Theoretical analysis of a Sinc-Nyström method for Volterra integro-differential equations and its improvement

摘要

A Sinc-Nyström method for Volterra integro-differential equations was developed by Zarebnia (2010). The method is quite efficient in the sense that exponential convergence can be obtained even if the given problem has endpoint singularity. However, its exponential convergence has not been proved theoretically. In addition, to implement the method, the regularity of the solution is required, although the solution is an unknown function in practice. This paper reinforces the method by presenting two theoretical results: (1) the regularity of the solution is analyzed, and (2) its convergence rate is rigorously analyzed. Moreover, this paper improves the method so that a much higher convergence rate can be attained, and theoretical results similar to those listed above are provided. Numerical comparisons are also provided.