cise301 : numerical methods topic 8 ordinary differential equations (odes) lecture 28-36

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CISE301: Numerical Methods

Topic 8 Ordinary Differential

Equations (ODEs)Lecture 28-36

KFUPM

Read 25.1-25.4, 26-2, 27-1

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Outline of Topic 8 Lesson 1: Introduction to ODEs Lesson 2: Taylor series methods Lesson 3: Midpoint and Heun’s method Lessons 4-5: Runge-Kutta methods Lesson 6: Solving systems of ODEs Lesson 7: Multiple step Methods Lesson 8-9: Boundary value Problems

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Lecture 35Lesson 8: Boundary Value

Problems

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Outlines of Lesson 8

Boundary Value Problem

Shooting Method

Examples

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Learning Objectives of Lesson 8 Grasp the difference between initial value

problems and boundary value problems.

Appreciate the difficulties involved in solving the boundary value problems.

Grasp the concept of the shooting method.

Use the shooting method to solve boundary value problems.

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Boundary-Value and Initial Value Problems

Boundary-Value Problems

The auxiliary conditions are not at one point of the independent variable

More difficult to solve than initial value problem

5.1)2(,1)0(2 2

xxexxx t

Initial-Value Problems

The auxiliary conditions are at one point of the independent variable

5.2)0(,1)0(2 2

xxexxx t

same different

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Shooting Method

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The Shooting Method

Target

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The Shooting Method

Target

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The Shooting Method

Target

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Solution of Boundary-Value Problems (BVP) Shooting Method for Boundary-Value Problems

1. Guess a value for the auxiliary conditions at one point of time.

2. Solve the initial value problem using Euler, Runge-Kutta, …

3. Check if the boundary conditions are satisfied, otherwise modify the guess and resolve the problem.

Use interpolation in updating the guess. It is an iterative procedure and can be

efficient in solving the BVP.

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Solution of Boundary-Value Problems Shooting Method

8.0)1(,2.0)0(2

)(2

yyxyyy

BVPsolvetoxyFind

Boundary-Value Problem

Initial-value Problemconvert

1. Convert the ODE to a system of first order ODEs.

2. Guess the initial conditions that are not available.

3. Solve the Initial-value problem.4. Check if the known boundary

conditions are satisfied.5. If needed modify the guess and

resolve the problem again.

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Example 1 Original BVP

2)1(,0)0(044

yyxyy

0 1 x

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Example 1 Original BVP

2)1(,0)0(044

yyxyy

2. 0

0 1 x

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Example 1 Original BVP

2)1(,0)0(044

yyxyy

2. 0

0 1 x

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Example 1 Original BVP

2)1(,0)0(044

yyxyy

to be determined

2. 0

0 1 x

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Example 1 Step1: Convert to a System of First Order ODEs

1 2 1

2 1 2

2

4 4 0(0) 0, (1) 2

Convert toa system of first order ODEsy y y (0) 0

,y 4(y ) y (0) ?

The problem will be solved using RK2 with h 0.01for different values of y (0) until we

y y xy y

x

have y(1) 2

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Example 1 Guess # 1

0)0(1#

yGuess

-0.7688

0 1 x

2)1(,0)0(044

yyxyy

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Example 1 Guess # 2

1)0(2#

yGuess

0.99

0 1 x

2)1(,0)0(044

yyxyy

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Example 1 Interpolation for Guess # 3

2)1(,0)0(044

yyxyy

)0(yGuess y(1)

1 0 -0.7688

2 1 0.9900

0.99

0 1 2 y’(0)

-0.7688

y(1)

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Example 1 Interpolation for Guess # 3

2)1(,0)0(044

yyxyy

)0(yGuess y(1)

1 0 -0.7688

2 1 0.9900

0.99

0 1 2 y’(0)

-0.7688

1.5743

2

y(1)

Guess 3

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Example 1 Guess # 3

5743.1)0(3#

yGuess

2.000

0 1 x

2)1(,0)0(044

yyxyy

Guess #3 should be used for the solution to the boundary value problem.

y(1)=2.000

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Summary of the Shooting Method1. Guess the unavailable values for the

auxiliary conditions at one point of the independent variable.

2. Solve the initial value problem.3. Check if the boundary conditions are

satisfied, otherwise modify the guess and resolve the problem.

