Spectral Collocation Methods for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels

Abstract

In this paper, a Jacobi spectral collocation approximation is proposed for the solution of second-order Volterra integro-differential equations with weakly singular kernels. The solution of such equations usually exhibits a singular behaviour at the origin. By using some suitable variable transformations, we obtain a new equation which is still weakly singular, but whose solution is as smooth as we like. Then the resulting equation is solved by standard spectral methods. We establish a rigorous error analysis for the proposed method, which shows that the numerical errors decay exponentially in $L^\infty$-norm and weighted $L^2$-norm. Finally, to perform the numerical simulation, a test example is considered with non-smooth solutions.