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```Contents
Preface i
1 First Order Partial Differential Equations 1
1.1 Curves and Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Genesis of First Order P.D.E. . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Classification of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Linear Equations of the First Order . . . . . . . . . . . . . . . . . . . 13
1.5 Pfaffian Differential Equations . . . . . . . . . . . . . . . . . . . . . . 18
1.6 Compatible Systems of First order P.D.E. . . . . . . . . . . . . . . . 24
1.7 Charpit¿s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.8 Jacobi¿s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.9 Integral Surfaces Through a Given Curve . . . . . . . . . . . . . . . . 42
1.10 Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.11 Non-linear First Order P.D.E. . . . . . . . . . . . . . . . . . . . . . . 59
2 Second Order Partial Differential Equations 77
2.1 Genesis of Second Order P.D.E. . . . . . . . . . . . . . . . . . . . . . 77
2.2 Classification of Second Order P.D.E. . . . . . . . . . . . . . . . . . . 80
2.3 One Dimensional Wave Equation . . . . . . . . . . . . . . . . . . . . 86
2.3.1 Vibrations of an Infinite String . . . . . . . . . . . . . . . . . 86
2.3.2 Vibrations of a Semi-infinite String . . . . . . . . . . . . . . . 90
2.3.3 Vibrations of a String of Finite Length . . . . . . . . . . . . . 92
2.3.4 Riemann¿s Method . . . . . . . . . . . . . . . . . . . . . . . . 94
2.3.5 Vibrations of a String of Finite Length
(Method of Separation of Variables) . . . . . . . . . . . . . . . 99
2.4 Laplace¿s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.4.1 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . 103
2.4.2 Maximum and Minimum Principles . . . . . . . . . . . . . . . 104
2.4.3 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . 107
2.4.4 The Dirichlet Problem for the Upper Half Plane . . . . . . . . 108
2.4.5 The Neumann Problem for the Upper Half Plane . . . . . . . 109
2.4.6 The Dirichlet Problem for a Circle . . . . . . . . . . . . . . . 110
2.4.7 The Dirichlet Exterior Problem for a Circle . . . . . . . . . . 114
2.4.8 The Neumann Problem for a Circle . . . . . . . . . . . . . . . 115
2.4.9 The Dirichlet Problem for a Rectangle . . . . . . . . . . . . . 116
2.4.10 Harnack¿s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 118
2.4.11 Laplace¿s Equation ¿ Green¿s Function . . . . . . . . . . . . 119
2.4.12 The Dirichlet Problem for a Half Plane . . . . . . . . . . . . . 120
2.4.13 The Dirichlet Problem for a Circle . . . . . . . . . . . . . . . 121
2.5 Heat Conduction Problem . . . . . . . . . . . . . . . . . . . . . . . . 122
2.5.1 Heat Conduction ¿ Infinite Rod Case . . . . . . . . . . . . . 122
2.5.2 Heat Conduction ¿ Finite Rod Case . . . . . . . . . . . . . . 124
2.6 Duhamel¿s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
2.6.1 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 128
2.6.2 Heat Conduction Equation . . . . . . . . . . . . . . . . . . . . 130
2.7 Classification in the Case of n-Variables . . . . . . . . . . . . . . . . . 132
2.8 Families of Equipotential Surfaces . . . . . . . . . . . . . . . . . . . . 137
2.9 Kelvin¿s Inversion Theorem . . . . . . . . . . . . . . . . . . . . . . . 139
A Fourier Transforms and Integrals 141
A.1 Fourier Integral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 141
A.2 Properties of the Fourier Transform . . . . . . . . . . . . . . . . . . . 141
A.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
A.4 Convolution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
A.5 Vibrations of an Infinite String . . . . . . . . . . . . . . . . . . . . . 144