the maximum error of the computed solutions arecalculated. The efficiency of the proposed NSFD scheme is shown through various tables and figures.The scheme gives good accuracy even for few spatial division. We also compute CPU time for allthe computations which reveal that our scheme gives an accurate result within few seconds whichsaves our time.

Sheerin Kayenat, Department of Mathematics, Indian Institute of Technology Patna, Bihta,Patna (BR) 801103, India, Email: [email protected].

Amit K. Verma, Department of Mathematics, Indian Institute of Technology Patna, Bihta, Patna(BR) 801103, India, Email: [email protected].



Analytical Approximate Solution of the Mathematical Models of Effectof Chemotherapy upon Glioblastoma Tumor Cells Growth in

Homogeneous Medium using KVIM

Shruti S. Sheth, Sarvajanik College of Engineering and Technology, Surat, India,email:[email protected]

Co-author 1: Dr. Twinkle R. Singh, Sardar Vallabhbhai National Institute of Technology, Surat,India.

ABSTRACT

Cancerous tumors from the mutations of one or more cells which usually undergo rapid uncontrolledgrowth thereby are impairing the functioning of normal tissues. There are many different types ofcancers, we concerned with the brain tumors particularly Glioblastoma, which make up about half ofall primary brain tumors diagnosed. It is very nasty tumors with a depressingly dismal prognosis forrecovery. We believe that model could be used to make predictions regarding the survival time of thepatient following the various types of treatments including resection, radiation and chemotherapy.For the solution of the models we have established the Kamal Variational Iteration Method (KVIM).Variational Iteration Method (VIM) has been established by J.Huan He in 1999, considered to solvethe non linear problems. It gives rapidly convergent successive approximations without any restric-tive assumptions or transformations which may change the physical behaviour of the problem. KamalTransform is a new arrival of an integral transform derived from Fourier Transform to solve the linearinitial value problems. Kamal Variational Iteration Method is established to solve the models.Our aim in this paper is to thrash out the current developments of the methods to crack the math-ematical modelling of effect of chemotherapy treatment upon Glioblastoma tumor cells growth andit might be unlock the doors for improvement.Objective of this research is to solve the basic mathematical models of the chemotherapy treatmentfor Glioblastoma tumor cells growth in homogeneous medium when spread of early cancer cells areknown and unknown, using new developed Kamal Variational Iteration Method.



A Review of Jaulent-Miodek Hierarchy

Amlan K Halder, Department of Mathematics, Pondicherry University,Puducherry - 605014, India


A Paliathanasis, Institute for Systems Science, Durban University of Technology, PO Box 1334,Durban 4000, Republic of South Africa.

PGL Leach, School of Mathematics, Statistics and Computer Science, University ofKwaZulu-Natal, Durban, South Africa and Institute for Systems Science, Durban University of

Technology, Durban, South Africa

ABSTRACT

We focus on the (1 + 1)- and hierarchy of the (1 + 2)- dimensional equations for the Jaulent-Miodeksystem through its point symmetries. We firstly discuss the (1 + 1)- dimensional coupled Jaulent-Miodek equation and its reductions through the solution symmetries. The reduced coupled odepossesses single symmetries which further reduces to third- and fourth-order ode with zero pointsymmetries. Hence we shift to singularity analysis to evaluate its integrability. The first four mem-bers of the (1+2)-dimensional hierarchy presents a different perspective. The reductions of the first,second and fourth members are quite similar for most of the symmetries whereas the third memberdiffers significantly. Certain closed-form solutions and well-known reductions are mentioned for eachof the members and, in particular, we focus on those symmetries which lead to varied differentialequations. Finally, we once again employ the singularity analysis to look for a series solution of eachof the mentioned PDEs.

Keywords: Symmetries, Singularities, Reduction of order, Jaulent-Miodek Hierarchy

References[1] Ablowitz M J, Ramani A & Segur H (1980), A connection between nonlinear evolution equations

and ordinary differential equations of P type I, Journal of Mathematical Physics, 21 715-721.

[2] Ablowitz M J, Ramani A & Segur H (1980), A connection between nonlinear evolution equationsand ordinary differential equations of P type II, Journal of Mathematical Physics, 21 1006-1015.

[3] Andriopoulos K & Leach P G L (2006), An interpretation of the presence of both positive andnegative nongeneric resonances in the singularity analysis, Physics Letters A, 359, 199-203.

[4] Wazwaz A M (2009), Multiple kink solutions and multiple singular kink solutions for (2+ 1)-dimensional nonlinear models generated by the JaulentMiodek hierarchy, Physics Letters A,373(21), 1844-1846.



A Finite Element Method for the Equation of Motion Arising in OldroydModel of Order One with Grad-Div Stabilization

Bikram Bir, Tezpur University, Assam, India, email: [email protected]

Deepjyoti Goswami, Tezpur University, Assam, India.

