Preface | |
Linear Partial Differential Equations | p. 1 |
Linear problems | p. 1 |
Classification | p. 5 |
Well-posed problems | p. 9 |
Method of general solutions | p. 12 |
Method of separation of variables | p. 14 |
The Wave Equation | p. 21 |
The vibrating string | p. 21 |
The initial value problem | p. 24 |
The nonhomogeneous wave equation | p. 29 |
Uniqueness of the initial value problem | p. 31 |
Initial-boundary value problems | p. 33 |
Initial-boundary value problems for semi-infinite string | p. 39 |
Green's Function and Sturm-Liouville Problems | p. 45 |
Solutions of second order linear equations | p. 45 |
Boundary value problems and Green's function | p. 50 |
Sturm-Liouville problems | p. 58 |
Convergence in the mean | p. 62 |
Integral operator with continuous, symmetric kernel | p. 66 |
Completeness of eigenfunctions of Sturm-Liouville problems | p. 75 |
Nonhomogeneous integral equation | p. 78 |
Further properties of eigenvalues and eigenfunctions | p. 81 |
Fourier Series and Fourier Transforms | p. 95 |
Trigonometric Fourier series | p. 95 |
Uniform convergence and completeness | p. 99 |
Other types of Fourier series | p. 102 |
Application to the wave equation | p. 105 |
Fourier integrals | p. 109 |
Fourier transforms | p. 113 |
Contour integration | p. 117 |
The Heat Equation | p. 127 |
Derivation of the heat equation | p. 127 |
Maximum principle | p. 130 |
The initial-boundary value problem | p. 133 |
Nonhomogeneous problems and finite Fourier transform | p. 135 |
The initial value problem | p. 138 |
The initial value problem for the nonhomogeneous equation | p. 142 |
Nonhomogeneous boundary conditions for initial-boundary value problems | p. 145 |
Laplace's Equation and Poisson's Equation | p. 151 |
Boundary value problems | p. 151 |
Green's identities and uniqueness theorems | p. 152 |
Maximum principle | p. 154 |
Laplace's equation in a rectangle | p. 155 |
Laplace's equation in disc | p. 158 |
Poisson's integral formula | p. 162 |
Green's function for Laplace's equation | p. 166 |
Poisson's equation in a disc | p. 173 |
Finite Fourier transform for Poisson's equation | p. 177 |
Dirichlet problem in the upper-half plane | p. 181 |
Problems in Higher Dimensions | p. 191 |
Classification | p. 191 |
Double Fourier series | p. 194 |
Laplace's equation in a cube | p. 200 |
The two-dimensional wave equation in a rectangular domain | p. 202 |
Bessel functions | p. 205 |
Singular Sturm-Liouville problem for Bessel's equation | p. 209 |
The two-dimensional wave equation in a circular domain | p. 212 |
Initial-boundary value problems for the heat equation | p. 217 |
Legendre's equation | p. 221 |
Properties of Legendre polynomials | p. 224 |
Legendre series and boundary value problems | p. 229 |
Laplace's equation in a sphere | p. 232 |
Poisson's integral formula in space | p. 235 |
Ascoli's Theorem | p. 243 |
App. B: Answers for Selected Problems | p. 245 |
Bibliography | p. 255 |
Index | p. 256 |
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