4. Repeat (3) until the boundary conditions are satisfied.

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Properties of the Shooting Method

1. Using interpolation to update the guess often results in few iterations before reaching the solution.

2. The method can be cumbersome for high order BVP because of the need to guess the initial condition for more than one variable.

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Lecture 36Lesson 9: Discretization

Method

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Outlines of Lesson 9

Discretization Method Finite Difference Methods for Solving Boundary

Value Problems

Examples

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Learning Objectives of Lesson 9

Use the finite difference method to solve BVP.

Convert linear second order boundary value problems into linear algebraic equations.

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Solution of Boundary-Value Problems Finite Difference Method

8.0)1(,2.0)0(2

)(2

yyxyyy

BVPsolvetoxyFind

y4=0.8

0 0.25 0.5 0.75 1.0 xx0 x1 x2 x3 x4

y

y0=0.2

y1=?y2=?

y3=?

Boundary-Value Problems

Algebraic Equationsconvert

Find the unknowns y1, y2, y3

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Solution of Boundary-Value Problems Finite Difference Method Divide the interval into n intervals. The solution of the BVP is converted to the

problem of determining the value of function at the base points.

Use finite approximations to replace the derivatives.

This approximation results in a set of algebraic equations.

Solve the equations to obtain the solution of the BVP.

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Finite Difference Method Example

8.0)1(,2.0)0(2 2

yyxyyy

y4=0.8

0 0.25 0.5 0.75 1.0 xx0 x1 x2 x3 x4

y

y0=0.2

Divide the interval [0,1 ] into n = 4 intervalsBase points are x0=0x1=0.25x2=.5x3=0.75x4=1.0

y1=?y2=?

y3=?

To be determined

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Finite Difference Method Example

8.0)1(,2.0)0(2 2

yyxyyy

2112

11

2

11

211

222

22

2Replace

iiiiiii

ii

iii

xyhyy

hyyy

Becomesxyyy

formuladifferencecentralhyyy

formuladifferencecentralh

yyyy

Divide the interval [0,1 ] into n = 4 intervalsBase points are x0=0x1=0.25x2=.5x3=0.75x4=1.0

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Second Order BVP

211

22

2

1

43210

22

2

2)()(2)(

)()(1,75.0,5.0,25.0,0

Points Base25.0

8.0)1(,2.0)0(2

hyyy

hhxyxyhxy

dxyd

hyy

hxyhxy

dxdy

xxxxx

hLet

yywithxydxdy

dxyd

iii

ii

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Second Order BVP

2

11

2111

43210

43210

212

11

22

2

163924

8216

8.0?,?,?,,2.01,75.0,5.0,25.0,0

3,2,122

2

iiii

iiiiiii

iiiiiii

xyyy

xyyyyyy

yyyyyxxxxx

ixyhyy

hyyy

xydxdy

dxyd

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Second Order BVP

0.74360.6477,0.4791,

)8.0(2475.05.0

)2.0(1625.0

3916024391602439

1639243

1639242

1639241

163924

321

2

2

2

3

2

1

23234

22123

21012

211

yyySolution

yyy

xyyyi

xyyyi

xyyyi

xyyy iiii

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Second Order BVP

2

11

2111

1003210

10099210

212

11

22

2

100002019910200

200210000

8.0?,?,?,,2.01,99.0,02.0,01.0,0

100,...,2,122

2

iiii

iiiiiii

iiiiiii

xyyy

xyyyyyy

yyyyyxxxxx

ixyhyy

hyyy

xydxdy

dxyd

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Summary of the Discretiztion Methods Select the base points. Divide the interval into n intervals. Use finite approximations to replace the

derivatives. This approximation results in a set of

algebraic equations. Solve the equations to obtain the solution

of the BVP.

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RemarksFinite Difference Method :

Different formulas can be used for approximating the derivatives.

Different formulas lead to different solutions. All of them are approximate solutions.

For linear second order cases, this reduces to tri-diagonal system.

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Summary of Topic 8Solution of ODEs

Lessons 1-3:• Introduction to ODE, Euler Method, • Taylor Series methods, • Midpoint, Heun’s Predictor corrector methodsLessons 4-5:• Runge-Kutta Methods (concept & derivation) • Applications of Runge-Kutta Methods To solve first order ODE

Lessons 6:• Solving Systems of ODE

Lessons 8-9:• Boundary Value Problems• Discretization method

Lesson 7:Multi-step methods