ABSTRACT

An error analysis of inf-sup stable finite element method for the equations of motion arising in the 2DOldroyd model of order one with a grad-div type stabilization is discussed. The numerical methodadds a stabilization term to the momentum equation and it makes the constants obtained in errorbounds do not dependent on negative powers of viscosity. Optimal error estimate for velocity inL∞(L2)-norm as well as in L2(L2)-norm and for the pressure in L2(L2)-norm in the semidiscretecase are established. Then, based on backward Euler method, a completely discrete scheme isanalyzed. All the analysis are carried out for both the case in which the solution is assume to besmooth and consequently has to satisfy nonlocal compatibility conditions as well as non-smoothinitial data in which the nonlocal compatibility conditions are not satisfied. Taking into account theloss of regularity suffered by the solution of the Oldroyd model of order one at the initial time in theabsence of nonlocal compatibility conditions of the data. Finally we present some numerical resultsto validate our theoretical results.



Convergence Analysis of Dirichlet-Neumann and Neumann-NeumannAlgorithms for Cahn-Hilliard Equation

Gobinda Garai, IIT Bhubaneswar, Bhubaneswar, India, [email protected]. Bankim C. Mandal, IIT Bhubaneswar, Bhubaneswar, India.

ABSTRACT

Cahn-Hilliard equation was originally proposed to describe the phase separation phenomenon forbinary alloy below the critical temperature and since then it has appeared in many other scientificfields, such as image processing and tumour growth. Because of its nonlinearity, it is of greatsignificance to develop efficient numerical methods to approximate the solution. Here, we solve acoupled second order elliptic system of constant coefficients which is resulted from many differentunconditionally gradient stable scheme using domain decomposition method at each time level. Weproved the convergence analysis of Dirichlet-Neumann and Neumann-Neumann method in the caseof two-subdomain. In the end, we show numerical results to illustrate our theoretical findings.



Non-Self-Adjoint Eigenvalue Problem for Wave Propagation in OpticalBent Waveguides

Rakesh Kumar, Indian Institute of technology, Jodhpur, India, email:[email protected]. Kirankumar R. Hiremath, Indian Institute of technology, Jodhpur, India,


ABSTRACT

Regular Strum-Liouville theory deals with the eigenvalue problems of differential operators withcontinuous coefficients and defined on the finite interval. One can find huge literature related to thistheory. This theory helps us to understand many applications in sciences such as simple harmonicmotion of pendulums, vibration of drums, heat conduction, etc. But there are situations where theconditions of the regular Strum-Liouville theory are not applicable. e.g. Schrodinger’s equation ofquantum mechanics, where the problem is defined on an infinite interval; optical wave propagationin waveguides, where the governing equation has discontinuous coefficients.

Theoretical analysis of electromagnetic wave propagation is a rich area of applied mathemat-ics. These days many researchers are investigating optical wave propagation in waveguides. Thesewaveguides are designed based on the geometric parameters, refractive indices, and wavelength. Thewave behavior in these waveguides is studied from numerical and theoretical perspectives based onthe simulation and semi-analytic analysis.

The behavior of the wave propagation in the waveguides is modeled as an eigenvalue problemon the infinite domain. In the case of the straight waveguides, it has been shown that the eigenvalueproblem is self-adjoint and has real eigenvalues. The behavior of solutions for the guided modes andother properties has been also addressed. But, in the case of the bent waveguides, there is a lack ofanalytical studies based on the function theoretic setting. The present work addresses this gap.

The challenges in the study of the bent waveguide eigenvalue problem are: The eigenvalueproblem is defined on the infinite domain [0,∞), and the differential operator has discontinuouscoefficients. Many questions regarding this eigenvalue problem came into mind. Is this eigenvalueproblem self-adjoint or non-self-adjoint? What is the nature of the eigenvalues corresponding tothis eigenvalue problem? What is the behavior of the eigenvalue problem corresponding to a bentwaveguide with the large bend radius, etc.?

In this contribution, we will address these issues related with the eigenvalue problem for theoptical bent waveguide with a piecewise-constant refractive index profile. We will show that thebent waveguide eigenvalue problem is non-self-adjoint and the eigenvalues are complex. We will alsoshow that as the bend radius tends to infinity, this non-self-adjoint problem becomes a self-adjointproblem. We investigate this asymptotic behavior, and show mathematically that for large bendradii, the bent waveguide behaves like a straight waveguide.



On Summation-Integral Type Operators with Their Quantitative Means

Rishikesh Yadav, Sardar Vallabhbhai National Institute of Technology Surat (Gujarat-395 007),India, email: [email protected]

Ramakanta Meher, Sardar Vallabhbhai National Institute of Technology Surat (Gujarat-395 007),India.

Vishnu Narayan Mishra, Indira Gandhi National Tribal University, Lalpur, Amarkantak 484 887,Anuppur, Madhya Pradesh, India.

ABSTRACT

In this article, the approximation properties of the summation-integral type operators defined byMishra et al. (Boll. Unione Mat. Ital. (2016) 8:297-305) are studied. The rate of convergence aswell as the convergence theorem are obtained for the defined operators. We develop the asymptoticformula to check the asymptotic behavior of the said operators; moreover, its quantitative means isestimated, as well as a Gruss-Voronovskaya-type theorem is determined. At last, graphical analysisis given in the support of approximation by the proposed operators.



