Error analysis of a residual-based Galerkin’s method for a system of Cauchy singular integral equations with vanishing endpoint conditions

Abstract

In this paper, we develop Galerkin’s residual-based numerical scheme for solving a system of Cauchy-type singular integral equations of index minus N using Chebyshev polynomials of the first and second kind, where N is the total number of Cauchy-type singular integral equations in the system. Without theoretical analysis, a numerical scheme is not justified. Therefore, first, we prove the well-posedness of the system of Cauchy-type singular integral equations with the help of the compactness of an operator. Further, we derive a theoretical error bound and the order of convergence. Also, we show that the resulting system of equations obtained by applying the algorithm is well-posed together with an explicit representation of the solution in matrix form. Finally, we give some illustrative examples to validate the theoretical error bounds numerically.