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The Approximate Solution of Singular Integral Equations

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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))

Abstract

We present a survey of numerical methods for solving Cauchy singular integral equations on both open and closed arcs in the plane. For completeness, necessary theory is reviewed, particularly the method of regularization. For closed arcs we discuss collocation methods based on piecewise polynomial or rational representations of the solution. Emphasis here, as for the open arc case, is on regularizable equations. For open arcs a detailed discussion is given of a degenerate kernel method developed recently by Dow and Elliott. In addition to this, a generalization of a Galerkin method due to Karpenko is presented. Attention is drawn to the relation of Cauchy singular equations and solving rectangular systems of linear equations. The possibility of exploiting this for the direct solution of such equations is discussed, and some direction for future research is given.

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Author information

Authors and Affiliations

  1. The University of Tasmania, Hobart, Tasmania, Australia

    D. Elliott (Professor of Applied Mathematics)

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  1. D. Elliott
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Editor information

Editors and Affiliations

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

    Michael A. Golberg

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© 1979 Springer Science+Business Media New York

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Elliott, D. (1979). The Approximate Solution of Singular Integral Equations. 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_3

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  • DOI: https://doi.org/10.1007/978-1-4757-1466-1_3

  • 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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