Mathematical Modeling of complex real-world problems in ...math.gmu.edu/~pseshaiy/PEER2013/PEER_GMU_NMAIST_INTRO.pdfMathematical Modeling of complex real-world problems in science

Padmanabhan (Padhu) Seshaiyer Professor, Mathematical Sciences

George Mason University Email: [email protected]

NM-AIST, Arusha September 2, 2013

Mathematical Modeling of complex real-world problems in science and engineering

mailto:[email protected]�

2

A Real-world Problem • All the faculty and staff attended the annual department

dinner. Everyone that attended the dinner shook the hand of every other person once. If one of them carried a very bad flu and there were a total of 190 handshakes then how many people were at the banquet? Explain your thinking. Any formula that you may use need to be explained as well.

Padmanabhan Seshaiyer George Mason University

http://fitnessandwellnessnews.com/flu-fear/doorknob-protection-pic/�

3

Geometric Approach

• Handshake with three people 2 + 1 = 1 + 2 = 3

• Handshake with four people 3 + 2 + 1 = 1 + 2 + 3 = 6

• Handshake with five people 4 + 3 + 2 + 1 = 1 + 2 + 3 + 4 = 10


Algebraic Approach • S = 1 + 2 + 3 + 4 +….. + n =

S = 1 + 2 + 3 + ……. + 99 + 100 S = 100 + 99 + 98 + ……. + 2 + 1 _____________________________________________ 2 S = 101 + 101 + 101 + ……. + 101 + 101 = 100 x 101

2)1( +nn

50502

)101(100==⇒ S

Carl Friedrich Gauss (1777-1855)


http://www.amazon.com/gp/product/images/B000QETOFW/ref=dp_image_0?ie=UTF8&n=284507&s=kitchen�

Christmas Math! • The popular Christmas carol starts innocuously enough. On day one, the

character in the song gets a single present (a partridge in a pear tree). But on day two, the beneficiary receives a new present (a pair of turtle doves) plus another partridge in a pear tree. Day three brings a second helping of day two's gifts, plus more new items (three French hens). By the twelfth day, the narrator is an undeniable pack rat - and maybe in violation of local zoning - after receiving 12 drummers drumming, and new copies of all the previous day's gifts. By the twelfth verse of the song, there are a lot of gifts. How many gifts did the character receive on the 12th day?

One 1 Two 1 + 2 Three 1 + 2 + 3 Four 1 + 2 + 3 + 4 ……. ……. Twelve 1 + 2 + 3 + 4 + 5 + ….. + 12

78


6

A Graphical (Numerical) Approach

• Consider solving the equation


1380−

=n

n

7

An iterative (Numerical) Approach

• Consider solving the equation:

i ni i ni 1 10 11 6 2 42 12 80 3 9 13 5 4 46 14 100 5 8 15 4 6 51 16 134 7 8 17 3 8 58 18 203 9 7 19 2 10 67 20 433 21 1

1380

1 −=+

ii n

n


8

Another iterative (Numerical) Approach

• Consider solving the equation:

i ni i ni 1 10 11 20 2 20 12 20 3 20 13 20 4 20 14 20 5 20 15 20 6 20 16 20 7 20 17 20 8 20 18 20 9 20 19 20 10 20 20 20

3801 +=+ ii nn


Modeling across the curriculum, Undergraduate Research & K-12 STEM Collaboration

Blood

Arterial Wall

1. URL: Undergraduate Research, http://about.gmu.edu/mason-hosts-nsf-student-research-experience-in-mathematics/ 2. Modeling, Analysis and Computation of Fluid Structure Interaction Models for Biological Systems, S. Minerva Venuti and P. Seshaiyer

(Mentor), SIAM Undergraduate Research Online, Vol. 3, pp 1-17 (2010). 3. Transforming Practice Through Undergraduate Researchers, P. Seshaiyer, Council on undergraduate research Quarterly, Fall 2012.

