Abstract
This paper is devoted to designing a practical algorithm to invert the Laplace transform by assuming that the transform possesses the Puiseux expansion at infinity. First, the general asymptotic expansion of the inverse function at zero is derived, which can be used to approximate the inverse function when the variable is small. Second, an inversion algorithm is formulated by splitting the Bromwich integral into two parts. One is the main weakly oscillatory part, which is evaluated by a composite Gauss–Legendre rule and its Kronrod extension, and the other is the remaining strongly oscillatory part, which is integrated analytically using the Puiseux expansion of the transform at infinity. Finally, some typical tests show that the algorithm can be used to invert a wide range of Laplace transforms automatically with high accuracy and the output error estimator matches well with the true error.
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References
Abate, J., Valkó, P.P.: Multi-precision Laplace transform inversion. Int. J. Numer. Methods Eng. 60, 979–993 (2004)
Antonelli, L., Corsaro, S., Marino, Z., Rizzardi, M.: Algorithm 944: Talbot Suite: parallel implementations of Talbot’s method for the numerical inversion of Laplace transforms. ACM Trans. Math. Soft. 40, 4 (2014). Article 29 18 pages
Aroca, F., Ilardi, G., Lopez de Medrano, L.: Puiseux power series solutions for systems of equations. Int. J. Math. 21, 1439–1459 (2010)
Calvetti, D., Golub, G.H., Gragg, W.B., Reichel, L.: Computation of Gauss–Kronrod quadrature rules. Math. Comp. 69, 1035–1052 (2000)
Coffey, M.W.: A set of identities for a class of alternating binomial sums arising in computing applications. Util. Math. 76, 79–90 (2008)
Cohen, A.M.: Numerical Methods for Laplace Transform Inversion. Springer Science+Business Media, New York (2007)
Connon, D.F.: Various applications of the (exponential) complete Bell polynomials. arXiv:1001.2835 (2010)
Cuomo, S., D´Amore, L., Murli, A., Rizzardi, M.: Computation of the inverse Laplace transform based on a collocation method which uses only real values. J. Comput. Appl. Math. 198, 98–115 (2007)
D´Amore, L., Lacetti, G., Murli, A.: An implementation of a Fourier-series method for the numerical inversion of the Laplace transform. ACM Trans. Math. Soft. 25, 279–305 (1999)
D´Amore, L., Campagna, R., Mele, V., Murli, A.: RelaTIve. An Ansi C90 software package for the real Laplace transform inversion. Numer. Algorithms 63, 187–211 (2013)
Davis, B., Martin, B.: Numerical inversion of the Laplace transform: a survey and comparison of methods. J. Comput. Phys. 33, 1–32 (1979)
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, San Diego (1984)
Dingfelder, B., Weideman, J.A.C.: An improved Talbot method for numerical Laplace transform inversion. Numer. Algorithms 68, 167–183 (2015)
Dubner, R., Abate, J.: Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. J. ACM 15, 115–123 (1968)
Duffy, D.G.: On the numerical inversion of Laplace transform, comparison of three new methods on characteristic problems from applications. ACM Trans. Math. Soft. 19, 333–359 (1993)
Duffy, D.G.: Transform Methods for Solving Partial Differential Equations, 2nd edn. Chapman and Hill, Boca Raton (2004)
Ehrich, S.: High order error constants of Gauss–Kronrod quadrature formulas. Analysis 16, 335–345 (1996)
Ehrich, S.: Stieltjes Polynomials and the Error of Gauss-Kronrod Quadrature Formulas. In: Gautschi, W., Golub, G.H., Opfer, G (eds.) Applications and Computation of Orthogonal Polynomials, 57–77. Proceedings of the Conf. Oberwolfach, International Series Numerical Mathematics, 131. Birkhäuser, Basel (1999)
Garbow, B.S., Giunta, G., Lyness, J.N., Murli, A.: Software for an implementation of Weeks’ method for the inverse Laplace transform problem. ACM Trans. Math. Soft. 14, 163–170 (1988)
Gaver, J.D.P.: Observing stochastic processes and approximate transform inversion. Oper. Res. 14, 444–459 (1966)
Gonnet, P.: A review of error estimation in adaptive quadrature. ACM Comput. Surv. 44, 4 (2012). Article 22 36 pages
Gzyl, H., Tagliani, A., Milev, M.: Laplace transform inversion on the real line is truly ill-conditioned. Appl. Math. Comput. 219, 9805–9809 (2013)
Handelsman, R.A., Olmstead, W.E.: Asymptotic solution to a class of nonlinear Volterra integral equations. SIAM J. Appl. Math. 22, 373–384 (1972)
Hassanzadeh, H., Pooladi-Darvish, M.: Comparison of different numerical Laplace inversion methods for engineering applications. Appl. Math. Comput. 189, 1966–1981 (2007)
Iqbal, M.: On a numerical technique regarding inversion of the Laplace transform. J. Comput. Appl. Math. 59, 145–154 (1995)
Kuhlman, K.L.: Review of inverse Laplace transform algorithms for Laplace-space numerical approaches. Numer. Algorithms 63, 339–355 (2013)
Laurie, D.P.: Calculation of Gauss–Kronrod quadrature rules. Math. Comput. 66, 1133–1145 (1997)
