Perturbative Linearization of Reaction-Diffusion Equations

TL;DR

This paper introduces a perturbative linearization method for reaction-diffusion equations, transforming nonlinear problems into linear hierarchies that rapidly converge to solutions, demonstrated on Fisher and TDGL equations.

Contribution

It presents a novel perturbative expansion approach starting from singular-perturbation solutions for reaction-diffusion systems.

Findings

Hierarchy of linear equations converges quickly to exact solutions

Method applied successfully to Fisher and TDGL equations

Numerical results confirm rapid convergence

Abstract

We develop perturbative expansions to obtain solutions for the initial-value problems of two important reaction-diffusion systems, viz., the Fisher equation and the time-dependent Ginzburg-Landau (TDGL) equation. The starting point of our expansion is the corresponding singular-perturbation solution. This approach transforms the solution of nonlinear reaction-diffusion equations into the solution of a hierarchy of linear equations. Our numerical results demonstrate that this hierarchy rapidly converges to the exact solution.