Cerebral Spinal Fluid (CSF)

u

CSF

F damping

F

mutt(0,t)

F spring

Blood

Wall

Engaging High school Teachers in Hands-on STEM Based Learning

Modeling Across the Curriculum Problem Solving Methodology

Engaging Undergraduates in multidisciplinary research Modeling Arterial-wall, blood flow interaction

Engaging High school Students through high school lesson study

Modeling

Computation Padmanabhan Seshaiyer (Padhu) Professor Mathematical Sciences George Mason University, Fairfax, VA Director, National Science Foundation REU SITE Director, VA Department of Education Program Director, COMPLETE Center Director, STEM Accelerator Program Email: [email protected] Web: http://math.gmu.edu/~pseshaiy/outreach.html

Engaging Students in Communicating Research

Engaging Students in modeling and computation NSF Research Experiences for Undergraduates

Posters on the Hill

http://about.gmu.edu/mason-hosts-nsf-student-research-experience-in-mathematics/�mailto:[email protected]�http://math.gmu.edu/~pseshaiy/outreach.html�

• NSF

– Research Experiences for Undergraduates (Summers 2011 & 2013) – Research Experiences for Undergraduates (Summers 2009 & 2010) – Research Experiences for Undergraduates (Summers 2006 & 2007)

• NSF – Computational Mathematics Program (2002 – 2005, 2006 – 2009) – CSUMS: Undergraduate Research Program on Computational

Mathematics (2007-2011) – Redraider Mini-symposium Series on Mathematical and Computational

Modeling of Biological Systems (2003 – 2004) • NIH, with Carnegie Mellon University (2009 – 2011) • THECB

– Advanced Research Program (2006 – 2008) • SCHEV

– Improving Teacher Quality Programs (2008 – 2014) • VA-DOE

– COMPLETE Center (2010 – 2014)


FUNDING AGENCIES

Multidisciplinary Research in Mathematics

• As concluded by the National Research Council: Undergraduate education will not change in a permanent way through the efforts of “Lone Rangers.” Change requires ongoing interaction among communities of people and institutions that will reinforce and drive reform.

• Research that happens across traditional mathematics

and at the edges of traditional disciplines. • Here is the problem, find the mathematics to solve it!


Global Health

Basic Scientific Research

Defense & National Security

Sustainability of the

Environment

Energy & Natural

resources

Multidisciplinary Research Impact Areas

Agriculture &

Food Security

Education STEM


Computational Biomechanics

Mathematical Biology

Materials Modeling

Nonlinear Dynamics

Stochastic Differential Equations

Fluid Dynamics

Fluid-Structure Interaction

Control Theory

Optimization

Porous Media

Mathematics in some Multidisciplinary Research Topics


Observe

Theorize

Formulate

Describe

Analyze

Simulate

Validate

Predict

Mathematical Modeling


About those “Models”

All models are wrong, but

some are more wrong than others.

C.W. Clark and M. Mangel (2000) Dynamic State Variable Models in Ecology.


Mathematics and Real-world Disasters

• On February 25, 1991, during the Gulf War, an American Patriot Missile battery in Dharan, Saudi Arabia, failed to intercept an incoming Iraqi Scud missile. The Scud struck an American Army barracks and killed 28 soldiers.

100......000001100100000000000.00000000Error =

sec0.3410606010050.00000009Time =××××=

...100.......011001100110011001100.00011001101=

≈ 50.00000009


Understanding Rupture of Saccular Aneurysms


MAV: Membrane Wing Deflection Padmanabhan Seshaiyer George Mason University

Some MAV Applications • Reconnaissance & Surveillance • Sensor placement • Navigating Building interiors • Biochemical Sensing • Search and Rescue Operations

• Traffic Monitoring • Border Surveillance • Fire and Rescue Operations • Forestry and wildlife surveys • Power-line inspection • Real-estate aerial photography • Many more…


Experimental Challenges in MAV Design

• Small size

• High surface-to-volume ratio

• Constrained weight and volume limitations

• Low Reynolds number regime

• Low aspect ratio fixed to rotary to flapping wings

• Longer flight time

• Better range-payload performance


Computational Challenges in MAV Design

Structural Dynamics

Fluid Dynamics

Guidance & Control

Propulsion & Power

FSI


Beam-Fluid Interaction Padmanabhan Seshaiyer George Mason University


Disease Dynamics Modeling Identifying the threshold or tipping point

(Kermack and McKendrick, 1927, 1932, 1933)

• The population is constant over the time interval of interest.