On Weakly L-Stable Time Integration Formula with an Application toNon-Linear Parabolic Partial Differential Equations

Mukesh Kumar Rawani, Indian Institute of Technology Patna, Patna, India, email:[email protected]

Amit Kumar Verma, Indian Institute of Technology Patna, Patna, India,.

ABSTRACT

In this work, we derive a high order weakly L-stable time integration formula. To derive the formula,we use fifth-order Hermite approximation polynomial and sixth-order backward explicit Taylor’sseries approximation to the initial value problem (IVP) u′(t) = f(t, u), u(t0) = η0. We apply thederived scheme to Burgers’ equation and Fisher equation. To show the efficiency of the method,we compute solutions of both Burgers’ equation and Fisher equation. Since for small values ofviscous coefficient, analytical solution of Burgers equations is not known. It is always interesting toconsider cases when the viscous coefficient is small. We have addressed this issue in this paper. Theconvergence and stability of the scheme is also discussed.



Mathematical Modeling and Numerical Simulation of a Time-FractionalPorous Medium Equation Arising in Fluid Flow through Porous Media

Juhi Kesarwani, Sardar Vallabhbhai National Institute of Technology, Surat, India, email:[email protected]

Ramakanta Meher, Sardar Vallabhbhai National Institute of Technology, Surat, India.

ABSTRACT

The prime objective of this paper is to obtain a time-fractional porous medium equation in theimbibition phenomenon through a water-wet porous medium. The Homotopy analysis method isused to solve the fractional-order porous medium equation and studied the behavior of time-fractionalon the saturation rate of the porous medium with different parametric effects such as viscosity ratio,inclined planes, and wettability of the medium. Finally, it is concluded that there is a significanteffect on the saturation and recovery rate of the hydrocarbon reservoir for a fixed value of fractional-orders.Keywords: Porous medium, Imbibition phenomenon, Time-fractional PDE, Inclined plane at anangle α.



Application of Chebyshev Polynomial for Numerical Approximation ofSome Real-Life Singular Differential Equation

Julee Shahni, Birla Institute of Technology, Mesra, Ranchi, IndiaEmail: [email protected]

Randhir Singh, Birla Institute of Technology, Mesra, Ranchi, India

ABSTRACT

In this work, we propose an efficient collocation technique based on Chebyshev polynomials fornumerical approximation of the nonlinear singular boundary value problems. Such equations areused to model various phenomena in mathematical physics and astrophysics such as the equilibriumof a uniformly charged gas [1], oxygen tension in spherical cell [2], heat sources in human head[3], the equilibrium of isothermal gas sphere [4], and radial stress on a shallow membrane cap [5].Firstly, we convert the singular boundary value problem into the equivalent integral equation. Thenthe collocation technique based on Chebyshev polynomial is applied to obtain a system of nonlinearequations which is then solved by the Newton-Raphson method. The exactness of the present methodis tested by comparing the obtained results with the exact solution and the results obtained by theother known techniques.

Keywords: Singular Boundary Value Problems; Integral Equation; Chebyshev Polynomials;Function Approximation.

References[1] J. B. Keller, Electrohydrodynamics: The equilibrium of a charged gas in a container, Journal of

Rational Mechanics and Analysis (1956) 715–724.

[2] S. Lin, Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics, Journal ofTheoretical Biology 60 (2) (1976) 449–457.

[3] R. Duggan, A. Goodman, Pointwise bounds for a nonlinear heat conduction model of the humanhead, Bulletin of Mathematical Biology 48 (2) (1986) 229–236.

[4] M. Chawla, R. Subramanian, H. Sathi, A fourth order method for a singular two-point boundaryvalue problem, BIT Numerical Mathematics 28 (1) (1988) 88–97.

[5] A. R. Kanth, K. Aruna, He’s variational iteration method for treating nonlinear singular bound-ary value problems, Computers & Mathematics with Applications 60 (3) (2010) 821–829.



Nonlinear Impulsive Dynamic Initial Value Problems with NonlocalConditions

Sanket Tikare, Department of Mathematics, Ramniranjan Jhunjhunwala College, Ghatkopar (W),Mumbai (M.S.) - 400 086, India,


Christopher C. Tisdell, School of Mathematics and Statistics, The University of New South Wales,Sydney NSW 2052, Australia.

ABSTRACT

The purpose of this paper is to introduce more general results on the existence of solutions fornonlinear impulsive dynamic equations on time scales with nonlocal initial conditions. We establishthe existence of solutions by applying a fixed point result due to O’Regan, while the uniqueness ofsolutions is obtained through the contraction mapping principle. Our results extend the previouswork in the literature. Examples are provided to illustrate the obtained results.



