Improvement of Sinc-collocation methods for Volterra-Fredholm integral equations of the second kind and their theoretical analysis
Tomoaki Okayama
TL;DR
The paper addresses the challenge of solving Volterra-Fredholm integral equations of the second kind with Sinc-collocation methods, focusing on improving endpoint handling and providing rigorous convergence results. It develops two easy-to-implement schemes based on SE and DE transformations, proving root-exponential convergence for the SE variant and almost exponential convergence for the DE variant under suitable analytic/Hölder conditions. The contributions include explicit convergence theorems, corrected interpolation properties to ensure stable discretizations, and clearer implementation guidance, together with numerical experiments showing superior performance of the DE-based approach. The work advances the practical reliability and efficiency of Sinc-collocation methods for these integral equations, with potential impact on applications requiring robust handling of endpoint singularities and high-accuracy solutions.
Abstract
Sinc-collocation methods for Volterra-Fredholm integral equations of the second kind were proposed independently by multiple authors: by Shamloo et al. in 2012 and by Mesgarani and Mollapourasl in 2013. Their theoretical analyses and numerical experiments suggest that the presented methods can attain root-exponential convergence. However, their convergence has not been strictly proved. This study improves these methods to facilitate implementation, and provides a convergence theorem for the improved method. For the same equations, another Sinc-collocation method was proposed in 2016 by John and Ogbonna, which is regarded as an improvement to the variable transformation employed by Shamloo et al. It may attain a higher rate than the previous methods, but its convergence has not yet been proved. Therefore, this study improves it to facilitate implementation, and provides a convergence theorem for the improved method.