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An improved Talbot method for numerical Laplace transform inversion

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Abstract

The classical Talbot method for the computation of the inverse Laplace transform is improved for the case where the transform is analytic in the complex plane except for the negative real axis. First, by using a truncated Talbot contour rather than the classical contour that goes to infinity in the left half-plane, faster convergence is achieved. Second, a control mechanism for improving numerical stability is introduced.

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References

  1. Abate, J., Valkó, P.P.: NumericalLaplaceInversion (2002). http://library.wolfram.com/infocenter/MathSource/4738/

  2. Abate, J., Valkó, P.P.: Multi-precision Laplace transform inversion. Int. J. Numer. Methods Fluids 60 (5), 979–993 (2004)

    Article  MATH  Google Scholar 

  3. Butcher, J.C.: On the numerical inversion of Laplace and Mellin transforms. Conference on Data Processing and Automatic Computing Machines,Salisbury,Australia (1957)

  4. Dingfelder, B.: Raten- und stabilitätsoptimierte Algorithmen zur Berechnung eines Integrals mit Hankelintegrationsweg. Bachelor’s thesis, Technische Universität München,Germany (2012)

  5. Dingfelder, B., Weideman, J.A.C.: ModifiedTalbot (2014). http://arxiv.org/e-print/1304.2505v3

  6. Duffy, D.G.: On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications. ACM Trans. Math. Software 19 (3), 333–359 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gavrilyuk, I.P., Makarov, V.L.: Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces. SIAM J. Numer. Ana.l 43 (5), 2144–2171 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kress, R.: Numerical analysis. Springer-Verlag, New York (1998)

    Book  MATH  Google Scholar 

  9. López-Fernández, M., Palencia, C.: On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51 (2-3), 289–303 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. López-Fernández, M., Palencia, C., Schädle, A.: A spectral order method for inverting sectorial Laplace transforms. SIAM J. Numer. Anal 44 (3), 1332–1350 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. McClure, T. talbot_inversion (2013). http://www.mathworks.com/matlabcentral/fileexchange/39035-numerical-inverse-laplace-transform/

  12. Murli, A., Rizzardi, M.: Algorithm 682: Talbot’s method for the Laplace inversion problem. ACM Trans. Math. Software 16 (2), 158–168 (1990)

    Article  MATH  Google Scholar 

  13. Rizzardi, M.: A modification of Talbot’s method for the simultaneous approximation of several values of the inverse Laplace transform. ACM Trans. Math. Software 21 (4), 347–371 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  14. Sheen, D., Sloan, I.H., Thomée, V.: A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature. IMA J. Numer. Anal. 23 (2), 269–299 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Strang, G.: Computational science and engineering. Wellesley-Cambridge Press, Wellesley (2007)

    MATH  Google Scholar 

  16. Talbot, A.: The accurate numerical inversion of Laplace transforms. J. Inst. Math. Appl 23 (1), 97–120 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  17. Trefethen, L.N.: Private communication (2006)

  18. Trefethen, L.N., Weideman, J.A.C., Schmelzer, T.: Talbot quadratures and rational approximations. BIT 46 (3), 653–670 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Weideman, J.A.C.: Optimizing Talbot’s contours for the inversion of the Laplace transform. SIAM J. Numer. Anal 44 (6), 2342–2362 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Weideman, J.A.C.: Improved contour integral methods for parabolic PDEs. IMA J. Numer. Anal. 30 (1), 334–350 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Weideman, J.A.C., Trefethen, L.N.: Parabolic and hyperbolic contours for computing the Bromwich integral. Math. Comp. 76 (259), 1341–1356 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Zentrum Mathematik, M3, Technische Universität München, Boltzmannstrasse 3, 85748, Garching bei München, Germany

    Benedict Dingfelder

  2. Department of Mathematical Sciences, Stellenbosch University, Stellenbosch, 7600, South Africa

    J. A. C. Weideman

Authors
  1. Benedict Dingfelder
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  2. J. A. C. Weideman
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Corresponding author

Correspondence to J. A. C. Weideman.

Additional information

The research of BD was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics.” He further acknowledges support from the graduate program TopMath of the Elite Network of Bavaria and the TopMath Graduate Center of TUM Graduate School at Technische Universität München. The research of JACW was supported by the National Research Foundation of South Africa.

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Dingfelder, B., Weideman, J.A.C. An improved Talbot method for numerical Laplace transform inversion. Numer Algor 68, 167–183 (2015). https://doi.org/10.1007/s11075-014-9895-z

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