Dynamics of an Eco-Epidemiological Model with Nonlinear IncidenceRate and Fear Effect

Ankur Jyoti Kashyap, Gauhati University, Guwahati, Assam, India.email: [email protected]

ABSTRACT

In this paper, a fractional-order eco-epidemiological model with nonlinear incidence rate and feareffect is studied. The populations are divided into susceptible prey, infected prey, and predator. Thefundamental mathematical results like the existence, uniqueness, non-negativity, and boundednessof the solutions of the system are discussed. Local and global asymptotic stability of all the possiblesteady states are discussed analytically. Finally, numerical simulation are conducted with the helpof some biologically feasible parameter values to validate our analytical findings. Dynamics of thesystem with respect to the induced rate of fear is discussed numerically. It is observed that themortality of the infected prey population has a stabilization effect on the fractional-order systemwhich helps in the coexistence of all the population. Besides, it is observed that the fractional ordercan help to control the coexistence of all the population.



Solution of a Differential Equation Using Fixed Point Results inGraphical Rectangular Metric Space

Pravin Baradol, Sardar Vallabhbhai National Institute of Technology, Surat, India,email:[email protected]

Dr. Dhananjay Gopal, Sardar Vallabhbhai National Institute of Technology, Surat, India.

ABSTRACT

In the present paper, we ensures the existence of a solution for an ordinary differential equation alongwith its boundary conditions by using the fixed point result in graphical rectangular metric space.For that, first we introduce a notion of graphical rectangular metric space, which is a graphicalversion of rectangular metric space. Utilizing the graphical Banach contraction mapping we provefixed point results in the aforesaid space. Appropriate examples are also presented which uphold ourresults.



Trajectory Controllability of Fractional Dynamical Systems - A Survey

Dibyajyoti Hazarika, North Lakhimpur College (Autonomous), North Lakhimpur, Assam, India,email: [email protected]

ABSTRACT

The main objective of this paper is to review the recent developments in the field of controllabilityand trajectory controllability of the dynamical systems governed by fractional differential equations.These dynamical systems may be finite or infinite dimensional as well as having linear/ nonlinearcomponents. The focus of this paper will lie around the trajectory controllability of these systems,which is a stronger notion than controllability. Roughly speaking, trajectory controllability meansthe ability to steer a dynamical system from an arbitrary initial state to a desired final state alonga pescribed trajectory. The derivatives involved in the differential equations of these dynamicalsystems are Riemann-Lioville and Caputo fractional derivatives.In this survey paper, I investigate the progresses made by various researchers in the recent years alsopresent few important results in this field.



Natural Convection in MHD Flow Past a Vertical Plate in Presence ofThermal Radiation

Kangkan Choudhury, Gauhati University, Guwahati-14, India, email:[email protected] Ahmed, Gauhati University, Guwahati-14, India.

ABSTRACT

The present paper deals with the analysis of unsteady MHD free convective flow past a movingvertical plate with variable suction in the presence of radiation in a slip flow regime. Slip flowconditions for the velocity, jump in temperature as well as concentration are taken into account inthe boundary conditions. The dimensionless governing equations are solved analytically by usingperturbation technique. The effects of various parameters on the velocity, temperature, concentrationfields as well as the transport properties are presented graphically. It is found that velocity increasesdue to the increase in magnetic parameter.



The Solution of Volterra Integro-Differential Equation of Second KindUsing Shehu Transform

Dr. Mamta Kumari,DCT’s Dhempe College of Arts & Science, Panaji, Goa, India, email:[email protected]

ABSTRACT

In this paper, Shehu transform is used for solving Volterra integro-differential equations of secondkind. The technique is described and illustrated with some applications.Keywords. Volterra integro-differential equation; Shehu transform; Convolution theorem; InverseShehu transform.



Numerical Technique for Singularly Perturbed Delay DifferentialEquations Using Gaussion Quadrature

M. Lalu, University College of Engineering, Osmania University, Hyderabad, India,email: [email protected]

K. Phaneendra, University College of Engineering, Osmania University, Hyderabad, India.

ABSTRACT

A quadrature method is suggested for the solution of singularly perturbed delay differential equa-tion. Initially, a first-order delay differential equation is achieved, which is asymptotically equivalentto the given singularly perturbed delay differential equation. Then Gaussian quadrature two-pointformula is implemented on the first order equation to get a tridiagonal system. Thomas algorithmis used to solve this system. The proposed method is implemented on model examples, for differentvalue of delay parameter and perturbation parameter. Maximum absolute errors are tabulated witha comparison to authorize the method. Theoretical convergence of the method is discussed. Thelayer behaviour is discussed using the graphical representation.

Keywards: singularly perturbed delay differential equations, Gaussian quadrature two-pointformula, layer behaviour, linear interpolation



Mixed Convection in Four-sided LID-Driven Sinusoidall Heated PorousCavity Using Stream Function-Vorticity Formulation

Manoj Kumar, Department of Mathematics, Faculty of Mathematical Sciences, University ofDelhi, Delhi-110007, India. email: [email protected]

Shobha Bagai, Cluster Innovation Centre, 3rd Floor University Stadium, G C Narang Road,University of Delhi, Delhi-110007, India.

Arvind Patel, Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi,Delhi-110007, India.

