Lecture on BVP

8/10/2019 Lecture on BVP

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BVP

Dr. Nasir M Mirza

Numerical Analysis

Email: [email protected]

mailto:[email protected]

mailto:[email protected]



1. Extensive application and use. Boundary Value Problemsoccur all physical problems where space variations aresought and are seen particularly in structural and solidmechanics.

2. Typical transport problems have BVP types. For example,

• thermal processing of food and biomaterials (drying,cooling, baking, microwaving, freezing,…).

• Movement of water and chemicals in soil and otherporous media.

Applications of BVP



2. The material properties in adjacent elements do nothave to be the same. This allows application of BVPto composite materials.

3. Irregularly shaped boundaries can be approximatedusing elements with straight sides or matched exactlyusing elements with curved boundaries. The finiteelement or finite difference method is used and it isnot limited to “nice” shapes with easily defined

boundaries (like many analytical solutions or someother numerical methods).

Applications of BVP



4. The size of the elements can be varied. This allowsthe element grid or mesh to be expanded or refinedas the need arises.

5. Boundary conditions such as discontinuous surfaceloadings present no difficulties for the method.Mixed boundary conditions can be easily handled.

Applications of BVP



8. BVP are an integral part of CAD/CAM (computeraided design/computer aided manufacturing) andCFD (computational fluid dynamic) systems.

9. Applications of BVP are also done using highperformance computing and parallel algorithms.

10. A a few test cases can include crash testing,explosion, combustion, fluid-structure interaction,large CFD problems, plasma fusion, biomedical

research, constructive surgery,….

10. Steady state and time-dependent problems arealso part of BV- problems.



6

Mixed Boundary Value Problems occur, in a natural way, in varieties ofbranches of Physics and Engineering and several mathematicalmethods have been developed to solve this class of problems ofApplied Mathematics.

While understanding applications of such boundary value problemsare of immense value to Physicists and Engineers, analyzing theseproblems mathematically and determining their solutions by utilizingthe most appropriate analytical or numerical methods are theconcerns of Applied Mathematicians.

Of the various analytical methods, which are useful to solve certainmixed boundary value problems arising in the theory of Scattering ofSurface Water Waves , the methods involving complex function theoryand singular integral equations will be examined in detail along withsome recent developments of such methods.

Applications of BVP





Ordinary differential Equations - Boundary Value Problems

Approximation of the solution to boundary value problems, differential equations with conditions

imposed at different points.

with the boundary conditions

ba yb y ya y

b xa y y x f y

and

,,,



Boundary Conditions

• Dirichlet boundary conditions

• Newman boundary conditions

• Mixed conditions

There are three types of boundary conditions:

b ya y ,

b ya y ,

b ycb ya yca y 21 ,



Linear ODE -Boundary ValueProblems

For simple equation with Dirichlet boundaryconditions between a < x < b

where, the p(x), q(x) and r(x) are functions of theindependent variable.

ba yb y ya y xr y xq y x p y and ,




Rewrite the equation in terms of two initial value problems

if the slope at the end, b, is not zero we can writeequation as:

1and0 ,

0and ,

avavv xqv x pv

au yau xr u xqu x pu a

x Av xu x y




At the boundary y(b) = y b,

Substitute back into the equations.

xvbv

bu y xu x y b

bvbu y

Ab Avbu yb y bb



Boundary Value Problems

Rewrite the equations as a set of first orderdifferential equations and solve:

v xqw x pw

vw

xr u xqw x pw

uw

22

2

11

1



The differential equations

• Taylor Series• Euler / Modified Euler• Runge-Kutta

• Adam Bashforth

The first order equations can be solved using thetechniques we developed in one dimensional initialvalue problems.



Higher-order differential ExampleProblem

Consider a simple second order differentialequation.

The initial conditions are x(1) =2, x(3) =-1 andD t = 0.2.

xt

t x

x

t

t dt

xd

5

1

51

2

2



Example Problem

The equation can be written as a set of two initialvalue problems.

11,01 ,5

1

01,21 ,5

1

2222

1111

x x xt

x

x x xt

t x



Example Problem

The equation can be written as a set of two initialvalue problems.

11,01 ,5

1

01,21 ,5

1

2222

22

1111

11

x x xt

w xw

x x xt

t w

xw



Example Problem

The two segments are determined:

t x[1] v[1] dv/dt[1] x*[1] v*[1] dv*/dt[1] x[2] v[2] dv/dt[2] x*[2] v*[2] dv*/dt[2] x[3] v[3]1 2 0 2.6 2 0.52 2.72 0 1 0 0.2 1 0.152 2 -3.493

1.2 2.052 0.532 2.7595 2.1584 1.0839 2.954 0.2 1.0152 0.152 0.403 1.0456 0.29019 1.3534 -3.0141.4 2.21359 1.1034 2.9938 2.4343 1.7021 3.2553 0.4061 1.0594 0.2924 0.618 1.1179 0.42022 0.7951 -2.5971.6 2.49414 1.7283 3.296 2.8398 2.3875 3.6175 0.6238 1.1307 0.4242 0.8499 1.2155 0.54397 0.3151 -2.2211.8 2.90571 2.4196 3.6597 3.3896 3.1515 4.0338 0.8584 1.2275 0.5494 1.1039 1.3374 0.66236 -0.093 -1.8682 3.46283 3.189 4.0777 4.1006 4.0045 4.4963 1.1149 1.3487 0.669 1.3847 1.4825 0.7754 -0.432 -1.522

2.2 4.18217 4.0464 4.542 4.9914 4.9548 4.9956 1.398 1.4931 0.7829 1.6967 1.6497 0.88226 -0.701 -1.1692.4 5.08228 5.0001 5.0428 6.0823 6.0087 5.5195 1.7123 1.6596 0.8904 2.0442 1.8377 0.98123 -0.899 -0.7972.6 6.18316 6.0563 5.5679 7.3944 7.1699 6.0536 2.062 1.8468 0.9898 2.4314 2.0447 1.06982 -1.02 -0.3952.8 7.50579 7.2185 6.1025 8.9495 8.439 6.5798 2.4512 2.0527 1.0785 2.8617 2.2684 1.1447 -1.056 0.04823 9.07154 8.4867 6.6286 10.769 9.8125 7.0768 2.8833 2.2751 1.1533 3.3383 2.5057 1.2018 -1 0.5398



Two Dimensional Steady-State..

Consider :

02

2

2

2

yT

xT

Laplace Equation



Analytical :

ab

neca

nb

a y

nnTo

T n

)12(cos)12)(1sinh(

)12sin(1214

0





Consider : System of grid-points (i,j)and its four immediate neighbors

i,ji+1,j

i,j-1

i,j+1

I-1,j D xD y




Difference equation:

02

2

2

2

y

T

x

T

02

2

2

1,,1,

2

,1,,1

D

D

y

T T T x

T T T

ji ji ji

ji ji ji




If D x = D y:

41,1,,1,1

, ji ji ji ji

jiT T T T

T




Example:What is the temperature at P

P= (100 + 0 + 0 + 0)/4= 25

not very accurate…grid is too coarse! 100

100

100

0

0

0

00




Apply the equation to each interior node :

100

100

100

0

0

0

00

100

0

0

0

a b

c d

Ta = (Tb + 0 + 100 + Tc)/4Tb = (0 + 0 + Ta + Td)/4Tc = (Td + Ta + 0 + 100)/4

Td = (0 + Tb + Tc + 0)/4

Four equations --> four unknown!



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Lecture on BVP