Parameter Identification in Traveling Wave Solutions of a Modified Fisher’s Equation

Abstract

In this work, we focus on the inverse problem of determining the parameters in a partial differential equation from given numerical solutions. For this purpose, we consider a modified Fisher’s equation that includes a relaxation time in relating the flux to the gradient of the density and an added cubic non-linearity. We show that such equations still possess traveling wave solutions by using standard methods for nonlinear dynamical systems in which fixed points in the phase plane are found and their stability characteristics are classified. A heteroclinic orbit in the phase plane connecting a saddle point to a node represents the traveling wave solution. We then design parameter estimation/discovery algorithms for this system including a few based on machine learning methods and compare their performance.