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Fully numerical Laplace transform methods

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Abstract

The role of the Laplace transform in scientific computing has been predominantly that of a semi-numerical tool. That is, typically only the inverse transform is computed numerically, with all steps leading up to that executed by analytical manipulations or table look-up. Here, we consider fully numerical methods, where both forward and inverse transforms are computed numerically. Because the computation of the inverse transform has been studied extensively, this paper focus mainly on the forward transform. Existing methods for computing the forward transform based on exponential sums are considered along with a new method based on the formulas of Weeks. Numerical examples include a nonlinear integral equation of convolution type, a fractional ordinary differential equation, and a partial differential equation with an inhomogeneous boundary condition.

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Acknowledgements

Annie Cuyt, Volker Mehrmann, Gerlind Plonka, Daniel Potts, and Rina-Marí Weideman generously shared advice and software related to exponential sum approximations.

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Authors and Affiliations

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

    J. A. C. Weideman

  2. Department of Applied Mathematics, University of Colorado, Boulder, CO, 80309, USA

    Bengt Fornberg

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Correspondence to J. A. C. Weideman.

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Appendix: MATLAB code for computing Laguerre coefficients

Appendix: MATLAB code for computing Laguerre coefficients

The following code computes the expansion coefficients (8). It uses Gauss-Laguerre quadrature, the computation of which is done here by a function from Chebfun [11]. The Laguerre polynomials are computed by the well-known three-term recursion [10, sect. 18.9], which is stable in the forward direction. For large n the improved methods of [14] should be considered.

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Weideman, J.A.C., Fornberg, B. Fully numerical Laplace transform methods. Numer Algor 92, 985–1006 (2023). https://doi.org/10.1007/s11075-022-01368-x

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