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Numerical Methods for Laplace Transform Inversion

:	Alan M. Cohen
:	Numerical Methods for Laplace Transform Inversion
:	Springer-Verlag
:	9780387688558
:	1
:	CHF 110.30
:

:	Analysis
:	English

:	252
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

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.

	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	106
	4.6 Dahlquist's	106
	110	106
	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