| Acknowledgments |
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v | |
| Preface |
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vii | |
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Potential, Heat, and Wave Equation |
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1 | (24) |
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1 | (1) |
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Classification of second order equations |
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2 | (1) |
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Laplace's and Poisson's equation |
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3 | (8) |
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11 | (7) |
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18 | (7) |
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Basic Approximation Theory |
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25 | (20) |
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26 | (3) |
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Projection and best approximation |
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29 | (5) |
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Important function spaces |
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34 | (11) |
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Sturm--Liouville Problems |
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45 | (22) |
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Sturm--Liouville problems for φ = μφ |
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45 | (8) |
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Sturm--Liouville problems for Lφ = μφ |
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53 | (6) |
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A Sturm--Liouville problem with an interface |
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59 | (8) |
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67 | (28) |
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67 | (1) |
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68 | (6) |
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Convergence of Fourier series |
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74 | (3) |
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77 | (3) |
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Operations on Fourier series |
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80 | (4) |
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Partial sums of the Fourier series and the Gibbs phenomenon |
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84 | (11) |
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Eigenfunction Expansions for Equations in Two Independent Variables |
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95 | (20) |
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One-Dimensional Diffusion Equation |
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115 | (46) |
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Applications of the eigenfunction expansion method |
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115 | (36) |
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Example 6.1 How many terms of the series solution are enough? |
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115 | (4) |
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Example 6.2 Determination of an unknown diffusivity from measured data |
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119 | (1) |
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Example 6.3 Thermal waves |
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120 | (5) |
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Example 6.4 Matching a temperature history |
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125 | (4) |
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Example 6.5 Phase shift for a thermal wave |
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129 | (2) |
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Example 6.6 Dynamic determination of a convective heat transfer coefficient from measured data |
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131 | (3) |
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Example 6.7 Radial heat flow in a sphere |
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134 | (3) |
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Example 6.8 A boundary layer problem |
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137 | (2) |
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Example 6.9 The Black--Scholes equation |
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139 | (3) |
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Example 6.10 Radial heat flow in a disk |
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142 | (4) |
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Example 6.11 Heat flow in a composite slab |
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146 | (2) |
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Example 6.12 Reaction-diffusion with blowup |
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148 | (3) |
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Convergence of uN (x, t) to the analytic solution |
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151 | (4) |
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Influence of the boundary conditions and Duhamel's solution |
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155 | (6) |
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One-Dimensional Wave Equation |
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161 | (34) |
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Applications of the eigenfunction expansion method |
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161 | (27) |
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Example 7.1 A vibrating string with initial displacement |
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161 | (5) |
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Example 7.2 A vibrating string with initial velocity |
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166 | (2) |
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Example 7.3 A forced wave and resonance |
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168 | (3) |
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Example 7.4 Wave propagation in a resistive medium |
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171 | (4) |
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Example 7.5 Oscillations of a hanging chain |
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175 | (2) |
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Example 7.6 Symmetric pressure wave in a sphere |
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177 | (3) |
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Example 7.7 Controlling the shape of a wave |
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180 | (2) |
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Example 7.8 The natural frequencies of a uniform beam |
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182 | (3) |
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Example 7.9 A system of wave equations |
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185 | (3) |
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Convergence of uN (x, t) to the analytic solution |
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188 | (2) |
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Eigenfunction expansions and Duhamel's principle |
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190 | (5) |
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Potential Problems in the Plane |
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195 | (60) |
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Applications of the eigenfunction expansion method |
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195 | (42) |
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Example 8.1 The Dirichlet problem for the Laplacian on a rectangle |
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195 | (6) |
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Example 8.2 Preconditioning for general boundary data |
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201 | (12) |
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Example 8.3 Poisson's equation with Neumann boundary data |
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213 | (2) |
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Example 8.4 A discontinuous potential |
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215 | (3) |
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Example 8.5 Lubrication of a plane slider bearing |
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218 | (2) |
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Example 8.6 Lubrication of a step bearing |
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220 | (1) |
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Example 8.7 The Dirichlet problem on an L-shaped domain |
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221 | (4) |
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Example 8.8 Poisson's equation in polar coordinates |
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225 | (5) |
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Example 8.9 Steady-state heat flow around an insulated pipe I |
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230 | (2) |
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Example 8.10 Steady-state heat flow around an insulated pipe II |
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232 | (2) |
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Example 8.11 Poisson's equation on a triangle |
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234 | (3) |
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Eigenvalue problem for the two-dimensional Laplacian |
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237 | (10) |
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Example 8.12 The eigenvalue problem for the Laplacian on a rectangle |
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237 | (2) |
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Example 8.13 The Green's function for the Laplacian on a square |
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239 | (4) |
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Example 8.14 The eigenvalue problem for the Laplacian on a disk |
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243 | (1) |
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Example 8.15 The eigenvalue problem for the Laplacian on the surface of a sphere |
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244 | (3) |
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Convergence of uN(x, y) to the analytic solution |
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247 | (8) |
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Multidimensional Problems |
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255 | (22) |
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Applications of the eigenfunction expansion method |
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255 | (10) |
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Example 9.1 A diffusive pulse test |
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255 | (3) |
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Example 9.2 Standing waves on a circular membrane |
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258 | (2) |
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Example 9.3 The potential inside a charged sphere |
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260 | (1) |
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Example 9.4 Pressure in a porous slider bearing |
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261 | (4) |
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The eigenvalue problem for the Laplacian in R3 |
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265 | (12) |
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Example 9.5 An eigenvalue problem for quadrilaterals |
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265 | (1) |
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Example 9.6 An eigenvalue problem for the Laplacian in a cylinder |
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266 | (1) |
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Example 9.7 Periodic heat flow in a cylinder |
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267 | (2) |
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Example 9.8 An eigenvalue problem for the Laplacian in a sphere |
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269 | (2) |
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Example 9.9 The eigenvalue problem for Schrodinger's equation with a spherically symmetric potential well |
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271 | (6) |
| Bibliography |
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277 | (2) |
| Index |
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279 | |
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