Cohen Numerical Methods for Laplace Transform Inversion

2007
ISBN: 978-0-387-68855-8
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 5, 252 Seiten

Reihe: Numerical Methods and Algorithms

This book gives background material on the theory of Laplace transforms, together with a fairly comprehensive list of methods that are available at the current time. Computer programs are included for those methods that perform consistently well on a wide range of Laplace transforms. Operational methods have been used for over a century to solve problems such as ordinary and partial differential equations.

Autoren/Hrsg.

Cohen, Alan M.

Inhaltsverzeichnis

1;Contents;6
2;Preface;9
3;Acknowledgements;13
4;Notation;14
5;Basic Results;16
5.1;1.1 Introduction;16
5.2;1.2 Transforms of Elementary Functions;17
5.3;1.3 Transforms of Derivatives and Integrals;20
5.4;1.4 Inverse Transforms;23
5.5;1.5 Convolution;24
5.6;1.6 The Laplace Transforms of some Special Functions;26
5.7;1.7 Difference Equations and Delay Differential Equations;29
5.8;1.8 Multidimensional Laplace T ransforms;33
6;Inversion Formulae and Practical Results;38
6.1;2.1 The Uniqueness Property;38
6.2;2.2 The Bromwich Inversion Theorem;41
6.3;2.3 The Post-Widder Inversion Formula;52
6.4;2.4 Initial and Final Value Theorems;54
6.5;2.5 Series and Asymptotic Expansions;57
6.6;2.6 Parseval's Formulae;58
7;The Method of Series Expansion;60
7.1;3.1 Expansion as a Power Series;60
7.2;3.2 Expansion in terms of Orthogonal Polynomials;64
7.3;3.3 Multi-dimensional Laplace transform inversion;81
8;Quadrature Methods;86
8.1;4.1 Interpolation and Gaussian type Formulae;86
8.2;4.2 Evaluation of Trigonometric Integrals;90
8.3;4.3 Extrapolation Methods;92
8.4;4.4 Methods using the Fast Fourier Transform ( FFT );96
8.5;4.5 Hartley Transforms;106
8.6;4.6 Dahlquist's "Multigrid" extension of FFT;110
8.7;4.7 Inversion of two-dimensional transforms;115
9;Rational Approximation Methods;117
9.1;5.1 The Laplace Transform is Rational;117
9.2;5.2 The least squares approach to rational Approximation;120
9.3;5.3 Pade, Pade-type and Continued Fraction Approximations;125
9.4;5.4 Multidimensional Laplace Transforms;133
10;The Method of Talbot;135
10.1;6.1 Early Formulation;135
10.2;6.2 A more general formulation;137
10.3;6.3 Choice of Parameters;139
10.4;6.4 Additional Practicalities;143
10.5;6.5 Subsequent development of Talbot's method;144
10.6;6.6 Multi-precision Computation;152
11;Methods based on the Post - Widder Inversion Formula;154
11.1;7.1 Introduction;154
11.2;7.2 Methods akin to Post-Widder;156
11.3;7.3 Inversion of Two-dimensional Transforms;159
12;The Method of Regularization;160
12.1;8.1 Introduction;160
12.2;8.2 Fredholm equations of the first kind - theoretical considerations;161
12.3;8.3 The method of Regularization;163
12.4;8.4 Application to Laplace Transforms;164
13;Survey Results;169
13.1;9.1 Cost's Survey;169
13.2;9.2 The Survey by Davies and Martin;170
13.3;9.3 Later Surveys;172
13.4;9.4 Test Transforms;180
14;Applications;181
14.1;10.1 Application 1. Transient solution for the Batch Service Queue M=MN=1;181
14.2;10.2 Application 2. Heat Conduction in a Rod;190
14.3;10.3 Application 3. Laser Anemometry;193
14.4;10.4 Application 4. Miscellaneous Quadratures;200
14.5;10.5 Application 5. Asian Options;204
15;Appendix;209
15.1;11.1 T able of Laplace T ransforms;210
15.2;11.2 The Fast Fourier Transform (FFT);216
15.3;11.3 Quadrature Rules;218
15.4;11.4 Extrapolation Techniques;224
15.5;11.5 Pade Approximation;232
15.6;11.6 The method of Steepest Descent;238
15.7;11.7 Gerschgorin's theorems and the Companion Matrix;239
16;Bibliography;242
17;Index;260