ABSTRACT

This study presents the mixed convection inside a four-sided lid-driven square porous cavity whoseright wall is maintained at a sinusoidal temperature condition, the left wall of the cavity is maintainedat a cold temperature, while the top and the bottom walls are adiabatic. We have discussed two dif-ferent cases depending upon the direction of the moving walls. Brinkmann-extended Darcy modelis represented in terms of ψ and ξ using the stream function-vorticity formulation to simulate themomentum transfer in the porous medium. This formulation is used to solve the governing equationsas a coupled system of equations which consists of the field variables, vorticity (ξ), stream function(ψ), and temperature (T ). The velocity components (u, v) are derived from the stream function (ψ)whereas the average Nusselt number is derived from temperature. The stability and consistency ofthe applied numerical scheme to the considered problem has been proven by matrix method. Thenumerical results are investigated by ranging the various dimensionless numbers such as Grashofnumber (103 ≤ Gr ≤ 105), Darcy number (10−1 ≤ Da ≤ 10−3), Reynolds number (10 ≤ Re ≤ 150)and keeping the Prandtl number (Pr = 0.7) fixed.

Highlights:

• A numerical study is presented on the combined convection in a four-sided lid-driven squareporous cavity with a sinusoidal variation of the temperature on the right wall.

• Two different cases depending on the direction of the motion of the walls with the sinusoidaltemperature distribution on the right vertical wall have been considered.

• The average Nusselt number along the left vertical wall of the cavity increases with an increaseof Grashof number (103 to 105), Darcy number (10−3 to 10−1), and Reynolds number (10 to150) in both cases.



Surfactant-laden Newtonian Falling Film Down a Wavy Channel: Linearand Nonlinear Stability Analysis

Md. Mouzakkir Hossain, Siluvai Antony Selvan and Dr. Harekrushna BeheraDepartment of Mathematics, SRM Institute of Science and Technoloy, Kattankulathur, India.

email:mouzakkir123gmail.com

ABSTRACT

An instability of surfactant-laden falling film down the wavy plane is analyzed using the methodof small aspect ratio and multiple scales. The falling film and insoluble surfactant are governedby the Navier–Stokes equation and surfactant-transport equation, respectively. Using the techniqueof small-aspect ratio, the non-linear evolution equation corresponding to the physical problem isobtained. Further, the method of multiple scales are employed directly in the evolution equationfor determining the nonlinear instability in the neighborhood of criticality. Moreover, the criticalconditions of the primary instability are determined, which depend on the surfactant properties. Thenumerical results carried out for different physical variables suggest that the behaviour of surfaceand surfactant modes on the primary instability are solely depend on the properties of an insolublesurfactant and wavy bottom. The amplitude of non-linear disturbance varies and shows anomalousbehaviour in the sub-critical and supercritical regime depending on the surfactant and amplitude ofwavy bottom. This work may find application in the coating and biomedical industries, where thefilm thickness is sufficiently small and the method of small aspect ratio holds good for estimatingthe primary instability.



Long Wave Stability Analysis of a Film Flow in the Presence of anInsoluble Surfactant

Muhammad Sani, Department of Mathematics College of Engineering and Technology, SRMInstitute of Science and Technology,

Kattankulathur-603 203, Tamil Nadu, [email protected]

Dr. Harekrushna Behera, Department of Mathematics College of Engineering and Technology,SRM Institute of Science and Technology,

Kattankulathur-603 203, Tamil Nadu, India.

ABSTRACT

Long wave stability analysis can serve as a powerful tool for determining stability/instability of fluidflowing horizontally with surfactant at the free surface which has an important effects on appliedMathematics, Physics, Ocean Engineering, etc.. This study investigates the long wave stability of ahorizontal film flow in the presence of an insoluble surfactant at the free surface. The motion of thefluid is described using the Navier-Stokes equation and the normal mode approach is employed forderiving the corresponding Orr-Sommerfeld system of equations. The effects of insoluble surfactantand other physical parameters are analyzed through the growth rate results. The contours of thebandwidth for varying Reynolds numbers against the Marongoni and Capillary numbers are elabo-rated in this work. It is observed that the insoluble surfactant plays a vital role to suppress the freesurface instability.



New Approximate Solutions of Time Fractional Klein Fock GordonEquation via HPJTM

Anil Kumar, Department of Mathematics, School of Physical and Decision Sciences, BabasahebBhimrao Ambedkar University, Lucknow-226025 (UP)India, [email protected]:

Brajesh Kumar SinghDepartment of Mathematics, School of Physical and Decision Sciences, Babasaheb Bhimrao

Ambedkar University Lucknow-226025 (UP) India.

ABSTRACT

The present article is concentrated on a new hybrid technique so called: homotopy purtubation J -transform method(HPJM), for analytical simulation of time fractional Klein-Fock Gordon equations(TF-KFGE), considering fractional derivative of Caputo type. TF-KFGE exhibit the behaviour ofspinless particle like Higgs Boson. Effectiveness and reliability of HPJM is shown by consideringthe three different test examples of TF-KFGE and in addition the numerical findings are depictedgraphically also. The findings demonstrate that the computed solutions are efficient, agreed wellwith the existing result and converges to the exact solutions rapidly.



