Complex variable and regularization methods of inversion of the Laplace transform
HTML articles powered by AMS MathViewer
- by D. D. Ang, John Lund and Frank Stenger PDF
- Math. Comp. 53 (1989), 589-608 Request permission
Abstract:
In this paper three methods are derived for approximating f, given its Laplace transform g on $(0,\infty )$, i.e., $\smallint _0^\infty {f(t)\exp ( - st) dt = g(s)}$. Assuming that $g \in {L^2}(0,\infty )$, the first method is based on a Sinc-like rational approximation of g, the second on a Sinc solution of the integral equation $\smallint _0^\infty {f(t)\exp ( - st) dt = g(s)}$ via standard regularization, and the third method is based on first converting $\smallint _0^\infty {f(t)\exp ( - st){\mkern 1mu} dt = g(s)}$ to a convolution integral over $\mathbb {R}$, and then finding a Sinc approximation to f via the application of a special regularization procedure to solve the Fourier transform problem. We also obtain bounds on the error of approximation, which depend on both the method of approximation and the regularization parameter.References
- Richard Bellman, Robert E. Kalaba, and Jo Ann Lockett, Numerical inversion of the Laplace transform: Applications to biology, economics, engineering and physics, American Elsevier Publishing Co., Inc., New York, 1966. MR 0205454 B. S. Berger, "Inversion of the n-dimensional Laplace transform," Math. Comp., v. 20, 1966, pp. 418-421. B. S. Berger, "The inversion of the Laplace transform with application to the vibrations of continuous elastic bodies," J. Appl. Mech., v. 35, 1968, pp. 837-839. B. S. Berger & S. Duangudom, "A technique for increasing the accuracy of the numerical inversion of the Laplace transform with applications," J. Appl. Mech., v. 40, 1973, pp. 1110-1112.
- Claude Brezinski and Jeannette Van Iseghem, Padé-type approximants and linear functional transformations, Rational approximation and interpolation (Tampa, Fla., 1983) Lecture Notes in Math., vol. 1105, Springer, Berlin, 1984, pp. 100–108. MR 783264, DOI 10.1007/BFb0072402 J. W. Cooley, P. A. W. Lewis & P. D. Welch, "The fast Fourier transform algorithm: Programming considerations in the calculation of sine, cosine, and Laplace transforms," J. Sound Vibration, v. 12, 1970, pp. 315-337.
- James W. Cooley and John W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comp. 19 (1965), 297–301. MR 178586, DOI 10.1090/S0025-5718-1965-0178586-1
- Kenny S. Crump, Numerical inversion of Laplace transforms using a Fourier series approximation, J. Assoc. Comput. Mach. 23 (1976), no. 1, 89–96. MR 436552, DOI 10.1145/321921.321931
- A. R. Davies, On the maximum likelihood regularization of Fredholm convolution equations of the first kind, Treatment of integral equations by numerical methods (Durham, 1982) Academic Press, London, 1982, pp. 95–105. MR 755345
- Brian Davies and Brian Martin, Numerical inversion of the Laplace transform: a survey and comparison of methods, J. Comput. Phys. 33 (1979), no. 1, 1–32. MR 549576, DOI 10.1016/0021-9991(79)90025-1
- H. Dubner and J. Abate, Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform, J. Assoc. Comput. Mach. 15 (1968), 115–123. MR 235726, DOI 10.1145/321439.321446 A. Erdélyi, "Inversion formula for the Laplace transform," Philos. Mag., v. 34, 1943, pp. 533-537.
- William Feller, An introduction to probability theory and its applications. Vol. II. , 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
- D. S. Gilliam, J. R. Schulenberger, and J. R. Lund, Spectral representation of the Laplace transform and related operators in $L_2(\textbf {R}_+)$, Computational and combinatorial methods in systems theory (Stockholm, 1985) North-Holland, Amsterdam, 1986, pp. 69–73. MR 923997
- Richard R. Goldberg and Richard S. Varga, Moebius inversion of Fourier transforms, Duke Math. J. 23 (1956), 553–559. MR 80800
- S. Ȧ. Gustafson and G. Dahlquist, On the computation of slowly convergent Fourier integrals, Methoden und Verfahren der mathematischen Physik, Band 6, B. I.-Hochschultaschenbücher, No. 725, Bibliographisches Inst., Mannheim, 1972, pp. 93–112. MR 0359377
- Peter Henrici, Applied and computational complex analysis. Vol. 2, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Special functions—integral transforms—asymptotics—continued fractions. MR 0453984
- F. R. de Hoog, J. H. Knight, and A. N. Stokes, An improved method for numerical inversion of Laplace transforms, SIAM J. Sci. Statist. Comput. 3 (1982), no. 3, 357–366. MR 667833, DOI 10.1137/0903022
- Yasuhiko Ikebe, The Galerkin method for the numerical solution of Fredholm integral equations of the second kind, SIAM Rev. 14 (1972), 465–491. MR 307515, DOI 10.1137/1014071 R. E. Jones, Solving Linear Algebraic Systems Arising in the Solution of Integral Equations of the First Kind, Ph.D. Dissertation, University of New Mexico, 1985.
