On spectral methods for Volterra integral equations and the convergence analysis. (English) Zbl 1174.65058

Summary: The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach (with theoretical justification) for Volterra type equations.


65R20 Numerical methods for integral equations
45D05 Volterra integral equations