A Handbook of Methods of Approximate Fourier Transformation and Inverse of the Laplace Transformation

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V. I. Krylov N.S.Skoblya

A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation Mir Publishers Moscow

B. H. K p b ljio n , H. G. CKOBJIH

METOflM UPHBJIMJKEHHOrO IIPE0 BPA3 0 BAHHH (PYPLE H OBPAmEHHH nP E 0 EPA3 0 BAHHH JIAnjIACA CIIPABOMHAJI K H H rA Ila^aTejitcTBO «HayKa» M o ck b s

V. I. Krylov and N. S. Skoblya

A HANDBOOK OF METHODS OF APPROXIMATE FOURIER TRANSFORMATION AND INVERSION OF THE LAPLACE TRANSFORMATION Translated from the Russian by George Yankovsky

MIR PUBLISHERS MOSCOW

First published 1977 Revised from the 1974 Russian edition

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© HaAaTejibCTBO «HayKa», 1974 © English translation, Mir Publishers, 1977

Contents

List of s y m b o l s ..................... Proface .............................................................

8 9

Part One INVERSION OF THE LAPLACE TRANSFORMATION Chapter 1. Introduction 1.1 Basic concepts in the theory of the Laplace transforma­ tion ........................................................ 1.2 Complex integral for computing inverse Laplace trans­ forms ........................................................ 1.3 Representing functions by the Laplace integral . . . . 1.4 Ill-conditioning of the problem of computing inverse Laplace transforms .....................................

15 22 25 30

Chapter 2. Some Analytical Methods for Computing Inverse Laplace Transforms 2.1 Finding the original function via the inversion formula 2.2 Expanding the original function into power series . . . 2.3 Expanding the original function into generalized power series .....................................................

32 36 38

Chapter 3. Methods of Numerical Inversion of Laplace Transforms Based on the Use of Special Expansions 3.1 Computing inverse Laplace transforms by polynomials or­ thogonal on a finite i n t e r v a l ............................. 3.2 Computing inverse Laplace transforms with the aid of the Fourier sine s e r i e s ................................... 3.3 Computing inverse Laplace transforms with the aid of series in terms of generalized Chebyshev-Laguerre po­ lynomials ..................................................

41 67 70 5

Chapter 4. Methods of Computing the Mellin Integral with the Aid of Interpolation Quadrature Formulas 4.1 4.2 4.3 4.4

The general theory of interpolation m e th o d s ........... The equal-interval interpolation m e t h o d ............. The unequal-interval interpolation m e t h o d ........... Other interpolation methods. Using the truncated Tay­ lor series .................................................. 4.5 Some theorems on convergence ofinterpolation . . . 4.6 Theorems on the convergence of interpolation methods ............................................. of inversion

75 79 80 90 92 105

Chapter 5. Methods of Numerical Inversion of Laplace Trans­ forms via Quadrature Formulas of Highest Accuracy 5.1 The theory of quadrature fo r m u la s........................ 5.2 Orthogonal polynomials connected with the quadrature formula of highest accuracy ........................... 5.3 Methods for computing the coefficients and points of a quadrature f o r m u l a ........................................

110 118 136

Chapter 6. Methods of Inverting Laplace Transforms via Quad­ rature Formulas with Equal Coefficients 6.1 Constructing a computation f o r m u l a ................... 6.2 Remark on the spacing of p o i n t s .....................

145 148

Part Two FOURIER TRANSFORMS AND THEIR APPLICATION TO INVERSION OF LAPLACE TRANSFORMS Chapter 7. Introduction ........................................ 7.1 Fourier transforms 7.2 Reducing integrals of the Mellin type to the Fourier transformation ...........................................

150 157

Chapter 8. Inversion of Laplace Transforms by Means of the Fourier Series 8.1 The case of a rapidly decreasing original function / (x) 8.2 The case of rapid decrease of the modulus of the image function F (p) 6

159 161

Clin pier 9. Interpolation Formulas for Computing Fourier Integ­ rals 9.1 Some preliminary remarks .............................. 9.2 Algebraic interpolation of the function f ( x ) ........... 9.3. Interpolation by rational fu n c tio n s..................

164 166 201

Cliupter 10. Highest-Accuracy Formulas for Computation 10.1 Introduction .......................................... 10.2 Constructing a formula of highest degree of accuracy .

233 236

Part Three ISOLATING SINGULARITIES OF A FUNCTION IN COMPUTATIONS Chapter 11. Isolating Singularities of the Image Function F (p ) 11.1 Introduction ........................................... 11.2 Removing and weakening the singularities of the image function F (p) 11.3 A remark on the increase in the rate of approach to zero of the image function F ( p ) ............................. 11.4 A table of image functions F (p) and the corresponding original functions / (x) for constructing the singular part of the image function Fx (p )

245 247 252 254

Chapter 12. Isolating Singularities of a Function in the Fourier Transformation 12.1 Removing discontinuities of the first k i n d ........... 12.2 Increasing the rate of approach to zero of the function undergoing tr a n sfo r m a tio n .............................

259

Bibliography ..................................................... Index .............................................................

266 268

263

List of Symbols

fit) F(P)