Gockenbach M.S. Partial Differential Equations. Analytical and Numerical Methods

Partial Differential

Equations

Analytical and Numerical Methods

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Partial Differential

Equations

Partial Differential

Equations

This page intentionally left blank

Partial Differential

Equations

Analytical

and

Numerical Methods

Mark

S.

Gockenbach

Michigan Technological University

Houghton, Michigan

siam

Society

fo r

Industrial

and

Applied

Mathematics

Philadelphia

Partial Differential

Equations

Analytical and Numerical Methods

Mark

S.

Gockenbach

Michigan Technological University

Houghton, Michigan

Society for Industrial and Applied Mathematics

Philadelphia

Copyright

©

2002

by the

Society

for

Industrial

and

Applied Mathematics.

1098765432

1

All

rights reserved. Printed

in the

United

States

of

America.

No

part

of

this book

may

be

reproduced, stored,

or

transmitted

in any

manner

without

the

written permission

of

the

publisher.

For

information, write

to the

Society

for

Industrial

and

Applied

Mathematics, 3600 University City

Science

Center, Philadelphia,

PA

19104-2688.

Library

of

Congress

Cataloging-in-Publication

Data

Gockenbach, Mark

S.

Partial

differential equations

:

analytical

and

numerical methods

/

Mark

S.

Gockenbach.

p.

cm.

Includes

bibliographical

references

and

index.

ISBN

0-89871-518-0

1.

Differential equations, Partial.

I.

Title.

QA377 .G63 2002

515'.353-dc21

2002029411

MATLAB

is a

registered trademark

of The

MathWorks, Inc.

Maple

is a

registered trademark

of

Waterloo Maple, Inc.

Mathematica

is a

registered trademark

of

Wolfram

Research,

Inc.

0-89871-518-0

is

a

registered trademark.

siam

10987654321

in

the United

States

of

America. No part

of

this

book

may

be reproduced, stored, or transmitted

in

any manner without the written permission

of

the publisher.

For

information, write to the Society for Industrial and Applied

Mathematics, 3600 University City Science Center, Philadelphia,

PA

19104-2688.

Library

of

Congress Cataloging-in-Publication

Data

Gockenbach,

Mark

S.

Partial differential equations : analytical and numerical methods / Mark

S.

Gockenbach.

p.cm.

Includes bibliographical references and index.

ISBN

0-89871-518-0

1.

Differential equations,

Partial.

I.

Title.

QA377

.G63 2002

515'.353-dc21

MATLAB

is

a registered trademark

of

The MathWorks,

Inc.

Maple

is

a registered trademark

of

Waterloo Maple,

Inc.

Mathematica

is

a registered trademark

of

Wolfram

Research,

Inc.

0-89871-518-0

•

5.laJ1l..

is

a registered trademark.

2002029411

Dedicated

to my

mother,

Joy

Gockenbach,

and to the

memory

of my

father, LeRoy Gockenbach.

Dedicated to

my

mother, Joy Gockenbach, and to the

memory

of

my

father, LeRoy Gockenbach.

This page intentionally left blank

Contents

Foreword

xiii

Preface

xvii

1

Classification

of

differential

equations

1

2

Models

in one

dimension

9

2.1

Heat

flow in a

bar; Fourier's

law 9

2.1.1 Boundary

and

initial conditions

for the

heat equation

13

2.1.2 Steady-state

heat

14

2.1.3

Diffusion

16

2.2

The

hanging

bar 21

2.2.1

Boundary conditions

for the

hanging

bar 24

2.3

The

wave

equation

for a

vibrating string

27

2.4

Suggestions

for

further

reading

30

3

Essential

linear

algebra

31

3.1

Linear systems

as

linear operator equations

31

3.2

Existence

and

uniqueness

of

solutions

to Ax = b 38

3.2.1 Existence

38

3.2.2

Uniqueness

42

3.2.3

The

Fredholm alternative

45

3.3

Basis

and

dimension

50

3.4

Orthogonal bases

and

projections

55

3.4.1

The L

2

inner product

58

3.4.2

The

projection theorem

61

3.5

Eigenvalues

and

eigenvectors

of a

symmetric matrix

68

3.5.1

The

transpose

of a

matrix

and the dot

product

... 71

3.5.2 Special properties

of

symmetric matrices

72

3.5.3

The

spectral method

for

solving

Ax = b 74

3.6

Preview

of

methods

for

solving ODEs

and

PDEs

77

3.7

Suggestions

for

further

reading

78

VII

fk

Contents

Foreword

xiii

Preface

xvii

1

Classification

of

differential

equations

1

2

Models

in

one

dimension

9

3

2.1

Heat

flow

in a bar; Fourier's

law.