• Births, deaths from causes other than disease in question, immigration and emigration are all ignored.

• The time scale is short • The population has no prior history of infections. • Individuals are found in three stages

– Susceptible – Infected – Recovered

• Design, Develop and Implement public health policy

Compartmental Models (Non-linear Dynamics)


http://www.google.co.tz/url?sa=i&rct=j&q=rift%20valley%20virus%20compartmental%20model&source=images&cd=&cad=rja&docid=ZAUdsNh9grIieM&tbnid=2n6sQ3T6s_2wOM:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Farticle%2Fpii%2FS002251931200210X&ei=C2YjUofbKMa30QXzsoG4Bw&psig=AFQjCNEO2tI-fnTLlSBWJ801AQNC4Tj1GA&ust=1378137941192272�

Modified Logistic Models (Fish Harvesting Models)

• Constant Harvesting • Proportional Harvesting • Restricted Proportional

Harvesting • Proportional Threshold

Harvesting • Seasonal Harvesting • Rotational Harvesting


http://www.google.co.tz/url?sa=i&rct=j&q=fish%20harvesting&source=images&cd=&cad=rja&docid=-r3qdc0NWdeFEM&tbnid=dUBEPmKMoenzGM:&ved=0CAUQjRw&url=http%3A%2F%2Fcollections.infocollections.org%2Fukedu%2Fen%2Fd%2FJii23we%2F9.1.html&ei=UWEjUu7VEOi70QW9rYDwBA&bvm=bv.51495398,d.ZGU&psig=AFQjCNGRFLsd7Pulhv7HS5NOQVSmzBDV8Q&ust=1378136726720579�

Transcription Models (Modeling Gene Expression)


http://www.google.co.tz/url?sa=i&rct=j&q=gene%20expression&source=images&cd=&cad=rja&docid=jol7DWCv915jdM&tbnid=EoNziE7HLfQDOM:&ved=0CAUQjRw&url=https%3A%2F%2Fwww.boundless.com%2Fbiology%2Fgene-regulation%2Fregulation-of-eukaryotic-gene-expression%2Fcells-vary-because-of-differential-gene-expression%2F&ei=bW0jUtmVM_Ob0AXgk4CIBA&psig=AFQjCNEGm60qE3LiKdXNXv6NWL1eJyy8SA&ust=1378139653745933�

Mathematical Modeling of the Symbiotic Relationship between Rhizobium Bacteria and Legumes Alexandra Lynn Zeller and Padmanabhan Seshaiyer

Abstract

Introduction

2012

Symbiosis is prevalent throughout the animal kingdom. There are three specific types of symbiosis: facilitative, obligate, and obligate-with thresholds. The symbiotic relationship between the rhizobium bacteria and the legume family is particularly unique due to the fact that in this symbiotic relationship is not reciprocal. Rhizobium is a nitrogen-fixing bacterium, which takes atmospheric nitrogen and chemically changes it to ammonia. Rhizobium belongs to the facilitative category. Therefore, it is able to live without the plant, but it is unable to fix nitrogen with out the legume. The legume belongs to the obligate category where it is not able to live without ammonia in the soil. Ammonia is vital in plant fertility and is an integral part of the world economy. Millions of dollars are spent annually on fertilizer to increase crop production. In 2010, the U.S. consumption of the plant nutrient nitrogen was 12,285 nutrient tons, having risen from 2,738 nutrient tons in 1960 to meet the growing food demand. Compared to a base of 1910-1914, fertilizer alone has risen to an indexed percent of 1277% in June of 2012. The symbiotic process requires no fertilizer, which would greatly reduce yearly expenditures on fertilizer. This scientist’s main goal is to mathematically model the symbiotic relationship between the rhizobium bacteria and its host legume. In this work, a mathematical model is proposed to predict the ideal conditions in which the legume could be grown and the rate at which the soil would become fertile. To accurately display a peaceful coexistence model, this work shows both analytically and numerically that the competition between the bacteria and legume must be less than that of the inhibition. This work also attempts to show various stages of the mutual relationship by developing suitable mathematical models and analyzing them.

