Application of Chebyshev Polynomials to the Approximate Solution of Singular Integral Equations of the First Kind with Cauchy Kernel on the Real Half-line

Abstract

In this paper, exact solution of the characteristic equation with Cauchy kernel on the real half-line is presented. Next, the Chebyshev polynomials of the second kind, $U_{n}(x)$, and fourth kind, $W_{n}(x)$, are used to derive numerical solutions of Cauchy-type singular integral equations of the first kind on the real half-line. The collocation points are chosen as the zeros of the Chebyshev polynomials of the first kind, $T_{n+2}(x)$, and third kind, $V_{n+1}(x)$. Moreover, estimations of errors of the approximated solutions are presented. The numerical results are given to show the accuracy of the methods presented.