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Introduction to the Numerical Solution of Cauchy Singular Integral Equations

  • Chapter
Numerical Solution of Integral Equations

Part of the book series: Mathematical Concepts and Methods in Science and Engineering ((MCSENG,volume 42))

Abstract

In this chapter we present a survey of many of the polynomial-based projection and quadrature methods that have been developed in the past 15 years for the numerical solution of Cauchy singular integral equations with constant coefficients on (−1, 1). Emphasis is placed on equations of the first kind, where an elementary proof of the inversion formula for the airfoil equation due to Peters enables one to develop the theory and subsequent numerical analysis using techniques which are well known for Fredholm equations of the second kind. Convergence proofs are given for many of the algorithms, and numerical examples are drawn from the literature to illustrate the efficiency of these methods when the data are smooth. However, in many practical problems where either the kernel and/or the right-hand side may have discontinuities, this is not the case, and further work needs to be done. Some possibilities for doing this are examined as well.

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    M. A. Golberg

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Golberg, M.A. (1990). Introduction to the Numerical Solution of Cauchy Singular Integral Equations. In: Golberg, M.A. (eds) Numerical Solution of Integral Equations. Mathematical Concepts and Methods in Science and Engineering, vol 42. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-2593-0_5

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