The nitrogen fixation process is a complex and vital part of soil fertility. The symbiotic relationship between the rhizobium bacteria and the legume family is one of a very small group that uses this as a means to survive. The bacterium first attaches itself to the roots of the legume and in return, the legume shields the bacteria’s nodules with its own roots. The legume then supplies various nutrients to the bacterium that are needed in the nitrogen fixing process, such as nitrogenase and hemoglobin. The plant then in turn becomes independent of soil nitrogen, providing it with a much more fertile soil to grow. Eventually, the plant dies and as the plant decays the nitrogen already in it gets reabsorbed into the topsoil. There are thirty different types of rhizobium bacteria with which only one specific legume will bond. Even if a plant decays in a particular ground place, the same bacteria must be introduced there later, otherwise no fixation can or will occur.

Mathematical Model and Analysis

A standard differential equation model for mutualism was used in this project. Upon manipulation of the two equations, only one specific case would hold for co-existence. For peaceful co-existence;

Competition < Inhibition

Nitrogen Cycle

Future Work

A unique system of differential equations with distinct variables needs to be developed specifically for this symbiotic relationship. This would then take into account various conditions of survival, necessary nutrients, as well as extreme situations. One could even take into account what would happen to the symbiotic relationship with and without fertilizer. Also, a large amount of data and observations need to be gathered and recorded. This data would then be used to further develop the model. Eventually, this project has the potential to significantly influence soil fertilization globally.

References 1. “Agricultural Prices”. National Agricultural Statistics Service (NASS), Agricultural Statistics Board, United States Department of Agriculture

(USDA)., June 28, 2012. http://usda.mannlib.cornell.edu/ 2. “Legume Seed Inoculants”, n.d. http://www.ext.colost.edu/ 3. Graves, Wendy, Bruce Peckham, and John Pastor. “A Bifurcation Analysis of a Differential Equations Model for Mutualism.” Bulletin of

Mathematical Biology 68, no. 8 (2006): 1851–1872.

Numerical Experiments

22202120

21101110

))1)(((

))1)(((

2

1

yaxerrrdtdy

xaxerrrdtdx

xk

yk

−−++=

−−++=

−

−

222021220

211011110

))((

))((

yayrrxkrdtdy

xaxrrykrdtdx

−+−=

−+−=

212121201110 ))(( aakkrrrr

Accurate Location of an Aircraft (Extending the work to Mobile Communications)


Modeling a Control System (Kerosene-Petrol Adulteration)


http://www.google.co.tz/url?sa=i&rct=j&q=petrol%20adulteration&source=images&cd=&cad=rja&docid=McM-DlR8U3kJyM&tbnid=Ap1J4RGN7I-3fM:&ved=0CAUQjRw&url=http%3A%2F%2Fguffadi.blogspot.com%2F2011%2F05%2Fsome-folks-use-visa-or-mastercard-or.html&ei=XXAjUvfFDoek0QXS7oGQDA&psig=AFQjCNEh2n7BWoocCfQZ2Pcs2rcEW8VJJA&ust=1378140620262372�

Pattern Recognition (Predicting Weather effects on crops)


Groundwater Flow Problem (Drinking water)


Modeling Hydraulic Conductivity

Chemical contaminant

and transport

Determining Optimal Shape

Identifying effective

Source terms

Advection-Diffusion-Convection Models (Tobacco Curing)

