Approximate solution of singular integral equations of the first kind with Cauchy kernel

In this work a study of efficient approximate methods for solving the Cauchy type singular integral equations (CSIEs) of the first kind, over a finite interval, is presented. In the solution, Chebyshev polynomials of the first kind, T n ( x ) , second kind, U n ( x ) , third kind, V n ( x ) , and fourth kind, W n ( x ) , corresponding to respective weight functions W ( 1 ) ( x ) = ( 1 − x 2 ) − 1 2 , W ( 2 ) ( x ) = ( 1 − x 2 ) 1 2 , W ( 3 ) ( x ) = ( 1 + x ) 1 2 ( 1 − x ) − 1 2 and W ( 4 ) ( x ) = ( 1 + x ) − 1 2 ( 1 − x ) 1 2 , have been used to obtain systems of linear algebraic equations. These systems are solved numerically. It is shown that for a linear force function the method of approximate solution gives an exact solution, and it cannot be generalized to any polynomial of degree n . Numerical results for other force functions are given to illustrate the efficiency and accuracy of the method.