Numerical Solutions of Ordinary Differential Equations

Lecture 13:Boundary Value Problems

MTH2212 – Computational Methods and Statistics

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Objectives

Introduction Shooting Method Finite Difference Method

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Introduction

An ODE is accompanied by auxiliary conditions. These conditions are used to evaluate the integral that result during the solution of the equation. An nth order equation requires n conditions.

If all conditions are specified at the same value of the independent variable, then we have an initial-value problem.

If the conditions are specified at different values of the independent variable, usually at extreme points or boundaries of a system, then we have a boundary-value problem.

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Introduction

Initial-value versus boundary-value problems

Initial-value problem where all the conditions are specified at the same value of the independent variable.

Boundary-value problem where the conditions are specified at different values of the independent variable.

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Introduction

Determination of eigenvalues: Special class of boundary-value problems that are common in engineering involving vibrations, elasticity, and other oscillating systems.

Two general approaches for solving BVP: Shooting method Finite-difference method

Both approaches will be illustrated by an example of heat balance.

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Heat balance problem

Heat balance of a long, thin rod Rod not insulated along its length and in a steady

state

aTTT 21

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Heat balance problem

Equation describing the problem

Boundary value conditions

Analytical solution

200)(

40)0(

2

1

TLT

TT

204523.534523.73 1.01.0 xx eeT

2

2

2

01.0

10

20

0)(

mh

mL

T

TThdx

Td

a

a

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The Shooting Method

Converts the boundary value problem to initial-value problem.

A trial-and-error approach is then implemented to solve the initial value approach.

For example, the 2nd order equation can be expressed as two first order ODEs:

An initial value is guessed, say z(0) = 10.

)( aTThdx

dz

zdx

dT

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The Shooting Method

The solution is then obtained by integrating the two 1st order ODEs simultaneously.

Using a 4th order RK method with a step size of 2:T(10)=168.3797.

This differs from T(10)=200. Therefore a new guess is made, z(0)=20 and the computation is performed again:T(10)=285.8980

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The Shooting Method

Because the original ODE is linear, the two sets of points, (z, T)1 and (z, T)2, are linearly related, a linear interpolation formula is used to compute the value of z(0) as

z(0) = 12.6907 is then used to determine the correct solution.

6907.12)3797.168200(3797.1688980.285

102010)0(

z

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The Shooting Method

First shotz(0) = 10 T(10) = 168.3797

Second shotz(0) = 20 T(10) = 285.8980

Final exact hitz(0) = 12.6907 T(10) = 200

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The Shooting Method

Nonlinear Two-Point Problems. For a nonlinear problem a better approach involves

recasting it as a roots problem.

Driving this new function, g(z0), to zero provides the solution.

200)()(

)(200

)(

00

0

010

zfzg

zf

zfT

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Finite Differences Methods

The most common alternatives to the shooting method.

Finite differences are substituted for the derivatives in the original equation.

211

2

2 2

x

TTT

dx

Td iii

aiii TxhTTxhT 21

21 )2(

0)(2

211

aiiii TTh

x

TTT

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Finite Differences Methods

Finite differences equation applies for each of the interior nodes. The first and last interior nodes, Ti-1 and Ti+1, respectively, are specified by the boundary conditions.

Thus, a linear equation transformed into a set of simultaneous algebraic equations.

It will be tridiagonal which can be solved efficiently.

aiii TxhTTxhT 21

21 )2(

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Finite Differences Methods

If we use a segment length Δx = 2 m (4 interior nodes)

Thus, a set of simultaneous algebraic equations.

which can be solved for

8.004.2 11 iii TTT

8.200

8.0

8.0

8.40

04.2100

104.210

0104.21

00104.2

4

3

2

1

T

T

T

T

4795.1595382.1247785.939698.65TT

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Assignment # 6

Computational Methods 27.4, 27.5

Statistics Check with Dr Faiz