5 pdes

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Unformatted text preview: . . . . . . . . . 81 . 81 . 85 . 92 . 94 . 98 . 103 . 107 . 111 iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 PDE’s in Higher Dimensions 5.1 The three most important linear partial differential equations 5.2 The Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . 5.3 Initial value problems for heat equations . . . . . . . . . . . . 5.4 Two derivations of the wave equation . . . . . . . . . . . . . . 5.5 Initial value problems for wave equations . . . . . . . . . . . . 5.6 The Laplace operator in polar coordinates . . . . . . . . . . . 5.7 Eigenvalues of the Laplace operator . . . . . . . . . . . . . . 5.8 Eigenvalues of the disk . . . . . . . . . . . . . . . . . . . . . . 5.9 Fourier analysis for the circular vibrating membrane* . . . . . . . . . . . . . . . . . . . . . . 115 115 118 123 129 134 136 141 144 150 A Using Mathematica to solve differential equations 156 Index 163 iv Chapter 1 Power Series 1.1 What is a power series? Functions are often represented efficiently by means of infinite series. Examples we have seen in calculu...
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