Equações diferenciais parciais e análise de Fourier, uma breve introdução-Tung K.K. em Inglês

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Partial Di�erential Equations
and Fourier Analysis | A
Short Introduction
Ka Kit Tung
Professor of Applied Mathematics
University of Washington
Preface
This short book is intended for a one-semester course for students in the
sciences and engineering after they have taken one year of calculus and one
term of ordinary di�erential equations. For universities on quarter systems,
sections labelled \optional" can be omitted without loss of continuity. A
course based on this book can be o�ered to sophomores and juniors.
Examples used in this book are drawn from traditional application areas
such as physics and engineering, as well as from biology, music, �nance
and geophysics. I have tried, whenever appropriate, to emphasize physical
motivation and have generally avoided theorems and proofs. I also have
tried to teach solution techniques, which will be useful in a student’s other
courses, instead of the theory of partial di�erential equations.
I believe that the subjects of partial di�erential equations and Fourier
analysis should be taught as early as feasible in an undergraduate’s curricu-
lum. Towards this end the book is written to present the subject matter
as simply as possible. Ample worked examples are given at the end of
the chapters as a further learning aid. Exercises are provided for the pur-
pose of reinforcing standard techniques learned in class. Tricky problems,
whose purpose is mainly to test the student’s mental dexterity, are generally
avoided.
i
ii
Contents
1 Introduction 1
1.1 Review of Ordinary Di�erential Equations . . . . . . . . . . . 1
1.1.1 First and Second Order Equations . . . . . . . . . . . 1
1.2 Nonhomogeneous Ordinary Di�erential Equations . . . . . . . 7
1.2.1 First-Order Equations: . . . . . . . . . . . . . . . . . . 7
1.2.2 Second-Order Equations: . . . . . . . . . . . . . . . . 10
1.3 Summary of ODE solutions . . . . . . . . . . . . . . . . . . . 11
1.4 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Exercise I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Solutions to Exercise I . . . . . . . . . . . . . . . . . . . . . . 14
1.7 Exercise II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8 Solutions to Exercise II . . . . . . . . . . . . . . . . . . . . . 18
2 Physical Origins of Some PDEs 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Conservation Laws: . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Di�usion of a tracer: . . . . . . . . . . . . . . . . . . . 20
2.2.2 Advection of a tracer: . . . . . . . . . . . . . . . . . . 22
2.2.3 Nonlinear advection: . . . . . . . . . . . . . . . . . . . 23
2.2.4 Heat conduction in a rod: . . . . . . . . . . . . . . . . 24
2.2.5 Ubiquity of the Di�usion Equation . . . . . . . . . . . 25
2.3 Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 A drunken sailor . . . . . . . . . . . . . . . . . . . . . 26
2.3.2 Price of stocks as a random walk . . . . . . . . . . . . 27
2.4 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Multiple Dimensions . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Types of second-order PDEs . . . . . . . . . . . . . . . . . . . 30
2.7 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 30
2.8 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 32
iii
iv CONTENTS
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.10 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Method of Similarity Variables (Optional) 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Similarity Variable . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 The \drunken sailor" problem solved . . . . . . . . . . . . . . 41
3.4 The fundamental solution . . . . . . . . . . . . . . . . . . . . 43
3.5 Examples from Fluid Mechanics . . . . . . . . . . . . . . . . . 44
3.5.1 The Rayleigh problem . . . . . . . . . . . . . . . . . . 44
3.5.2 Di�usion of vorticity . . . . . . . . . . . . . . . . . . . 46
3.6 The Age of the Earth . . . . . . . . . . . . . . . . . . . . . . 47
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Simple Plane-Wave Solutions 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Linear homogeneous equations . . . . . . . . . . . . . . . . . 53
4.2.1 Three di�erent types of behaviors . . . . . . . . . . . . 54
4.2.2 Rossby waves in the atmosphere . . . . . . . . . . . . 56
4.2.3 Laplace’s equation in a circular disk . . . . . . . . . . 57
4.3 Further developments (optional) . . . . . . . . . . . . . . . . 62
4.3.1 The wave equation . . . . . . . . . . . . . . . . . . . . 62
4.3.2 The di�usion equation . . . . . . . . . . . . . . . . . . 64
4.4 Forced oscillation . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.2 Subsoil temperature . . . . . . . . . . . . . . . . . . . 66
4.4.3 Why the earth does not have a corona . . . . . . . . . 67
4.5 A comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.6 Exercise I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7 Solutions to Exercise I . . . . . . . . . . . . . . . . . . . . . . 71
4.8 Exercise II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.9 Solutions to Exercise II . . . . . . . . . . . . . . . . . . . . . 74
5 d’Alembert’s Solution 77
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 d’Alembert’s approach . . . . . . . . . . . . . . . . . . . . . . 77
5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
CONTENTS v
5.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 Separation of Variables 89
