On a class of noncompact weakly singular Volterra integral equations: theory and application to fractional differential equations with variable coefficient

Abstract

The purpose of this paper is twofold. We first carry out an analysis of a class of noncompact weakly singular Volterra integral equations whose kernels possess both an end-point and diagonal singularities. A numerical method based on piecewise collocation scheme on uniform meshes including its convergence analysis are presented. We will then indicate the usefulness of our main concerned equation in the numerical study of an initial-value problem for fractional differential equations with variable coefficient. As a result, we reformulate the linear fractional differential equations with Erdélyi–Kober derivative to a particular type of the underlying Volterra equations with weakly singular kernels. Under certain verifiable conditions on the coefficient, the existence and uniqueness results as well as the numerical solution of the resulting equation by spline collocation method on piecewise polynomial space are analyzed. The reliability and efficiency of this approach are finally demonstrated by some numerical experiments.