Cover......Page 1
Numerical Methods and Algorithms Volume 5......Page 3
Numerical Methods for Laplace Transform Inversion......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 9
Acknowledgements......Page 13
Notation......Page 14
1.1 Introduction......Page 16
1.2 Transforms of Elementary Functions......Page 17
1.2.1 Elementary Properties of Transforms......Page 18
1.3 Transforms of Derivatives and Integrals......Page 20
1.4 Inverse Transforms......Page 23
1.5 Convolution......Page 24
1.6 The Laplace Transforms of some Special Functions......Page 26
1.7 Difference Equations and Delay Differential Equations......Page 29
1.7.1 z-Transforms......Page 31
1.8 Multidimensional Laplace Transforms......Page 33
2.1 The Uniqueness Property......Page 38
2.2 The Bromwich Inversion Theorem......Page 41
2.3 The Post-Widder Inversion Formula......Page 52
2.4 Initial and Final Value Theorems......Page 54
2.5 Series and Asymptotic Expansions......Page 57
2.6 Parseval's Formulae......Page 58
3.1 Expansion as a Power Series......Page 60
3.2 Expansion in terms of Orthogonal Polynomials......Page 64
3.2.1 Legendre Polynomials......Page 65
3.2.2 Chebyshev Polynomials......Page 67
3.2.3 Laguerre Polynomials......Page 70
3.2.4 The method of Weeks......Page 73
3.3 Multi-dimensional Laplace transform inversion......Page 81
4.1 Interpolation and Gaussian type Formulae......Page 86
4.2 Evaluation of Trigonometric Integrals......Page 90
4.3.1 The P -transformation of Levin......Page 92
4.3.2 The Sidi mW-Transformation for the Bromwich integral......Page 93
4.4 Methods using the Fast Fourier Transform (FFT)......Page 96
4.5 Hartley Transforms......Page 106
4.6 Dahlquist's "Multigrid" extension of FFT......Page 110
4.7 Inversion of two-dimensional transforms......Page 115
5.1 The Laplace Transform is Rational......Page 118
5.2 The least squares approach to rational Approximation......Page 121
5.2.1 Sidi's Window Function......Page 123
5.2.2 The Cohen-Levin Window Function......Page 124
5.3 Pade, Pade-type and Continued Fraction Approximations......Page 126
5.3.1 Prony's method and z-transforms......Page 131
5.3.2 The Method of Grundy......Page 133
5.4 Multidimensional Laplace Transforms......Page 134
6.1 Early Formulation......Page 136
6.2 A more general formulation......Page 138
6.3 Choice of Parameters......Page 140
6.4 Additional Practicalities......Page 144
6.5.1 Piessens' method......Page 145
6.5.2 The Modification of Murli and Rizzardi......Page 147
6.5.3 Modifications of Evans et al......Page 148
6.5.4 The Parallel Talbot Algorithm......Page 152
6.6 Multi-precision Computation......Page 153
7.1 Introduction......Page 156
7.2 Methods akin to Post-Widder......Page 158
7.3 Inversion of Two-dimensional Transforms......Page 161
8.1 Introduction......Page 162
8.2 Fredholm equations of the first kind -- theoretical considerations......Page 163
8.3 The method of Regularization......Page 165
8.4 Application to Laplace Transforms......Page 166
9.1 Cost's Survey......Page 172
9.2 The Survey by Davies and Martin......Page 173
9.3.1 Narayanan and Beskos......Page 175
9.3.3 D'Amore, Laccetti and Murli......Page 176
9.3.4 Cohen......Page 177
9.4 Test Transforms......Page 183
10.1 Application 1. Transient solution for the Batch Service Queue M/M^N/1......Page 184
10.2 Application 2. Heat Conduction in a Rod......Page 193
10.3 Application 3. Laser Anemometry......Page 196
10.4 Application 4. Miscellaneous Quadratures......Page 203
10.5 Application 5. Asian Options......Page 207
11. Appendix......Page 212
11.1 Table of Laplace Transforms......Page 213
11.1.1 Table of z-Transforms......Page 218
11.2 The Fast Fourier Transform (FFT)......Page 219
11.3 Quadrature Rules......Page 221
11.4 Extrapolation Techniques......Page 227
11.5 Pade Approximation......Page 235
11.5.1 Continued Fractions. Thiele's method......Page 238
11.6 The method of Steepest Descent......Page 241
11.7 Gerschgorin's theorems and the Companion Matrix......Page 242
Bibliography......Page 246
Index......Page 264