Contributed talk for ADENA2020 on Numerical solution of system of nonlinear equations associatedwih ordinary differential equations

Numerical Solution of System of Nonlinear Equations Associated wihOrdinary Differential Equations

Bijaya Mishra, Gandhi Institute for Technological Advancement, Bhubaneswar, India,email:[email protected]

Salila Dutta, Utkal University, Bhubaneswar, India.Ambit Kumar Pany, Siksha O Anusandhan University, Bhubaneswar, India.

ABSTRACT

In this paper, we have developed an efficient multistep iterative method to compute the numericalsolution of system of nonlinear equations associated with ordinary differential equations.The com-putational cost arising due to use of higher order Frechet derivatives in the constuction of multistepiterative methods for the solution of nonlinear system of eqations has been avoided.Maximum ef-fort has been given in achieveing higher order of convergence .The efficiency index of the proposedmethod is calculated and compared with the same class of methods.

This is an abstract. It may include the main results and few references, if required. Maximumlength is one page. The given format has to be strictly followed.



Impact of Magnetic Field on Polar Fluid with Newtonian Heat Transferin Vertical Concentric Annuli

Lipika Panigrahi, Department of Mathematics, Veer Surendra Sai University of Technology, Burla,Odisha -768018, India. email: [email protected]

J.P.Panda, Department of Mathematics, Veer Surendra Sai University of Technology, Burla,Odisha -768018, India.

ABSTRACT

A fully developed laminar free convective flow of polar fluid between two concentric vertical cylindershas been considered. The outer surface of the inner cylinders is subjected to Newtons law of heattransfer in the presence of an adjustable transverse magnetic field. The magnetic lines of forceinteract with conducting flowing fluid to generate a force act-at-a-distance. The analytic solutions ofthe governing equations are obtained in the form of modified Bessel function. The important findingsare: Newtonian heating/cooling as well as vortex viscosity reduces the velocity distribution; micro-rotation and linear velocity having opposite signs has an experimental bearing; Magnetic force, act-at-a-distance, reduces the primary flow which may have a therapeutic application; Reynolds analogyholds good with respect to the annular spacing. Most importantly, high Newtonian heating of thebounding surface may lead to back flow as well as instability to flow and heat transfer processes.



A Haar Scale-3 Wavelet Technique for Solving a MEMS Based FractionalDifferential Equation

Harpreet Kaur, I.K. Gujral Punjab Technical University, Mohali Campus-I, Indiaemail:[email protected],[email protected]

ABSTRACT

A mathematical model as a nonlinear fractional differential equation(FDE) based on micro-electromechanical system(MEMS)[1] is solved by using Haar scale-3 wavelet numerical technique [2,3,4]. Amodel is designed primarily to determine the viscosity of fluids that are encountered during oil wellexploration. For nonlinearity, quasi-linearization process is applied and obtained numerical resultsare compared with the available results in literature.

References[1] X.S. He, Q.X. Liu, X.C. Huang and Y.M. Chen, Dynamic Response Analysis of the Fractional-

Order System of MEMS Viscometer, Comp. Modell. Engg. Sci., 108(2015),159-169.

[2] G. Bachman, L. Narici and E. Beckenstein, Fourier and Wavelet Analysis, Springer, 2005.

[3] U. Lepik and H. Hein, Haar Wavelets with Applications, Springer, 2014.

[4] H. Kaur, R.C. Mittal and V. Mishra, Haar Wavelet Approximate Solutions for the GeneralizedLane-Emden Equations Arising in Astrophysics, Comp. Phys. Commu., 184(2013), 2169-2177.



Comparison of Shooting Technique for Boundary Value Problems usingDifferent Runge-kutta Methods

B. MALLIKARJUNA, BMS College of engineering, Affiliate to VTU, Belagavi, Bangalore, Indiaemail: [email protected]

ABSTRACT

In this paper, a numerical comparison of the shooting technique for boundary value problems hasbeen presented. In the process of shooting technique, first convert boundary value problems intoinitial value problems by assuming some initial conditions and then integrate using various Rungekutta (RK) methods, namely fourth order Runge-Kutta, 4th and 5th order Runge-Kutta, Runge-Kutta Fehlberg, Runge-Kutta Gill. Assumed initial conditions are refined using Newton-Raphsonmethod. These results are also compared with bvp4c MATLAB solver. For each RK method, erroris estimated through adjustment of step size and presented the best solution.

KEYWORDS: Boundary Value Problems; RK Methods; MATLAB bvp4c; Error Estimation.



Optimization Free Neural Network Approach for Solving Ordinary andPartial Differential Equations

Shagun Panghal, Motilal Nehru National Institute of Technology Allahabad, India, email:[email protected]

Manoj Kumar, Motilal Nehru National Institute of Technology Allahabad, India

ABSTRACT

Current work introduces a fast converging neural network-based approach for solution of ordinaryand partial differential equations. Proposed technique eliminates the need of time-consuming op-timization procedure for training of neural network. Rather, it uses the extreme learning machinealgorithm for calculating the neural network parameters so as to make it satisfy the differentialequation and associated boundary conditions. Various ordinary and partial differential equations aretreated using this technique, and accuracy and convergence aspects of the procedure are discussed.