. . . . . . . . . . . . . . .

..

9

2.1.1 Boundary

and

initial conditions for the heat equation

13

2.1.2 Steady-state

heat

flow

14

2.1.3

Diffusion.

. . . . . . . . . . . . . . . . .

16

2.2

The hanging

bar

. . . . . . . . . . . . . . . . . . . . .

21

2.2.1 Boundary conditions for the hanging

bar

24

2.3 The wave equation for a vibrating string

27

2.4 Suggestions for further

reading.

. . . . . . . . . . . . 30

Essential

linear

algebra

3.1 Linear systems as linear operator

equations.

3.2 Existence and uniqueness of solutions

to

Ax

= b

3.2.1 Existence

........

.

3.2.2 Uniqueness

.......

.

3.2.3

The

Fredholm alternative

3.3 Basis and dimension. . . . . . . .

3.4 Orthogonal bases and projections .

3.4.1

The

L2

inner product .

3.4.2

The

projection theorem

3.5 Eigenvalues and eigenvectors of a symmetric matrix

3.5.1 The transpose

of

a matrix and the dot product

3.5.2 Special properties of symmetric matrices

3.5.3 The spectral method for solving

Ax

= b

3.6 Preview of methods for solving ODEs and

PDEs

.

3.7 Suggestions for further reading . . . . . . . . . . . . .

VII

31

38

42

45

50

55

58

61

68

71

72

74

77

78

viii

Contents

4

Essential ordinary

differential

equations

79

4.1

Converting

a

higher-order equation

to a first-order

system

... 79

4.2

Solutions

to

some simple ODEs

82

4.2.1

The

general solution

of a

second-order homogeneous

ODE

with

constant

coefficients

82

4.2.2

A

special inhomogeneous second-order linear

ODE . 85

4.2.3

First-order

linear ODEs

87

4.3

Linear systems with

constant

coefficients

91

4.3.1 Homogeneous systems

92

4.3.2

Inhomogeneous systems

and

variation

of

parameters

96

4.4

Numerical methods

for

initial value problems

101

4.4.1 Euler's method

102

4.4.2

Improving

on

Euler's method: Runge-Kutta methods

104

4.4.3 Numerical methods

for

systems

of

ODEs

108

4.4.4

Automatic step control

and

Runge-Kutta-Fehlberg

methods

110

4.5

Stiff

systems

of

ODEs

115

4.5.1

A

simple example

of a

stiff

system

117

4.5.2

The

backward Euler method

118

4.6

Green's

functions

123

4.6.1

The

Green's

function

for a first-order

linear

ODE . . 123

4.6.2

The

Dirac

delta

function

125

4.6.3

The

Green's

function

for a

second-order

IVP

....

126

4.6.4

Green's

functions

for

PDEs

127

4.7

Suggestions

for

further

reading

128

5

Boundary value problems

in

statics

131

5.1 The

analogy between BVPs

and

linear algebraic systems

....

131

5.1.1

A

note

about

direct integration

141

5.2

Introduction

to the

spectral method;

eigenfunctions

144

5.2.1 Eigenpairs

of

—

-j^

under Dirichlet conditions

....

144

5.2.2

Representing

functions

in

terms

of

eigenfunctions

. . 146

5.2.3 Eigenfunctions under other boundary conditions; other

Fourier

series

150

5.3

Solving

the BVP

using Fourier series

155

5.3.1

A

special case

155

5.3.2

The

general case

156

5.3.3 Other boundary conditions

161

5.3.4 Inhomogeneous boundary conditions

164

5.3.5 Summary

166

5.4

Finite

element methods

for

BVPs

172

5.4.1

The

principle

of

virtual

work

and the

weak

form

of

a BVP 173

5.4.2

The

equivalence

of the

strong

and

weak

forms

of the

BVP 177

5.5

The

Galerkin method

180

VIII

4

5

Contents

Essential

ordinary

differential

equations

79

82

4.1 Converting a higher-order equation

to

a first-order system

4.2 Solutions

to

some simple ODEs

..............