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 An example of heat conduction in a rod: . . . . . . . . . . . . 89
6.3 Separation of variables: . . . . . . . . . . . . . . . . . . . . . 90
6.4 Physical interpretation of the solution: . . . . . . . . . . . . . 96
6.5 A vibrating string problem: . . . . . . . . . . . . . . . . . . . 97
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7 Fourier Sine Series 103
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 Finding the Fourier coe�cients . . . . . . . . . . . . . . . . . 103
7.3 An Example: . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.4 Some comments: . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.5 A mathematical curiosity . . . . . . . . . . . . . . . . . . . . 110
7.6 Representing the cosine by sines . . . . . . . . . . . . . . . . 110
7.7 Application to the Heat Conduction Problem . . . . . . . . . 111
7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8 Fourier Cosine Series 121
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.2 Finding the Fourier coe�cients . . . . . . . . . . . . . . . . . 121
8.3 Application to PDE with Neumann Boundary Conditions . . 123
9 Eigenfunction Expansion 127
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.2 Eigenfunctions and boundary conditions . . . . . . . . . . . . 128
9.3 Orthognality Condition for Xn . . . . . . . . . . . . . . . . . 132
10 Nonhomogeneous Partial Differential Equations 135
10.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10.2 Eigenfunction expansion . . . . . . . . . . . . . . . . . . . . . 135
10.3 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
11 Collapsing Bridges 141
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
11.2 Marching soldiers on a bridge, a simple model . . . . . . . . . 141
11.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
vi CONTENTS
11.4 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
11.5 A di�erent forcing function . . . . . . . . . . . . . . . . . . . 145
11.6 Tacoma Narrows Bridge . . . . . . . . . . . . . . . . . . . . . 147
11.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
11.8 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
12 Fourier Series 151
12.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
12.2 Periodic Eigenfunctions . . . . . . . . . . . . . . . . . . . . . 151
12.3 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
12.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
12.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
12.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
12.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
12.5 Complex Fourier series . . . . . . . . . . . . . . . . . . . . . . 162
13 Fourier Series, Fourier Transform and Laplace Transform 165
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
13.2 Dirichlet Theorem . . . . . . . . . . . . . . . . . . . . . . . . 165
13.3 Fourier integrals . . . . . . . . . . . . . . . . . . . . . . . . . 166
13.4 Fourier transform and inverse transform . . . . . . . . . . . . 167
13.5 Laplace transform and inverse transform . . . . . . . . . . . . 168
14 Fourier Transform and Its Application to Partial Differential
Equations 171
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
14.2 Fourier transform of some simple functions . . . . . . . . . . 171
14.3 Application to PDEs . . . . . . . . . . . . . . . . . . . . . . . 174
14.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
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ii
Contents
1 Introduction 1
1.1 Review of Ordinary Di�erential Equations . . . . . . . . . . . 1
1.1.1 First and Second Order Equations . . . . . . . . . . . 1
1.2 Nonhomogeneous Ordinary Di�erential Equations . . . . . . . 7
1.2.1 First-Order Equations: . . . . . . . . . . . . . . . . . . 7
1.2.2 Second-Order Equations: . . . . . . . . . . . . . . . . 10
1.3 Summary of ODE solutions . . . . . . . . . . . . . . . . . . . 11
1.4 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Exercise I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Solutions to Exercise I . . . . . . . . . . . . . . . . . . . . . . 14
1.7 Exercise II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8 Solutions to Exercise II . . . . . . . . . . . . . . . . . . . . . 18
2 Physical Origins of Some PDEs 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Conservation Laws: . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Di�usion of a tracer: . . . . . . . . . . . . . . . . . . . 20
2.2.2 Advection of a tracer: . . . . . . . . . . . . . . . . . . 22
2.2.3 Nonlinear advection: . . . . . . . . . . . . . . . . . . . 23
2.2.4 Heat conduction in a rod: . . . . . . . . . . . . . . . . 24
2.2.5 Ubiquity of the Di�usion Equation . . . . . . . . . . . 25
2.3 Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.1 A drunken sailor . . . . . . . . . . . . . . . . . . . . . 26
2.3.2 Price of stocks as a random walk . . . . . . . . . . . . 27
2.4 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Multiple Dimensions . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Types of second-order PDEs . . . . . . . . . . . . . . . . . . . 30
2.7 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 30
2.8 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 32
iii
iv CONTENTS
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.10 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Method of Similarity Variables (Optional) 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Similarity Variable . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 The \drunken sailor" problem solved . . . . . . . . . . . . . . 41
