Title: Numerical solution of Fredholm integral equations of the first kind with singular logarithmic kernel and singular unknown function via monic Chebyshev polynomials

Authors: E.S. Shoukralla; M. Kamel; M.A. Markos

Addresses: Faculty of Engineering and Technology, Future University, 90th St, Cairo Governorate 11835, Egypt ' Faculty of Electronic Engineering, Menoufia University, Al Minufya, Egypt ' Faculty of Electronic Engineering, Menoufia University, Al Minufya, Egypt

Abstract: This paper provides a new method for the numerical solution of Fredholm integral equation of the first kind whose unknown function is singular at the end-points of the integration domain and has a weakly singular logarithmic kernel. The method is based on monic Chebyshev polynomials approximation. The singular behaviour of the unknown function is isolated by replacing it with a product of two functions; the first is a well-behaved unknown function, while the second is a badly-behaved known function. Furthermore, the singularity of the kernel is treated by creating two asymptotic expressions. This method, in addition to its simplicity, has a very important advantage, namely its ability to compute the functional values of the unknown function at the end-points of the integration domain, whereas the exact solution and other methods failed to find these values. It turns out from the two illustrated examples that the presented method significantly simplifies the computations, saves time, and ensures a superior accuracy of the solution.

Keywords: monic Chebyshev polynomials; weakly singular; Fredholm integral equations; first kind; potential-type equations.

DOI: 10.1504/IJCSM.2021.10041739

International Journal of Computing Science and Mathematics, 2021 Vol.14 No.1, pp.77 - 88

Received: 22 Jan 2019
Accepted: 31 Jan 2019

Published online: 12 Oct 2021 *

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