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A Numerical Solution of Singular Integral Equation
This article gives numerical method to solve the Cauchy type singular integral equation of the first kind over the finite interval [−1, 1] in two cases: the solution is unbounded and bounded at the end points x = ±1. The solution is obtained by approximating the unknown function by the weighted Chebyshev polynomials of the first and second kinds for unbounded and bounded cases, respectively, and then calculating the obtained Cauchy singular integral analytically. Lagrange-Chebyshev interpolation is used to approximate the regular kernel. The collocation points are chosen to be the zeros of Chebyshev polynomials. Numerical result shows the efficiency and accuracy of the method. The proposed algorithm provide a shortcut for programming the approximate solution of Cauchy type singular integral equation of the first kind.
Singular integral equations, Cauchy kernel, Lagrange-Chebyshev interpolation.
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