.

4.2.1

The

general solution of a second-order homogeneous

ODE with constant coefficients

..........

.

4.2.2 A special inhomogeneous second-order linear

ODE

4.2.3 First-order linear ODEs

...

4.3 Linear systems with constant coefficients . . . . . . . . . . . .

82

85

87

91

92

96

4.3.1 Homogeneous systems

...............

.

4.3.2 Inhomogeneous systems

and

variation of parameters

4.4 Numerical methods for initial value problems

..........

.

101

102

4.4.1 Euler's

method

....................

.

4.4.2 Improving

on

Euler's method:

Runge-Kutta

methods 104

4.4.3 Numerical methods for systems

of

ODEs

......

108

4.4.4 Automatic step control

and

Runge-Kutta-Fehlberg

methods.

. . . . . . . . . . . . . . 110

4.5 Stiff systems of

ODEs.

. . . . . . . . . . . . . 115

4.5.1 A simple example of a stiff system 117

4.5.2

The

backward Euler

method

. . . 118

4.6 Green's functions

.......

. . . . . . . . . 123

4.6.1

The

Green's function for a first-order linear

ODE.

123

4.6.2

The

Dirac

delta

function

...........

125

4.6.3

The

Green's function for a second-order

IVP

126

4.6.4 Green's functions for

PDEs

127

4.7 Suggestions for further reading . . 128

Boundary

value

problems

in

statics

131

5.1

The

analogy between

BVPs

and

linear algebraic systems 131

5.1.1 A

note

about

direct

integration.

. . . . . . . 141

5.2 Introduction

to

the

spectral method; eigenfunctions . . . . 144

5.2.1 Eigenpairs

of

-

::2

under Dirichlet

conditions.

144

5.2.2 Representing functions in terms of eigenfunctions. 146

5.2.3 Eigenfunctions under other

boundary

conditions; other

Fourier series . . . . . . . 150

5.3 Solving

the

BVP

using Fourier series 155

5.3.1 A special case

......

155

5.3.2

The

general case . . . . . 156

5.3.3

Other

boundary

conditions 161

5.3.4 Inhomogeneous

boundary

conditions 164

5.3.5 Summary . . . . . . . . . . . . . . . 166

5.4 Finite element methods for

BVPs

. . . . . . . . 172

5.4.1

The

principle of virtual work

and

the

weak form

of

a

BVP

..........................

173

5.4.2

The

equivalence of

the

strong

and

weak forms of

the

BVP.

. . 177

5.5

The

Galerkin

method

........................

180

Contents

ix

5.6

Piecewise polynomials

and the

finite

element method

188

5.6.1 Examples using piecewise linear

finite

elements

. . . 193

5.6.2

Inhomogeneous Dirichlet conditions

197

5.7

Green's

functions

for

BVPs

202

5.7.1

The

Green's

function

and the

inverse

of a

differential

operator

207

5.8

Suggestions

for

further

reading

210

6

Heat

flow and

diffusion

211

6.1

Fourier series methods

for the

heat

equation

211

6.1.1

The

homogeneous

heat

equation

214

6.1.2 Nondimensionalization

217

6.1.3

The

inhomogeneous

heat

equation

220

6.1.4 Inhomogeneous boundary conditions

222

6.1.5

Steady-state

heat

flow and

diffusion

224

6.1.6 Separation

of

variables

225

6.2

Pure Neumann conditions

and the

Fourier cosine series

229

6.2.1

One end

insulated; mixed boundary conditions

. . . 229

6.2.2

Both ends insulated; Neumann boundary conditions

231

6.2.3

Pure Neumann conditions

in a

steady-state

BVP . . 237

6.3

Periodic boundary conditions

and the

full

Fourier series

245

6.3.1 Eigenpairs

of

—

-j^

under periodic boundary conditions247

6.3.2 Solving

the BVP

using

the

full

Fourier series ....