A Study on Fractal Theory and its Application in Mathematical Physics

Priyanka, Rungta College of Engineering and Technology/CSVTU, Bhilai, India. email:

[email protected]

S.C.Shrivastava, Rungta College of Engineering and Technology/CSVTU, Bhilai, India.R.Shrivastava, Shri Shankaracharya Engineering College/CSVTU, Bhilai, India .

ABSTRACT

The study of Fractal theory is an interesting area that offers research prospectus in numerous branchesof physics, computer Science, engineering, Pure and Applied Science. Fractal was characterized byrough or non-regular geometric shape that can be split into pieces where each smaller pieces is coun-terpart of the whole. Fractal is endless patterns. A Fractals is an infinitely complex objects generallydisplays self-similarity on different scales. They are made by rehashing a simple technique again andagain like a feedback loop. Especially, fractals plays an important role in applications such as signaland image compression, soil mechanics, fluid mechanics, computer graphics etc. The key role inthe image compression is the theory of IFS. Iterated function system defines mathematically someconcepts of chaos and irregularity. Fractal dimensional analysis has wide applications in the field ofsurface analysis and study of structure of different materials. The fractal dimension analysis are usedto characterized complicated and chaotic shapes. The fractal dimension analysis offers a possibilityof a comparison between complicated and complex shapes.

The main purpose of this study was to describe Fractal Theory in consequence of applicationin mathematical physics. Especially, we describe Fractal dimensional analysis by iterated functionsystem.

REFERENCES:

[1] B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman New York (1982).

[2] J. E. Hutchinson, Fractals and self similarity, Indiana University Mathematics Journal, 30(1981), pp.713-747.

[3] M. Barnsley, Fractals Everywhere, New york: Academic Press, USA (1993).

[4] M. F. Barnsley, S .Demko, Iterated function systems and the globle construction of fractals,Proc. Roy. Soc. London Ser A, 399 (1985), pp.243-275.

[5] N.Sarkar, B. B. Chaudhari, An efficient differential box counting approach to compute fractalgeometry, IEEE Trans. System, Man,and Cybernetics 24, (1994), pp.115-120.

[6] N. Sarkar, B.B. Chaudhari, Texture segmentation using fractal dimension, IEEE Transactionon Pattern Analysis and Machine Intelligence 17,(1995), pp.72-77.



Construction of Bivariate Fractal Interpolation Function and itsFractional Calculus

Subhash Chandra, Indian Institute of Technology, Mandi, India, email:[email protected] Chandra, Indian Institute of Technology, Mandi, India.

Syed Abbas, Indian Institute of Technology, Mandi, India.

ABSTRACT

This is an abstract. In this article, we construct bivariate fractal interpolation function (FIF)from existing literatures and then we discuss fractional calculus corresponding to this bivariateFIF. We investigate partial integrals and partial derivatives of bivariate fractal interpolation func-tion. We prove also that the mixed Riemann-Liouville fractional integral and derivative of orderγ = (p, q); p > 0, q > 0, of bivariate fractal interpolation function are again bivariate interpolationfunction corresponding to some iterated function system (IFS).



Stability of Double diffusive mixed Convective Flow in Vertical PipeFilled With Porous Medium under LTNE Model

S.Kapoor, Department of Education in Science and Mathematics, Regional Institute of Education(National council of Education Research and Training), Bhubneshwar , India, email:

[email protected]

ABSTRACT

In the present work the Mathematical Modeling of the fully developed double-diffusive mixed convec-tion in a vertical pipe under local thermal non-equilibrium state has been developed. The numericalinvestigation is done for the governing physical model. The non-Darcy Brinkman-Forchheimer-extended model has been used and solved numerically by spectral collocation method. Specialattention is given to understand the effect of buoyancy ratio (N) and thermal non-equilibrium pa-rameters: inter phase heat transfer coefficient (H) as well as porosity scaled thermal conductivityratio (γ) on the flow profiles as well as on rates of heat and solute transfer. Judged from the influenceof buoyancy ratio on velocity profile, when both the buoyancy forces: thermal as well as solutal arein favor of each other and for given any value of (H), it has been found that for (N) equal to 10 aswell as 100, the basic velocity profile shows back flow for small sub domain of the domain of the flow.When two buoyancy forces are opposing to each other (RaT = −1000) velocity profile possesses akind of distortion, in which the number of zeroes increases on increasing (N). Corresponding varia-tion of heat transfer rate in the (N ,Nuf ) -plane shows a sinusoidal pattern. The flow Separation onthe flow profile dies out on increasing H for (N=0) . It has also been found 3 that for each N , whenN < 0.7, there exists a minimum value of H such that the velocity profile becomes free from flowseparation. Influence of H on the profiles of solid temperature as well as solute, in both situations aresimilar. Overall, the impact of LTNE parameters, specially , on heat transfer rate of double-diffusiveconvection is not straight forward.