3.4 The fundamental solution . . . . . . . . . . . . . . . . . . . . 43
3.5 Examples from Fluid Mechanics . . . . . . . . . . . . . . . . . 44
3.5.1 The Rayleigh problem . . . . . . . . . . . . . . . . . . 44
3.5.2 Di�usion of vorticity . . . . . . . . . . . . . . . . . . . 46
3.6 The Age of the Earth . . . . . . . . . . . . . . . . . . . . . . 47
3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Simple Plane-Wave Solutions 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Linear homogeneous equations . . . . . . . . . . . . . . . . . 53
4.2.1 Three di�erent types of behaviors . . . . . . . . . . . . 54
4.2.2 Rossby waves in the atmosphere . . . . . . . . . . . . 56
4.2.3 Laplace’s equation in a circular disk . . . . . . . . . . 57
4.3 Further developments (optional) . . . . . . . . . . . . . . . . 62
4.3.1 The wave equation . . . . . . . . . . . . . . . . . . . . 62
4.3.2 The di�usion equation . . . . . . . . . . . . . . . . . . 64
4.4 Forced oscillation . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.2 Subsoil temperature . . . . . . . . . . . . . . . . . . . 66
4.4.3 Why the earth does not have a corona . . . . . . . . . 67
4.5 A comment . . . . . . . . . . . . . . . . . . . . . . . .. . . . 69
4.6 Exercise I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7 Solutions to Exercise I . . . . . . . . . . . . . . . . . . . . . . 71
4.8 Exercise II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.9 Solutions to Exercise II . . . . . . . . . . . . . . . . . . . . . 74
5 d’Alembert’s Solution 77
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 d’Alembert’s approach . . . . . . . . . . . . . . . . . . . . . . 77
5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
CONTENTS v
5.5 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6 Separation of Variables 89
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 An example of heat conduction in a rod: . . . . . . . . . . . . 89
6.3 Separation of variables: . . . . . . . . . . . . . . . . . . . . . 90
6.4 Physical interpretation of the solution: . . . . . . . . . . . . . 96
6.5 A vibrating string problem: . . . . . . . . . . . . . . . . . . . 97
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.7 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7 Fourier Sine Series 103
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2 Finding the Fourier coe�cients . . . . . . . . . . . . . . . . . 103
7.3 An Example: . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.4 Some comments: . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.5 A mathematical curiosity . . . . . . . . . . . . . . . . . . . . 110
7.6 Representing the cosine by sines . . . . . . . . . . . . . . . . 110
7.7 Application to the Heat Conduction Problem . . . . . . . . . 111
7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8 Fourier Cosine Series 121
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.2 Finding the Fourier coe�cients . . . . . . . . . . . . . . . . . 121
8.3 Application to PDE with Neumann Boundary Conditions . . 123
9 Eigenfunction Expansion 127
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
9.2 Eigenfunctions and boundary conditions . . . . . . . . . . . . 128
9.3 Orthognality Condition for Xn . . . . . . . . . . . . . . . . . 132
10 Nonhomogeneous Partial Differential Equations 135
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10.2 Eigenfunction expansion . . . . . . . . . . . . . . . . . . . . . 135
10.3 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
11 Collapsing Bridges 141
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
11.2 Marching soldiers on a bridge, a simple model . . . . . . . . . 141
11.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
vi CONTENTS
11.4 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
11.5 A di�erent forcing function . . . . . . . . . . . . . . . . . . . 145
11.6 Tacoma Narrows Bridge . . . . . . . . . . . . . . . . . . . . . 147
11.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
11.8 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
12 Fourier Series 151
12.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
12.2 Periodic Eigenfunctions . . . . . . . . . . . . . . . . . . . . . 151
12.3 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
12.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
12.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
12.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
12.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
12.5 Complex Fourier series . . . . . . . . . . . . . . . . . . . . . . 162
13 Fourier Series, Fourier Transform and Laplace Transform 165
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
13.2 Dirichlet Theorem . . . . . . . . . . . . . . . . . . . . . . . . 165
13.3 Fourier integrals . . . . . . . . . . . . . . . . . . . . . . . . . 166
13.4 Fourier transform and inverse transform . . . . . . . . . . . . 167
13.5 Laplace transform and inverse transform . . . . . . . . . . . . 168
14 Fourier Transform and Its Application to Partial Differential
Equations 171
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
14.2 Fourier transform of some simple functions . . . . . . . . . . 171
14.3 Application to PDEs . . . . . . . . . . . . . . . . . . . . . . . 174
14.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
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