Convergence analysis of the generalized Euler-Maclaurin quadrature rule for solving weakly singular integral equations

Abstract

In the present article we show that the generalized Euler-Maclaurin summation formula is a natural theoretical and practical approach for numerical solving of integral equations. We use the formula to study the convergence analysis for solving weakly singular (Fredholm and Volterra) integral equations for the uniform mesh and a class of the nonuniform meshes. In each case of Fredholm and Volterra equations, it is shown that the rate of convergence is $O(n^{-2})$. Since these equations have different nature, the proposed convergence analysis for each equation has a different structure. Moreover, as an application of this summation formula, we consider the numerical solution of fractional ordinary differential equations (FODEs) by transforming FODEs into the associated weakly singular Volterra integral equations of the first kind. Some numerical illustrations are designed to depict the accuracy and versatility of the idea.