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

Abstract

Stenger and Rashidinia--Zarebnia independently proposed two Sinc-collocation methods for Volterra integral equations of the second kind, but the relation between the methods has not been clarified. This study reformulates Stenger's method for general two-variable kernels and rigorously establishes its applicability and convergence. We prove that the approximate functions produced by the two methods are not generally identical, although they coincide at every collocation point. We also provide a rigorous convergence proof for the Rashidinia--Zarebnia method and show that both methods attain the same root-exponential convergence rate. Because Stenger's method is simpler to implement than the Rashidinia--Zarebnia method, we adopt it as the basis for further improvement. By replacing the tanh transformation with the double-exponential transformation, we develop a new Sinc-collocation method and prove its almost exponential convergence. Numerical examples support the theoretical results and demonstrate the favorable balance between accuracy and computational cost of the proposed method.

Keywords: Sinc numerical method, tanh transformation, double-exponential transformation, collocation method, Nyström method