Numerical solution of the time fractional nonlinear Fisher-KPP diffusion-reaction equation using the local domain boundary element method

TL;DR

This paper introduces an extended Local Domain Boundary Element Method (LD-BEM) to efficiently solve nonlinear time fractional Fisher-KPP equations, enabling accurate modeling of systems with memory effects in biological, chemical, and thermal phenomena.

Contribution

The paper presents a novel extension of LD-BEM tailored for nonlinear time fractional Fisher-KPP equations, reducing computational costs and handling nonlocal fractional derivatives effectively.

Findings

Successfully applied to 2D Fisher-KPP problems with various fractional derivatives.

Demonstrated improved efficiency over conventional BEM.

Validated accuracy through numerical experiments.

Abstract

The Fisher-KPP partial differential equation has been employed in science to model various biological, chemical, and thermal phenomena. Time fractional extensions of Fisher's equation have also appeared in the literature, aiming to model systems with memory. The solution of the time fractional Fisher-KPP equation is challenging due to the interplay between the nonlinearity and the nonlocality imposed by the fractional derivatives. An accurate method that for the solution of time fractional diffusion problems is the Boundary Element Method (BEM). The conventional BEM has a high computational cost and memory requirements since it leads to dense coefficient matrices. For nonlinear transient problems, its efficiency is further reduced due to the appearance of volume integrals. In the present work an extension of the recently proposed Local Domain Boundary Element Method (LD-BEM) is presented for the solution of nonlinear time fractional Fisher-KPP problems. The implemented numerical method is used to examine various two-dimensional problems related to the Fisher-KPP equation using different definitions of the fractional derivative.