Convergence analysis of the product integration method for solving the fourth kind integral equations with weakly singular kernels

In this paper, we consider product integration method based on orthogonal polynomials to solve mixed system of Volterra integral equations of the first and second kind with weakly singular kernels. For investigation of the theoretical and numerical analysis of the mixed systems, the notions of the tractability index and nu-smoothing property are extended for a weakly singular Volterra integral operator. Convergence analysis of the product integration method is derived. Finally, the proposed method is illustrated by two examples, which confirm the theoretical prediction of the error estimation.

Automated summary

This paper investigates a product integration method based on orthogonal polynomials for solving mixed systems of Volterra integral equations with weakly singular kernels. Theoretical and numerical analysis, including tractability index and nu-smoothing property extension, are conducted for the mixed systems. Convergence analysis of the method is derived, and two examples are provided to illustrate the theoretical error estimation.