S DOSE I

NICOTINE FROM TOBACCO INDUSTRIES

NICOTINE FROM SMOKERS

NICOTINE FROM TOBACCO CURING

β α


http://www.google.co.tz/url?sa=i&rct=j&q=advection+diffusion&source=images&cd=&cad=rja&docid=aPpVJQ076-DaBM&tbnid=rUlVALcOtiP4ZM:&ved=0CAUQjRw&url=http%3A%2F%2Fcde.nwc.edu%2FSCI2108%2Fcourse_documents%2Fearth_moon%2Fearth%2Fearth_science%2Fconvection%2Fpollution_transfer.htm&ei=X18jUt7aBOKl0wWonYFo&bvm=bv.51495398,d.ZGU&psig=AFQjCNHrn3dBu2zyESnAXSIVzRHEjHbvkQ&ust=1378136271096317�http://www.google.co.tz/url?sa=i&rct=j&q=tobacco%20curing%20chamber%20tanzania&source=images&cd=&cad=rja&docid=valklv8eKuMNwM&tbnid=ZKFhMx9TrRa21M:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.zimbabwesituation.com%2Fnov8_2012.html&ei=418jUs2MHeGT0AXok4GYAQ&bvm=bv.51495398,d.ZGU&psig=AFQjCNErxk8QjKjKTLEou1Rhtt4QfG_nfw&ust=1378136402259682�

Population Dynamics Registering births-deaths


http://www.google.co.tz/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=0ubq4LMUe7DxoM&tbnid=fGGOp1VdGSC_uM:&ved=0CAUQjRw&url=http://www.sciencedirect.com/science/article/pii/S0006320707004648&ei=e1UjUozaLsLW0QXrgoCQDg&bvm=bv.51495398,d.ZGU&psig=AFQjCNH2IpGZD4_hOVYqajNJZ2OvlQailQ&ust=1378133699129274�http://www.google.co.tz/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=6fJ8rOPNKoBdEM&tbnid=MS3yjXlXo8cYMM:&ved=0CAUQjRw&url=http://www.biology.iupui.edu/biocourses/N100/2k4ch39pop.html&ei=SVYjUu2gEOKW0AW7yoGIAg&bvm=bv.51495398,d.ZGU&psig=AFQjCNE7Cu5ebp_dYIlTFWXS7MImW9vz4w&ust=1378133796342657�http://www.google.co.tz/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=hdiGaY21Hx-QGM&tbnid=oKmH8muUPO5xIM:&ved=0CAUQjRw&url=http://www.population-growth-migration.info/population.html.old&ei=vlYjUq8fhs7RBc-QgIgN&bvm=bv.51495398,d.ZGU&psig=AFQjCNHkq-D5-Q3Gs_2ktAprpNmpiej-Bg&ust=1378134039040488�http://www.google.co.tz/url?sa=i&source=images&cd=&cad=rja&docid=oD5ONjDF1M0LAM&tbnid=TuZiwlf90p6oRM:&ved=&url=http%3A%2F%2Fwww.theogm.com%2F2010%2F10%2F05%2Fmaking-education-count%2F&ei=iFsjUs2TAsLD7AaGtIGgCA&psig=AFQjCNEONnIVkfHf9E9Wnk0LQUCJFVsy9A&ust=1378135304082374�

Technology Acceptance Models (Kay Connelly, 2007)

Optimization (Allocation of natural resources)

http://www.ce.utexas.edu/prof/mckinney/ce385d/papers/GAMS-Tutorial.pdf

Discrete Time Population Models (Age-structure Models)


=

)()()()(

)(

4

3

2

1

tntntntn

tN )()1( tNLtN =+

)0()( NLtN t=

Typical First Step

Physical System

Mathematical Model


Modeling Population

tyd td y )(α

yKdtdy

=⇒


Let y(t) be a homogeneous population density and y0 be the initial population.

Kteyty 0)( =⇒y0

Predator-Prey Problem

Kteyty

yKdtdyty

dtdy

0)(

)(

=⇒

=⇒α3α

XYYdtdY

=

XY

1α 2α

4α

__

__ +

Y(0) = Y0

X(0) = X0

XdtdX

=


Displacement of a Linear Elastic Bar

)(xfdxd

dxduK

dxdu

−=

=⇒

σ

σασ

X=a X=b

f(x)

u(x)

0)(0)(

)(

==

=

−

buau

xfdxduK

dxd


Solution Methodology

Physical System

Mathematical Model

Analytical Solution

Compare


Finding an Analytical Solution

0)(0)(

)(

==

=

−

buau

xfdxduK

dxd

3α

XYYdtdY

=

XY

1α 2α

4α

__

__ +

Y(0) = Y0

X(0) = X0

XdtdX

=

BVP IVP


Solution Methodology

Physical System

Mathematical Model

Numerical Solution Analytical Solution

Experimental Data Compare Compare

Compare


BVP or IVP

Finite Element Methods

Finite Difference Methods

Shooting Methods

Runge Kutta

Methods ….............

