Recovery of high order accuracy in spectral collocation method for linear Volterra integral equations of the third-kind with non-smooth solutions

Abstract

This paper aims to provide a rigorous analysis of exponential convergence of the Chebyshev collocation method for third kind linear Volterra integral equations. Different from Volterra integral equations of the second kind, the integral operator in third kind equations is noncompact under certain conditions, which brings special challenges to numerical analysis. The key idea of the proposed method is to adopt a smoothing transformation for the Chebyshev collocation method to circumvent the curse of singularity at the beginning of time. Therefore, the solution of the resulting equation will possess better regularity and then the numerical method can achieve the spectral accuracy. Moreover, in order to show the applicability and efficiency of the method, several examples with non-smooth solutions are illustrated.