General Solution For Generalised Newell-Whitehead-Segel Equations

TL;DR

This paper develops a general canonical solution framework for a broad class of nonlinear parabolic equations with medium responses, applicable across physical, biological, and neurological systems, using Green's functions and iterative convolution methods.

Contribution

It introduces a unified solution approach for generalized Newell-Whitehead-Segel equations with nonlinear medium responses, expanding analytical tools for complex systems modeling.

Findings

Provides canonical solutions involving Green's functions.

Applicable to diverse systems like biological and neurological models.

Facilitates further analysis of nonlinear parabolic PDEs.

Abstract

In this monograph, two sets of parabolic differential equations are studied, each with nonlinear medium response. The equations are generally referred to as "Newell-Whitehead-Segel equation," which model a wide variety of nonlinear physical, mechanical and biological systems. Nonlinear medium response can be viewed in many perspectives, such as, memory response from the medium, whereby, the medium "remembers" earlier influences; reactive responses, whereby, the medium is actively responsive to input, such as, chemical reactivity, turbulence and many other circumstances; these equations arise often in the biological sciences when modeling population dynamics, whether the population be genomic, such as, alleles, or animal species in the environment; finally, these sets of equations are often employed to model neurological responses from excitable cellular media. The solutions provided are of a very general nature, indeed, whereby, a canonical set of solutions are given for a class of nonlinear parabolic partial differential equations with nonlinear medium response expressed as either a p-times iterative convolution or p-times multiplicative response. The advantage of canonical solution sets are these solutions involve classic representative forms, such as, Green's function or Green's heat kernel and aid researchers in further complication, analysis and understanding of the systemic behavior of these types of nonlinear systems.