Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Parabolic and hyperbolic contours for computing the Bromwich integral


Authors: J. A. C. Weideman and L. N. Trefethen
Journal: Math. Comp. 76 (2007), 1341-1356
MSC (2000): Primary 65D30, 44A10
DOI: https://doi.org/10.1090/S0025-5718-07-01945-X
Published electronically: March 7, 2007
MathSciNet review: 2299777
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: 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.


References [Enhancements On Off] (What's this?)

  • 1. I. P. Gavrilyuk and V. L. Makarov, Exponentially convergent parallel discretization methods for the first order evolution equations, Comput. Methods Appl. Math. 1 (2001), no. 4, 333-355. MR 1892950 (2003f:65174)
  • 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. MR 2091405 (2005e:65210)
  • 3. M. López-Fernández, C. Palencia, and A. Schädle, A spectral order method for inverting sectorial Laplace transforms, SIAM J. Numer. Anal. 44 (2006), no. 3, 1332-1350 (electronic). MR 2231867
  • 4. F. Mainardi, G. Pagnini, and R. K. Saxena, Fox $ H$ functions in fractional diffusion, J. Comput. Appl. Math. 178 (2005), no. 1-2, 321-331. MR 2127888 (2005m:26008)
  • 5. E. Martensen, Zur numerischen Auswertung uneigentlicher Integrale, Z. Angew. Math. Mech. 48 (1968), T83-T85. MR 0256565 (41:1221)
  • 6. W. McLean and V. Thomée, Time discretization of an evolution equation via Laplace transforms, IMA J. Numer. Anal. 24 (2004), no. 3, 439-463. MR 2068831 (2005d:47072)
  • 7. M. 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. MR 1364695 (96k:65084)
  • 8. H.-R. Schwarz, Numerical analysis, John Wiley & Sons Ltd., Chichester, 1989. MR 1005534 (90g:65003)
  • 9. D. Sheen, I.H. Sloan, and V. 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. MR 1975267 (2004b:65161)
  • 10. A. Talbot, The accurate numerical inversion of Laplace transforms, J. Inst. Math. Appl. 23 (1979), 97-120. MR 526286 (80c:65244)
  • 11. L. N. Trefethen, Spectral methods in MATLAB, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. MR 1776072 (2001c:65001)
  • 12. L. N. Trefethen, J. A. C. Weideman, and T. Schmelzer, Talbot quadratures and rational approximations, BIT 46 (2006), no. 3, 653-670. MR 2265580
  • 13. J. A. C. Weideman, Computing special functions via inverse Laplace transforms, International Conference on Numerical Analysis and Applied Mathematics 2005 (Rhodes) (T.E. Simos, G. Psihoyios, and Ch. Tsitouras, eds.), Wiley-VCH, 2005, pp. 702-704.
  • 14. -, Optimizing Talbot's contours for the inversion of the Laplace transform, SIAM J. Numer. Anal. 44 (2006), no. 6, 2342-2362.
  • 15. J. A. C. Weideman and S. C. Reddy, A MATLAB differentiation matrix suite, ACM Trans. Math. Software 26 (2000), no. 4, 465-519. MR 1939962 (2003g:65004)
  • 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. MR 972454 (90a:65228)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65D30, 44A10

Retrieve articles in all journals with MSC (2000): 65D30, 44A10


Additional Information

J. A. C. Weideman
Affiliation: Department of Applied Mathematics, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa
Email: weideman@dip.sun.ac.za

L. N. Trefethen
Affiliation: Oxford University Computing Laboratory, Wolfson Bldg., Parks Road, Oxford OX1 3QD, United Kingdom
Email: LNT@comlab.ox.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-07-01945-X
Keywords: Laplace transform, Talbot's method, trapezoidal rule, fractional differential equation
Received by editor(s): December 9, 2005
Published electronically: March 7, 2007
Additional Notes: The first author was supported by the National Research Foundation in South Africa under grant FA2005032300018
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society