A novel method to find an exact solution of Cauchy integral equations

The purpose of this paper is to provide 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. To do this, it will suffice to employ The Chebyshev polynomials of the second kind with the corresponding weight function to approximate the density function. More specifically, the Chebyshev polynomials of the first kind are good enough to be used for approximating the force function. Considering force function as a cubic function, it can be concluded that the numerical solution of characteristic singular integral equation does agree with the exact solution. Furthermore, grounded results reveal that this numerical method is appreciated giving the exact solution for other singular integral equations with degenerate kernels.

Keywords:

Cauchy Integral Equations (CIE) , exact solution , Chebyshev polynomials