Lee, J., Sheen, D.: An accurate numerical inversion of Laplace transforms based on the location of their poles. Comput. Math. Appl. 48, 1415–1423 (2004)
Levin, D.: Numerical inversion of the Laplace transform by accelerating the convergence of Bromwich’s integral. J. Comput. Appl. Math. 1, 247–250 (1975)
Lin, F.R., Liang, F.: Application of high order numerical quadratures to numerical inversion of the Laplace transform. Adv. Comput. Math. 36, 267–278 (2012)
Liu, Z.F., Wang, T.K., Gao, G.H.: A local fractional Taylor expansion and its computation for insufficiently smooth functions. E. Asian J. Appl. Math. 5, 176–191 (2015)
Lyness, J.N., Giunta, G.: A modification of the Weeks method for the numerical inversion of the Laplace transform. Math. Comp. 47, 313–322 (1986)
Massouros, P.G., Genin, G.M.: Algebraic inversion of the Laplace transform. Comput. Math. Appl. 50, 179–185 (2005)
Milovanović, G.V., Cvetković, A.S.: Numerical inversion of the Laplace transform. Facta Universitatis-series: Electronics and Energetics 20, 295–310 (2005)
Monegato, G.: An overview of the computational aspects of Kronrod quadrature rules. Numer. Algorithms 26, 173–196 (2001)
Murli, A., Rizzardi, M.: Algorithm 682: Talbot’s method for the Laplace inversion problem. ACM Trans. Math. Soft. 16, 158–168 (1990)
Naeeni, M.R., Campagna, R., Eskandari-Ghadi, M., Ardalan, A.A.: Performance comparison of numerical inversion methods for Laplace and Hankel integral transforms in engineering problems. Appl. Math. Comput. 250, 759–775 (2015)
Oberhettinger, F., Badii, L.: Tables of Laplace Transforms. Springer-Verlag, Berlin (1973)
Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010). http://dlmf.nist.gov
Petras, K.: On the computation of the Gauss–Legendre quadrature formula with a given precision. J. Comput. Appl. Math. 112, 253–267 (1999)
Piessens, R.: Gaussian quadrature formulas for the numerical integration of Bromwich’s integral and the inversion of the Laplace transform. J. Eng. Math. 5, 1–9 (1971)
Poteaux, A., Rybowicz, M.: Good reduction of Puiseux series and applications. J. Symb. Comput. 47, 32–63 (2012)
Rizzardi, M.: A modification of Talbot’s method for the simultaneous approximation of several values of the inverse Laplace transform. ACM Trans. Math. Soft. 21, 347–371 (1995)
Schiff, J.L.: The Laplace Transform: Theory and Applications. Springer-Verlag, New York (1999)
Sellier, A.: Asymptotic expansions of a class of integrals. Proc. R. Soc. Lond. A Math. Phys. 445, 693–710 (1994)
Sidi, A.: Practical Extrapolation Methods–Theory and Applications. Cambridge University Press, Cambridge (2003)
Swarztrauber, P.N.: On computing the points and weights for Gauss–Legendre quadrature. SIAM J. Sci. Comput. 24, 945–954 (2002)
Talbot, A.: The accurate numerical inversion of Laplace transforms. IMA J. Appl. Math. 23, 97–120 (1979)
Valkó, P.P., Abate, J.: Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion. Comput. Math. Appl. 48, 629–636 (2004)
Wang, T.K., Li, N., Gao, G.H.: The asymptotic expansion and extrapolation of trapezoidal rule for integrals with fractional order singularities. Int. J. Comput. Math. 92, 579–590 (2015)
Wang, T.K., Liu, Z.F., Zhang, Z.Y.: The modified composite Gauss-type rules for singular integrals using Puiseux expansions. Math. Comp. 86, 345–373 (2017)
Wang, T.K., Zhang, Z.Y., Liu, Z.F.: The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions. Adv. Comput. Math. 43, 319–350 (2017)
Weeks, W.T.: Numerical inversion of the Laplace transform using Laguerre functions. J. ACM 13, 419–429 (1966)
Weideman, J.A.C.: Algorithms for parameter selection in the Weeks method for inverting the Laplace transform. SIAM J. Sci. Comput. 21, 111–128 (1999)
Weideman, J.A.C.: Optimizing Talbot’s contours for the inversion of the Laplace transform. SIAM J. Numer. Anal. 44, 2342–2362 (2006)
Weideman, J.A.C., Trefethen, L.N.: Parabolic and hyperbolic contours for computing the Bromwich integral. Math. Comp. 76, 1341–1356 (2007)
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The authors would like to thank the anonymous referees for their useful suggestions and valuable remarks, which significantly improve the quality of the paper.
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This project was partially supported by the National Natural Science Foundation of China (grant No. 11471166), Natural Science Foundation of Jiangsu Province (grant No. BK20141443) and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).
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Wang, T., Gu, Y. & Zhang, Z. An algorithm for the inversion of Laplace transforms using Puiseux expansions. Numer Algor 78, 107–132 (2018). https://doi.org/10.1007/s11075-017-0369-y
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DOI: https://doi.org/10.1007/s11075-017-0369-y
Keywords
- Inverse Laplace transform
- Bromwich integral
- Puiseux expansion
- Gauss–Legendre rule
- Gauss–Kronrod rule
- Error estimate