Spectral Lanczos' tau method for systems of nonlinear integro-differential equations

TL;DR

This paper extends the spectral Lanczos' tau method to nonlinear integro-differential systems, introducing techniques for stability and accuracy, and demonstrating superior performance over existing methods through numerical experiments.

Contribution

It introduces a novel extension of the spectral Lanczos' tau method for nonlinear integro-differential equations, including new techniques for stability and efficiency.

Findings

Method significantly outperforms existing numerical approaches.

Enhanced stability and accuracy in solving nonlinear integro-differential systems.

Numerical results validate the effectiveness of the proposed techniques.

Abstract

In this paper an extension of the spectral Lanczos' tau method to systems of nonlinear integro-differential equations is proposed. This extension includes (i) linearization coefficients of orthogonal polynomials products issued from nonlinear terms and (ii) recursive relations to implement matrix inversion whenever a polynomial change of basis is required and (iii) orthogonal polynomial evaluations directly on the orthogonal basis. All these improvements ensure numerical stability and accuracy in the approximate solution. Exposed in detail, this novel approach is able to significantly outperform numerical approximations with other methods as well as different tau implementations. Numerical results on a set of problems illustrate the impact of the mathematical techniques introduced.