A pseudo-spectral method for a non-local KdV-Burgers equation posed on $\mathbb R$

TL;DR

This paper introduces a novel pseudo-spectral numerical method for solving a non-local KdV-Burgers equation with fractional derivatives on the real line, transforming it into a bounded domain for efficient computation.

Contribution

The paper develops a new spectral approach that accurately computes fractional derivatives by mapping the unbounded domain to a finite interval.

Findings

Effective transformation of the real line to a finite interval

Accurate computation of fractional derivatives in spectral space

Potential for improved numerical solutions of non-local PDEs

Abstract

In this paper, we present a new pseudo-spectral method to solve the initial value problem associated to a non-local KdV-Burgers equation involving a Caputo-type fractional derivative. The basic idea is, using an algebraic map, to transform the whole real line into a bounded interval where we can apply a Fourier expansion. Special attention is given to the correct computation of the fractional derivative in this setting.