Filtering the Tau method with Frobenius-Pad\'e Approximants

TL;DR

This paper introduces a Padé approximation-based method to enhance the accuracy of spectral solutions for differential equations, especially near singularities, by improving the Tau method with rational approximants.

Contribution

It presents a novel approach combining Padé approximants with the Tau method to better handle solutions with nearby singularities in differential equations.

Findings

Improved spectral solution accuracy near singularities.

Effective rational approximation of solutions with close singularities.

Enhanced convergence properties of the Tau method.

Abstract

In this work, we use rational approximation to improve the accuracy of spectral solutions of differential equations. When working in the vicinity of solutions with singularities, spectral methods may fail their propagated spectral rate of convergence and even they may fail their convergence at all. We describe a Pad\'e approximation based method to improve the approximation in the Tau method solution of ordinary differential equations. This process is suitable to build rational approximations to solutions of differential problems when their exact solutions have singularities close to their domain.