ALL 12 CHAPTERS COVERED
SOLUTION MANUAL
, Contents
Preface v
Errata vi
1 A Preview of Applications and Techniques 1
1.1 What Is a Partial Differential Equation? 1
1.2 Solving and Interpreting a Partial Differential Equation 2
2 Fourier Series 4
2.1 Periodic Functions 4
2.2 Fourier Series 6
2.3 Fourier Series of Functions with Arbitrary Periods 10
2.4 Half-Range Expansions: The Cosine and Sine Series 14
2.5 Mean Square Approximation and Parseval’s Identity 16
2.6 Complex Form of Fourier Series 18
2.7 Forced Oscillations 21
Supplement on Convergence
2.9 Uniform Convergence and Fourier Series 27
2.10 Dirichlet Test and Convergence of Fourier Series 28
3 Partial Differential Equations in Rectangular Coordinates 29
3.1 Partial Differential Equations in Physics and Engineering 29
3.3 Solution of the One Dimensional Wave Equation:
The Method of Separation of Variables 31
3.4 D’Alembert’s Method 35
3.5 The One Dimensional Heat Equation 41
3.6 Heat Conduction in Bars: Varying the Boundary Conditions 43
3.7 The Two Dimensional Wave and Heat Equations 48
3.8 Laplace’s Equation in Rectangular Coordinates 49
3.9 Poisson’s Equation: The Method of Eigenfunction Expansions 50
3.10 Neumann and Robin Conditions 52
4 Partial Differential Equations in
Polar and Cylindrical Coordinates 54
4.1 The Laplacian in Various Coordinate Systems 54
, Contents iii
4.2 Vibrations of a Circular Membrane: Symmetric Case 79
4.3 Vibrations of a Circular Membrane: General Case 56
4.4 Laplace’s Equation in Circular Regions 59
4.5 Laplace’s Equation in a Cylinder 63
4.6 The Helmholtz and Poisson Equations 65
Supplement on Bessel Functions
4.7 Bessel’s Equation and Bessel Functions 68
4.8 Bessel Series Expansions 74
4.9 Integral Formulas and Asymptotics for Bessel Functions 79
5 Partial Differential Equations in Spherical Coordinates 80
5.1 Preview of Problems and Methods 80
5.2 Dirichlet Problems with Symmetry 81
5.3 Spherical Harmonics and the General Dirichlet Problem 83
5.4 The Helmholtz Equation with Applications to the Poisson, Heat,
and Wave Equations 86
Supplement on Legendre Functions
5.5 Legendre’s Differential Equation 88
5.6 Legendre Polynomials and Legendre Series Expansions 91
6 Sturm–Liouville Theory with Engineering Applications 94
6.1 Orthogonal Functions 94
6.2 Sturm–Liouville Theory 96
6.3 The Hanging Chain 99
6.4 Fourth Order Sturm–Liouville Theory 101
6.6 The Biharmonic Operator 103
6.7 Vibrations of Circular Plates 104
, iv Contents
7 The Fourier Transform and Its Applications 105
7.1 The Fourier Integral Representation 105
7.2 The Fourier Transform 107
7.3 The Fourier Transform Method 112
7.4 The Heat Equation and Gauss’s Kernel 116
7.5 A Dirichlet Problem and the Poisson Integral Formula 122
7.6 The Fourier Cosine and Sine Transforms 124
7.7 Problems Involving Semi-Infinite Intervals 126
7.8 Generalized Functions 128
7.9 The Nonhomogeneous Heat Equation 133
7.10 Duhamel’s Principle 134
12 Green’s Functions and Conformal Mappings 136
12.1 Green’s Theorem and Identities 136
12.2 Harmonic Functions and Green’s Identities 138
12.3 Green’s Functions 139
12.4 Green’s Functions for the Disk and the Upper Half-Plane 140
12.5 Analytic Functions 141
12.6 Solving Dirichlet Problems with Conformal Mappings 146
12.7 Green’s Functions and Conformal Mappings 151