A New Mathematical Technique for the Analytical Treatment ofDifferential Equations

Saurabh Tomar, Department of Mathematics, Indian Institute of Technology Kharagpur,Kharagpur 721302, India, email:[email protected]

R. K. Pandey, Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur721302, India.

ABSTRACT

Various analytical techniques have been introduced to solve differential equations, so far. Mostof these techniques are computationally intensive due to complicated symbolic computations. Inthis study, we proposed a novel technique to get the analytical solution of linear and nonlineardifferential equations. This novel approach requires only the evaluation of the first iteration to solvethe problems, minimize the computational work compare to the existing methods, and overcomevarious shortcomings of the existing methods. Some well-known differential equations, includingthe Riccati equation, the logistic equation, the Van der Pol’s oscillator equation, the pantographequation, the Lane-Emden equation, and the Klein-Gordon equation are considered to demonstratethe applicability of the technique. The implementation of the technique to various problems revealsthat the new proposal is a highly promising, effective, and reliable tool to solve differential equations.



On the Approximate Solutions of a Class of Fractional Order NonlinearVolterra Integro-Differential Initial Value Problems and Boundary Value

Problems of First Kind and their Convergence Analysis

Subrata Rana, Department of Mathematics, Indian Institute of Technology, Patna, India, email:[email protected]

ABSTRACT

In this work we consider a class of fractional order Volterra integro-differential equations of firstkind where the fractional derivative is considered in the Caputo sense. Here, we consider the initialvalue problem and the boundary value problem separately. For simplicity of the analysis, we reduceeach of these problems to the fractional order Volterra integro-differential equation of second kindby using the Leibniz’s rule. We have obtained sufficient conditions for the existence and uniquenessof the solutions of initial and the boundary value problems. An operator based method has beenconsidered to approximate their solutions. In addition, we provide a convergence analysis of theadopted approach. Several numerical experiments are presented to support the theoretical results.



Stability of Double Diffusive Mixed Convective Flow in Vertical PipeFilled With Porous Medium Under LTNE Model

Name of presenting author, S.Kapoor, Department of Education in Science and Mathematics,Regional Institute of Education (National council of Education Research and Training),

Bhubneshwar , India, email: [email protected]

ABSTRACT

In the present work the Mathematical Modeling of the fully developed double-diffusive mixed convec-tion in a vertical pipe under local thermal non-equilibrium state has been developed. The numericalinvestigation is done for the governing physical model. The non-Darcy Brinkman-Forchheimer-extended model has been used and solved numerically by spectral collocation method. Specialattention is given to understand the effect of buoyancy ratio (N) and thermal non-equilibrium pa-rameters: inter phase heat transfer coefficient (H) as well as porosity scaled thermal conductivityratio (γ) on the flow profiles as well as on rates of heat and solute transfer. Judged from the influenceof buoyancy ratio on velocity profile, when both the buoyancy forces: thermal as well as solutal arein favor of each other and for given any value of (H), it has been found that for (N) equal to 10 aswell as 100, the basic velocity profile shows back flow for small sub domain of the domain of the flow.When two buoyancy forces are opposing to each other (RaT = −1000) velocity profile possesses akind of distortion, in which the number of zeroes increases on increasing (N). Corresponding varia-tion of heat transfer rate in the (N ,Nuf ) -plane shows a sinusoidal pattern. The flow Separation onthe flow profile dies out on increasing H for (N=0) . It has also been found 3 that for each N , whenN < 0.7, there exists a minimum value of H such that the velocity profile becomes free from flowseparation. Influence of H on the profiles of solid temperature as well as solute, in both situations aresimilar. Overall, the impact of LTNE parameters, specially , on heat transfer rate of double-diffusiveconvection is not straight forward.



Spectral Study of reiner Philippof Fluid Flow from an InclinedStretching Sheet

Y.S.KALYAN CHAKRAVARTHY, M.S.Ramaiah Institute of Technology, Bangalore, Indiaemail:[email protected]

B. Mallikarjuna1, BMS College of engineering, Affiliate to VTU, Belagavi, Bangalore, India

ABSTRACT

This paper is aimed to study the hydrodynamic heat transfer flow of a Reiner Philoppff fluid past avertically stretchable inclined sheet. A set of similarity transformations are used to reduce the partialdifferential equations into ordinary differential equations and then solved the governing equationsnumerically by the spectral-quasilinearization method. The effect of physical parameters like Bing-ham number, Thermal buoyancy parameter, and inclination angle parameter have been discussedgraphically for velocity and temperature profiles. The heat transfer rate and skin friction coefficientare tabulated and interpreted for various physical parameters. It is observed that dilatant fluid andpseudoplastic fluid shows diverse behavior as Bingham’s number increases. With the enhancementin the inclination angle parameter, the fluid temperature raises for dilatant fluid, pseudoplastic fluid,and Newtonian fluid. The study of flow model over the stretching sheet is important in fluid dy-namics due to their enormous applications in engineering, environment such as metal and polymerextrusion, drawing of plastic sheets, the liquid coating on photographic films etc.

KEYWORDS: Non Newtonian Fluid; Numerical Method; Boundary Layer; Inclined StretchingSheet;


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Book of Abstract of ADENA 2020 International Conference on … · 2020. 10. 22. · Book of Abstract of ADENA 2020 International Conference on Advances in Di erential Equations and