Abstract
This paper aims to present a Clenshaw–Curtis–Filon quadrature to approximate thesolution of various cases of Cauchy-type singular integral equations (CSIEs) of the second kind witha highly oscillatory kernel function. We adduce that the zero case oscillation (k = 0) proposed methodgives more accurate results than the scheme introduced in Dezhbord at el. (2016) and Eshkuvatovat el. (2009) for small values of N. Finally, this paper illustrates some error analyses and numericalresults for CSIEs.
Highlights
Integral equations have broad roots in branches of science and engineering [1,2,3,4,5,6]
Cauchy-type singular integral equations (CSIEs) of the second kind occur in electromagnetic scattering and quantum mechanics [7] and are defined as: b 1 u(y)K(x, y) au(x) + π ⨍
A singular integral equation with a Cauchy principal value is a generalized form of an airfoil equation ik(y−x)
Summary
Integral equations have broad roots in branches of science and engineering [1,2,3,4,5,6]. The solution to the above-mentioned Equation (1) contains boundary singularities w(x) = (x + 1) (1 − x) , i.e., u(x) = w(x)g(x) and g(x) is a smooth function [9,10]. Α = −β, solution cases of the CSIE of the second type depending on values of κ are:. Z.K. Eshkuvatov [10] introduced the method taking Chebyshev polynomials of all four kinds for all four different solution cases of the CSIE. This research work introduces the Clenshaw–Curtis–Filon quadrature to approximate the solution for various cases of a Cauchy singular integral equation of the second kind, Equation (1), at spaced points xi. The rest of the paper is organised as follows; Section 2 defines the numerical evaluation of the Cauchy integral in CSIE and approximates the solution at spaced points xi.
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