A skew symmetry-preserving computational technique for obtaining the positive and the bounded solutions of a time-delayed advection–diffusion–reaction equation

Abstract

In this work, we consider a one-dimensional, time-delayed, advective version of the well-known Fisher–Kolmogorov–Petrovsky–Piscounov equation from population dynamics, which extends several models from mathematical physics, including the classical wave equation, the nonlinear Klein–Gordon equation, a FitzHugh–Nagumo equation from electrodynamics, and the Burgers–Huxley equation and the Newell–Whitehead–Segel equation from fluid mechanics. We propose a skew symmetry-preserving, finite-difference scheme for approximating the solutions of the model under investigation, and establish conditions on the model coefficients and the numerical parameters under which the method provides positive or bounded approximations for initial data which are likewise positive or bounded, respectively. The derivation of the conditions under which the positivity and the boundedness of the approximations is guaranteed is based on the properties of the inverses of M-matrices; in fact, the conditions obtained here assure that the iterative method is described in vector form through the multiplication by a matrix of this type. We provide simulations in order to show that the technique is indeed conditionally positivity-preserving and boundedness-preserving.