Numerical solution of Cauchy singular integral equations of the first kind with index V=1

Abstract

A method based on Gauss-Chebyshev quadrature and barycentric interpolation is used to obtain the numerical solution of Cauchy singular integral equations of the first kind with index equal to 1 at non-Chebyshev nodes. The unknown function in the equation is first expressed as a product of an appropriate weight function and a truncated weighted series of Chebyshev polynomial of the first kind. Some properties of Chebyshev polynomials are then used to reduce the equation to a system of linear equations. On solving the linear system, the numerical solution of the Cauchy singular integral equation is obtained at Chebyshev nodes, after which barycentric interpolation is used to obtain the numerical solution at non-Chebyshev nodes. When the numerical solution obtained is compared with the analytical solution and the absolute error computed, the results are found to be satisfactory.