- I. M. Longman, Note on a method for computing infinite integrals of oscillatory functions, Proc. Cambridge Philos. Soc. 52 (1956), 764–768. MR 82193
- Yudell L. Luke, On the computation of oscillatory integrals, Proc. Cambridge Philos. Soc. 50 (1954), 269–277. MR 62518 J. Ross MacDonald, "Accelerated convergence, divergence, iteration, extrapolation, and curve fitting," J. Appl. Phys., v. 35, 1964, pp. 3034-3041.
- Martin J. Marsden and Gerald D. Taylor, Numerical evaluation of Fourier integrals, Numerische Methoden der Approximationstheorie, Band 1 (Tagung, Oberwolfach, 1971) Internat. Schriftenreihe Numer. Math., Band 16, Birkhäuser, Basel, 1972, pp. 61–76. MR 0386234
- J. T. Marti, On a regularization method for Fredholm equations of the first kind using Sobolev spaces, Treatment of integral equations by numerical methods (Durham, 1982) Academic Press, London, 1982, pp. 59–66. MR 755342
- Piero de Mottoni and Giorgio Talenti, Stabilization and error bounds for the inverse Laplace transform, Numer. Funct. Anal. Optim. 3 (1981), no. 3, 265–283. MR 629946, DOI 10.1080/01630568108816090
- M. Z. Nashed and Grace Wahba, Some exponentially decreasing error bounds for a numerical inversion of the Laplace transform, J. Math. Anal. Appl. 52 (1975), no. 3, 660–668. MR 431668, DOI 10.1016/0022-247X(75)90087-6 National Bureau of Standards, Handbook of Mathematical Functions, vol. 55, Applied Math. Series, Pitman, Boston, 1964.
- Athanasios Papoulis, A new method of inversion of the Laplace transform, Quart. Appl. Math. 14 (1957), 405–414. MR 82734, DOI 10.1090/S0033-569X-1957-82734-2
- R. Piessens, Numerical inversion of the Laplace transform, IEEE Trans. Automatic Control AC-14 (1969), 299–301. MR 0245178, DOI 10.1109/tac.1969.1099180
- Emil L. Post, Generalized differentiation, Trans. Amer. Math. Soc. 32 (1930), no. 4, 723–781. MR 1501560, DOI 10.1090/S0002-9947-1930-1501560-X
- Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
- Herbert E. Salzer, Tables for the numerical calculation of inverse Laplace transforms, J. Math. and Phys. 37 (1958), 89–109. MR 102907, DOI 10.1002/sapm195837189
- Herbert E. Salzer, Additional formulas and tables for orthogonal polynomials originating from inversion integrals, J. Math. and Phys. 40 (1961), 72–86. MR 129576
- R. A. Schapery, Approximate methods of transform inversion for viscoelastic stress analysis, Proc. 4th U.S. Nat. Congr. Appl. Mech. (Univ. California, Berkeley, Calif., 1962) Amer. Soc. Mech. Engrs., New York, 1962, pp. 1075–1085. MR 0153175
- C. J. Shirtliffe and D. G. Stephenson, A computer oriented adaption of Salzer’s method for inverting Laplace transforms, J. Math. and Phys. 40 (1961), 135–141. MR 150949
- Kishore Singhal and Jiri Vlach, Computation of time domain response by numerical inversion of the Laplace transform, J. Franklin Inst. 299 (1975), 109–126. MR 375744, DOI 10.1016/0016-0032(75)90133-7 M. Silverberg, "Improving the efficiency of Laplace transform inversion for network analysis," Electron. Lett., v. 6, 1970, pp. 105-106.
- Frank Stenger, Numerical methods based on Whittaker cardinal, or sinc functions, SIAM Rev. 23 (1981), no. 2, 165–224. MR 618638, DOI 10.1137/1023037
- Frank Stenger, Explicit, nearly optimal, linear rational approximation with preassigned poles, Math. Comp. 47 (1986), no. 175, 225–252. MR 842132, DOI 10.1090/S0025-5718-1986-0842132-0
- Andrey N. Tikhonov and Vasiliy Y. Arsenin, Solutions of ill-posed problems, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; John Wiley & Sons, New York-Toronto, Ont.-London, 1977. Translated from the Russian; Preface by translation editor Fritz John. MR 0455365
- Manfred R. Trummer, A method for solving ill-posed linear operator equations, SIAM J. Numer. Anal. 21 (1984), no. 4, 729–737. MR 749367, DOI 10.1137/0721049 D. V. Widder, The Laplace Transform, Princeton Univ. Press, Princeton, N.J., 1936. V. Zakian, "Numerical inversion of Laplace transform," Electron. Lett., v. 5, 1969, pp. 120-121. V. Zakian, "Optimization of numerical inversion of Laplace transform," Electron. Lett., v. 6, 1970, pp. 677-679.
- V. Zakian and D. R. Gannon, Least-squares optimisation of numerical inversion of Laplace transforms, Electron. Lett. 7 (1971), 70–71. MR 319370, DOI 10.1049/el:19710048
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Math. Comp. 53 (1989), 589-608
- MSC: Primary 65R10; Secondary 44A10
- DOI: https://doi.org/10.1090/S0025-5718-1989-0983558-7
- MathSciNet review: 983558