249

6.3.3 Solving

the

IBVP using

the

full

Fourier series ....

252

6.4

Finite

element methods

for the

heat

equation

256

6.4.1

The

method

of

lines

for the

heat equation

260

6.5

Finite

elements

and

Neumann conditions

266

6.5.1

The

weak

form

of a BVP

with Neumann conditions

266

6.5.2

Equivalence

of the

strong

and

weak

forms

of a BVP

with

Neumann conditions

267

6.5.3 Piecewise linear

finite

elements with Neumann con-

ditions

269

6.5.4

Inhomogeneous Neumann conditions

273

6.5.5

The finite

element

method

for an

IBVP

with Neu-

mann conditions

274

6.6

Green's

functions

for the

heat

equation

279

6.6.1

The

Green's

function

for the

one-dimensional

heat

equation under Dirichlet conditions

280

6.6.2

Green's functions under other boundary conditions

. 281

6.7

Suggestions

for

further

reading

283

7

Waves

285

7.1 The

homogeneous wave equation without boundaries

285

7.2

Fourier series methods

for the

wave equation

291

7.2.1 Fourier series solutions

of the

homogeneous wave

equation

293

Contents

IX

6

5.6 Piecewise polynomials

and

the

finite element method . . . 188

5.6.1 Examples using piecewise linear finite elements 193

5.6.2 Inhomogeneous Dirichlet conditions

..

. . . . 197

5.7 Green's functions for

BVPs.

. . . . . . . . . . . . . . . . . 202

5.7.1

The

Green's function

and

the

inverse of a differential

operator.

. . . . . . 207

5.8 Suggestions for further

reading.

210

Heat

flow

and

diffusion

6.1 Fourier series methods for the heat equation

6.1.1

The

homogeneous heat equation

6.1.2 Nondimensionalization

.....

.

211

214

217

6.1.3

The

inhomogeneous heat equation 220

6.1.4 Inhomogeneous boundary conditions

222

6.1.5 Steady-state heat

flow

and

diffusion 224

6.1.6 Separation of variables. . . . . . . .

225

6.2

Pure

Neumann conditions and

the

Fourier cosine series 229

6.2.1 One end insulated; mixed boundary conditions 229

6.2.2 Both ends insulated; Neumann boundary conditions

231

6.2.3

Pure

Neumann conditions in a steady-state

BVP

. . 237

6.3 Periodic boundary conditions

and

the

full Fourier series . . . . .

245

6.3.1 Eigenpairs of -

-/l;x

under periodic boundary conditions247

6.3.2 Solving

the

BVP

using

the

full Fourier series . 249

6.3.3 Solving

the

IBVP using

the

full Fourier series . 252

6.4 Finite element methods for

the

heat equation 256

6.4.1

The

method of lines for

the

heat

equation.

. .

260

6.5 Finite elements

and

Neumann conditions . . . . . . . . . .

266

6.5.1

The

weak form of a BVP with Neumann conditions

266

6.5.2 Equivalence of

the

strong

and

weak forms of a

BVP

with Neumann conditions

...............

267

6.5.3 Piecewise linear finite elements with Neumann con-

ditions.

. . . . . . . . . . . . . . . . . . . . . . . . .

269

6.5.4 Inhomogeneous Neumann conditions

.........

273

6.5.5

The

finite element method for

an

IBVP

with Neu-

mann

conditions . . . . . . . . . . . . . . . . . . . . 274

6.6 Green's functions for

the

heat equation . . . . . . . . . . . . . .

279

6.6.1

The

Green's function for

the

one-dimensional heat

equation under Dirichlet conditions

.........

280

6.6.2 Green's functions under other boundary

conditions.

281

6.7 Suggestions for further

reading.

. . . . . . . . . . . . . . . . . .

283

7 VVaves

285

7.1

The

homogeneous wave equation without boundaries

......

285

7.2 Fourier series methods for

the

wave equation

...........

291

7.2.1 Fourier series solutions of

the

homogeneous wave

equation.

. . . . . . . . . . . . . . . . . . . . . . . . 293

Gockenbach M.S. Partial Differential Equations. Analytical and Numerical Methods

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