Solution to a System of Equations

Direct Methods

Indirect Methods

Error Analysis

Experimental Data

Solution to BVP/IVP


FEM: Weak Variational Form

0)1(0)0(

10:

==

FEM: Discretization

)(),( wFwuaVwallforthatsuchVuFind

=∈∈


N

NNN

=∈∈

VVN

N

vuCuuVvallFor

||||||||:

−≤−∈


FEM: Matrix Formulation


N

NNN

=∈∈

∑=

=n

iiiNu

1

φξ

)(,

..1

1jj

n

iii

i

Fa

njallforthatsuchFind

φφφξ

ξ

=

=

∑=

bA

=ξ


Fluid

Flow-structure interaction (FSI)

0.

).(

),0(:

=∇

=∇+∇+∆−∂∂

×Ω

u

fpuuutu

TFluid

f

f

νρ

Γ

Solid Fluid Solid

])([5.0~~2)~(~

~.

),0(:

2

2

T

s

s

wwt r

btw

TS o l i d

∇+∇=

+=

=∇−∂∂

×Ω

εεµελσ

σρ

Γ



Beam–fluid coupled model

What type of transformative research and training can one do?

Modify Key Assumptions Build Realistic Geometry Optimize Mathematical Techniques Enhance Mathematical Software Match Experimental Data Refine Mathematical Model Perform Parameter Estimation Studies


My Contacts! Padmanabhan Seshaiyer Professor, Mathematical Sciences Director, STEM Accelerator Director, COMPLETE Center George Mason University Fairfax, Virginia 22030 USA Email: [email protected] Web: http://math.gmu.edu/~pseshaiy

mailto:[email protected]�http://math.gmu.edu/~pseshaiy�

Mathematical Modeling of complex real-world problems in science and engineeringA Real-world Problem Geometric ApproachAlgebraic ApproachChristmas Math!A Graphical (Numerical) ApproachAn iterative (Numerical) ApproachAnother iterative (Numerical) ApproachModeling across the curriculum, Undergraduate Research & K-12 STEM CollaborationFUNDING AGENCIESMultidisciplinary Research in MathematicsSlide Number 12Slide Number 13Slide Number 14About those “Models”Mathematics and �Real-world DisastersUnderstanding Rupture of �Saccular AneurysmsSlide Number 18MAV: Membrane Wing DeflectionSome MAV ApplicationsExperimental Challenges �in MAV DesignComputational Challenges �in MAV DesignBeam-Fluid InteractionDisease Dynamics Modeling�Identifying the threshold or tipping point�(Kermack and McKendrick, 1927, 1932, 1933)Slide Number 25Compartmental Models�(Non-linear Dynamics)Modified Logistic Models�(Fish Harvesting Models)Transcription Models�(Modeling Gene Expression)Slide Number 29Slide Number 30Modeling a Control System�(Kerosene-Petrol Adulteration)Pattern Recognition�(Predicting Weather effects on crops)Slide Number 33Advection-Diffusion-Convection Models�(Tobacco Curing)Population Dynamics�Registering births-deathsTechnology Acceptance Models�(Kay Connelly, 2007)Optimization�(Allocation of natural resources)Discrete Time Population Models�(Age-structure Models)Slide Number 39Modeling PopulationPredator-Prey ProblemDisplacement of a Linear Elastic BarSlide Number 43Finding an Analytical SolutionSlide Number 45Slide Number 46FEM: Weak Variational FormFEM: DiscretizationFEM: Matrix FormulationFlow-structure interaction (FSI)Beam–fluid coupled modelSlide Number 52What type of transformative research and training can one do?My Contacts!

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Mathematical Modeling of complex real-world problems in ...math.gmu.edu/~pseshaiy/PEER2013/PEER_GMU_NMAIST_INTRO.pdfMathematical Modeling of complex real-world problems in science