Improvement of Sinc-collocation methods for Volterra-Fredholm integral equations of the second kind and their theoretical analysis

Abstract: Sinc-collocation methods for Volterra-Fredholm integral equations of the second kind were proposed in 2012 and 2013 by multiple authors independently. Their theoretical analyses and numerical experiments suggest that the presented methods can attain root-exponential convergence. However, the convergence of these methods has not been strictly proved. This study first improves their methods to make them easier to implement, and provides a convergence theorem of the improved method. For the same equations, another Sinc-collocation method was proposed in 2016. This method is regarded as an improvement in the variable transformation employed in the method proposed in 2012. The method in 2016 may attain a higher rate than the previous methods, but its convergence has not yet been proved. For the method in 2016 as well, this study improves it to be easy to implement, and provides a convergence theorem.