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On an Initial-Value Method for Quickly Solving Volterra Integral Equations: A Review

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Solution Methods for Integral Equations

Part of the book series: Mathematical Concepts and Methods in Science and Engineering ((MCSENG,volume 18))

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Abstract

A method of converting nonlinear Volterra equations to systems of ordinary differential equations is compared with a standard technique, the method of moments, for linear Fredholm equations. The method amounts to constructing a Galerkin approximation when the kernel is either finitely decomposable or approximated by a certain Fourier sum. Numerical experiments from recent work by Bownds and Wood serve to compare several standard approximation methods as they apply to smooth kernels. It is shown that, if the original kernel decomposes exactly, then the method produces a numerical solution which is as accurate as the method used to solve the corresponding differential system. If the kernel requires an approximation, the error is greater, but in examples seems to be around 0.5% for a reasonably small number of approximating terms. In any case, the problem of excessive kernel evaluations is circumvented by the conversion to the system of ordinary differential equations.

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Tucson, Arizona, USA

    J. M. Bownds (Associate Professor)

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  1. J. M. Bownds
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Editors and Affiliations

  1. Department of Mathematics, University of Nevada at Las Vegas, Las Vegas, Nevada, USA

    Michael A. Golberg

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Bownds, J.M. (1979). On an Initial-Value Method for Quickly Solving Volterra Integral Equations: A Review. In: Golberg, M.A. (eds) Solution Methods for Integral Equations. Mathematical Concepts and Methods in Science and Engineering, vol 18. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-1466-1_9

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  • DOI: https://doi.org/10.1007/978-1-4757-1466-1_9

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-1468-5

  • Online ISBN: 978-1-4757-1466-1

  • eBook Packages: Springer Book Archive

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