Relation between two Sinc-collocation methods for Volterra integral equations of the second kind and further improvement

TL;DR

This paper compares two Sinc-collocation methods for Volterra integral equations, reveals their relation, extends their applicability, and proposes an improved method with faster convergence, supported by numerical validation.

Contribution

It theoretically relates two existing methods, justifies Stenger's method for general kernels, and introduces an enhanced method with near-exponential convergence.

Findings

Both methods achieve root-exponential convergence.

Stenger's method is valid for kernels of two variables.

The improved method attains near-exponential convergence.

Abstract

Two different Sinc-collocation methods for Volterra integral equations of the second kind have been independently proposed by Stenger and Rashidinia--Zarebnia. However, their relation remains unexplored. This study theoretically examines the solutions of these two methods, and reveals that they are not generally equivalent, despite coinciding at the collocation points. Strictly speaking, Stenger's method assumes that the kernel of the integral is a function of a single variable, but this study theoretically justifies the use of his method in general cases, i.e., the kernel is a function of two variables. Then, this study rigorously proves that both methods can attain the same, root-exponential convergence. In addition to the contribution, this study improves Stenger's method to attain significantly higher, almost exponential convergence. Numerical examples supporting the theoretical results are also provided.