Numerical Methods for Laplace Transform Inversion (eBook)

Contents 6
Preface 9
Acknowledgements 13
Notation 14
Basic Results 16
1.1 Introduction 16
1.2 Transforms of Elementary Functions 17
1.3 Transforms of Derivatives and Integrals 20
1.4 Inverse Transforms 23
1.5 Convolution 24
1.6 The Laplace Transforms of some Special Functions 26
1.7 Difference Equations and Delay Differential Equations 29
1.8 Multidimensional Laplace T ransforms 33
Inversion Formulae and Practical Results 38
2.1 The Uniqueness Property 38
2.2 The Bromwich Inversion Theorem 41
2.3 The Post-Widder Inversion Formula 52
2.4 Initial and Final Value Theorems 54
2.5 Series and Asymptotic Expansions 57
2.6 Parseval's Formulae 58
The Method of Series Expansion 60
3.1 Expansion as a Power Series 60
3.2 Expansion in terms of Orthogonal Polynomials 64
3.3 Multi-dimensional Laplace transform inversion 81
Quadrature Methods 86
4.1 Interpolation and Gaussian type Formulae 86
4.2 Evaluation of Trigonometric Integrals 90
4.3 Extrapolation Methods 92
4.4 Methods using the Fast Fourier Transform ( FFT ) 96
4.5 Hartley Transforms 106
4.6 Dahlquist's "Multigrid" extension of FFT 110
4.7 Inversion of two-dimensional transforms 115
Rational Approximation Methods 117
5.1 The Laplace Transform is Rational 117
5.2 The least squares approach to rational Approximation 120
5.3 Pade, Pade-type and Continued Fraction Approximations 125
5.4 Multidimensional Laplace Transforms 133
The Method of Talbot 135
6.1 Early Formulation 135
6.2 A more general formulation 137
6.3 Choice of Parameters 139
6.4 Additional Practicalities 143
6.5 Subsequent development of Talbot's method 144
6.6 Multi-precision Computation 152
Methods based on the Post - Widder Inversion Formula 154
7.1 Introduction 154
7.2 Methods akin to Post-Widder 156
7.3 Inversion of Two-dimensional Transforms 159
The Method of Regularization 160
8.1 Introduction 160
8.2 Fredholm equations of the first kind - theoretical considerations 161
8.3 The method of Regularization 163
8.4 Application to Laplace Transforms 164
Survey Results 169
9.1 Cost's Survey 169
9.2 The Survey by Davies and Martin 170
9.3 Later Surveys 172
9.4 Test Transforms 180
Applications 181
10.1 Application 1. Transient solution for the Batch Service Queue M=MN=1 181
10.2 Application 2. Heat Conduction in a Rod 190
10.3 Application 3. Laser Anemometry 193
10.4 Application 4. Miscellaneous Quadratures 200
10.5 Application 5. Asian Options 204
Appendix 209
11.1 T able of Laplace T ransforms 210
11.2 The Fast Fourier Transform (FFT) 216
11.3 Quadrature Rules 218
11.4 Extrapolation Techniques 224
11.5 Pade Approximation 232
11.6 The method of Steepest Descent 238
11.7 Gerschgorin's theorems and the Companion Matrix 239
Bibliography 242
Index 260