A Note on the Numerical Solution of Singular Integral Equations of Cauchy Type

Authors: M. Abdulkawi, Z. K. Eshkuvatov, N. M. A. Nik Long

Abstract:

This manuscript presents a method for the numerical solution of the Cauchy type singular integral equations of the first kind, over a finite segment which is bounded at the end points of the finite segment. The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density function. The force function is approximated by using the Chebyshev polynomials of the first kind. It is shown that the numerical solution of characteristic singular integral equation is identical with the exact solution, when the force function is a cubic function. Moreover, it also shown that this numerical method gives exact solution for other singular integral equations with degenerate kernels.

Keywords: Singular integral equations, Cauchy kernel, Chebyshev polynomials, interpolation.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1075048

References:

[1] Z. K. Eshkuvatov, N. M. A. Nik Long and M. Abdulkawi, Approximate solution of singular integral equations of the first kind with Cauchy kernel. Appl. Math. Lett. 2008. In Press, doi:10.1016/j.aml.2008.08.001.
[2] I. K. Lifanov, Singular Integral Equation and Discrete Vortices. VSO. The Netherlands, 1996.
[3] P. A. Martin and F. J. Rizzo, On boundary integral equations for crack problems. Proc. Roy. Soc. A. 421, 341-345, 1989.
[4] J. C. Mason and D. C. Handscomb, Chebyshev polynomials. CRC Press LLC, 2003.
[5] N. I. Muskhelishvili, Singular Integral Equations. Edited by J.R.M. Radok, Noordhoff International publishing Leyden, 1977.
[6] K. Prem. Kythe and R. Michael. Sch¨aferkotter, Handbook of computational methods for integration. Chapman&Hall/CRC Press 2005.
[7] M. Sheshko, Singular integral equations with Cauchy and Hilbert kernels and theirs approximated solutions. The Learned Society of the Catholic University of Lublin, 2003. (in Russian)