×

Parabolic and hyperbolic contours for computing the Bromwich integral. (English) Zbl 1113.65119

Summary: Some of the most effective methods for the numerical inversion of the Laplace transform are based on the approximation of the Bromwich contour integral. The accuracy of these methods often hinges on a good choice of contour, and several such contours have been proposed in the literature. Here we analyze two recently proposed contours, namely a parabola and a hyperbola. Using a representative model problem, we determine estimates for the optimal parameters that define these contours. An application to a fractional diffusion equation is presented.

MSC:

65R10 Numerical methods for integral transforms
44A10 Laplace transform
45K05 Integro-partial differential equations
35K05 Heat equation
26A33 Fractional derivatives and integrals
35A22 Transform methods (e.g., integral transforms) applied to PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ivan P. Gavrilyuk and Vladimir L. Makarov, Exponentially convergent parallel discretization methods for the first order evolution equations, Comput. Methods Appl. Math. 1 (2001), no. 4, 333 – 355. · Zbl 0998.65056
[2] M. López-Fernández and C. Palencia, On the numerical inversion of the Laplace transform of certain holomorphic mappings, Appl. Numer. Math. 51 (2004), no. 2-3, 289 – 303. · Zbl 1059.65120
[3] María López-Fernández, César Palencia, and Achim Schädle, A spectral order method for inverting sectorial Laplace transforms, SIAM J. Numer. Anal. 44 (2006), no. 3, 1332 – 1350. · Zbl 1124.65120
[4] Francesco Mainardi, Gianni Pagnini, and R. K. Saxena, Fox \? functions in fractional diffusion, J. Comput. Appl. Math. 178 (2005), no. 1-2, 321 – 331. · Zbl 1061.33012
[5] Erich Martensen, Zur numerischen Auswertung uneigenlicher Integrale, Z. Angew. Math. Mech. 48 (1968), T83 – T85 (German).
[6] William McLean and Vidar Thomée, Time discretization of an evolution equation via Laplace transforms, IMA J. Numer. Anal. 24 (2004), no. 3, 439 – 463. · Zbl 1068.65146
[7] Mariarosaria Rizzardi, A modification of Talbot’s method for the simultaneous approximation of several values of the inverse Laplace transform, ACM Trans. Math. Software 21 (1995), no. 4, 347 – 371. · Zbl 0887.65133
[8] H.-R. Schwarz, Numerical analysis, John Wiley & Sons, Ltd., Chichester, 1989. A comprehensive introduction; With a contribution by J. Waldvogel; Translated from the German. · Zbl 0715.65003
[9] Dongwoo Sheen, Ian H. Sloan, and Vidar Thomée, A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature, IMA J. Numer. Anal. 23 (2003), no. 2, 269 – 299. · Zbl 1022.65108
[10] A. Talbot, The accurate numerical inversion of Laplace transforms, J. Inst. Math. Appl. 23 (1979), no. 1, 97 – 120. · Zbl 0406.65054
[11] Lloyd N. Trefethen, Spectral methods in MATLAB, Software, Environments, and Tools, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. · Zbl 0953.68643
[12] L. N. Trefethen, J. A. C. Weideman, and T. Schmelzer, Talbot quadratures and rational approximations, BIT 46 (2006), no. 3, 653 – 670. · Zbl 1103.65030
[13] J. A. C. Weideman, Computing special functions via inverse Laplace transforms, International Conference on Numerical Analysis and Applied Mathematics 2005 (Rhodes) , Wiley-VCH, 2005, pp. 702-704. · Zbl 1088.65517
[14] -, Optimizing Talbot’s contours for the inversion of the Laplace transform, SIAM J. Numer. Anal. 44 (2006), no. 6, 2342-2362. · Zbl 1131.65105
[15] J. A. C. Weideman and S. C. Reddy, A MATLAB differentiation matrix suite, ACM Trans. Math. Software 26 (2000), no. 4, 465 – 519.
[16] J. A. C. Weideman and L. N. Trefethen, The eigenvalues of second-order spectral differentiation matrices, SIAM J. Numer. Anal. 25 (1988), no. 6, 1279 – 1298. · Zbl 0666.65063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.