SOLUTION MANUAL
, Contents
Preface v
Errata vi
1 A Preview of Applications and Techniques 1
1.1 What Is a Partial Differential Equation? 1
1.2 Solving and Interpreting a Partial Differential Equation 2
2 Fourier Series 4
2.1 Periodic Functions 4
2.2 Fourier Series 6
2.3 Fourier Series of Functions with Arbitrary Periods 10
2.4 Half-Range Expansions: The Cosine and Sine Series 14
2.5 Mean Square Approximation and Parseval’s Identity 16
2.6 Complex Form of Fourier Series 18
2.7 Forced Oscillations 21
Supplement on Convergence
2.9 Uniform Convergence and Fourier Series 27
2.10 Dirichlet Test and Convergence of Fourier Series 28
3 Partial Differential Equations in Rectangular Coordinates 29
3.1 Partial Differential Equations in Physics and Engineering 29
3.3 Solution of the One Dimensional Wave Equation:
The Method of Separation of Variables 31
3.4 D’Alembert’s Method 35
3.5 The One Dimensional Heat Equation 41
3.6 Heat Conduction in Bars: Varying the Boundary Conditions 43
3.7 The Two Dimensional Wave and Heat Equations 48
3.8 Laplace’s Equation in Rectangular Coordinates 49
3.9 Poisson’s Equation: The Method of Eigenfunction Expansions 50
3.10 Neumann and Robin Conditions 52
4 Partial Differential Equations in
Polar and Cylindrical Coordinates 54
4.1 The Laplacian in Various Coordinate Systems 54
, Contents iii
4.2 Vibrations of a Circular Membrane: Symmetric Case 79
4.3 Vibrations of a Circular Membrane: General Case 56
4.4 Laplace’s Equation in Circular Regions 59
4.5 Laplace’s Equation in a Cylinder 63
4.6 The Helmholtz and Poisson Equations 65
Supplement on Bessel Functions
4.7 Bessel’s Equation and Bessel Functions 68
4.8 Bessel Series Expansions 74
4.9 Integral Formulas and Asymptotics for Bessel Functions 79
5 Partial Differential Equations in Spherical Coordinates 80
5.1 Preview of Problems and Methods 80
5.2 Dirichlet Problems with Symmetry 81
5.3 Spherical Harmonics and the General Dirichlet Problem 83
5.4 The Helmholtz Equation with Applications to the Poisson, Heat,
and Wave Equations 86
Supplement on Legendre Functions
5.5 Legendre’s Differential Equation 88
5.6 Legendre Polynomials and Legendre Series Expansions 91
6 Sturm–Liouville Theory with Engineering Applications 94
6.1 Orthogonal Functions 94
6.2 Sturm–Liouville Theory 96
6.3 The Hanging Chain 99
6.4 Fourth Order Sturm–Liouville Theory 101
6.6 The Biharmonic Operator 103
6.7 Vibrations of Circular Plates 104
, iv Contents
7 The Fourier Transform and Its Applications 105
7.1 The Fourier Integral Representation 105
7.2 The Fourier Transform 107
7.3 The Fourier Transform Method 112
7.4 The Heat Equation and Gauss’s Kernel 116
7.5 A Dirichlet Problem and the Poisson Integral Formula 122
7.6 The Fourier Cosine and Sine Transforms 124
7.7 Problems Involving Semi-Infinite Intervals 126
7.8 Generalized Functions 128
7.9 The Nonhomogeneous Heat Equation 133
7.10 Duhamel’s Principle 134
12 Green’s Functions and Conformal Mappings 136
12.1 Green’s Theorem and Identities 136
12.2 Harmonic Functions and Green’s Identities 138
12.3 Green’s Functions 139
12.4 Green’s Functions for the Disk and the Upper Half-Plane 140
12.5 Analytic Functions 141
12.6 Solving Dirichlet Problems with Conformal Mappings 146
12.7 Green’s Functions and Conformal Mappings 151
