Convergence to the traveling wave solution of a biological-physical model through a differential equation with piecewise constant argument

Abstract

Abstract Getting to know explicitly or approximately the traveling wave solutions of the diffusive delay logistic equation, commonly known as the delayed Kolmogorov-Petrovsky-Piscounov-Fisher equation, is of major importance for understanding various biological and physical phenomena. In this study, we discretize the delay argument of the equation that satisfies the traveling wave and we obtain a second order delay differential equation with piecewise constant argument. We prove the existence and uniqueness of a solution for the discretized equation, and then prove that this solution converges uniformly along the whole straight towards the traveling wave. The methodology posed is based on the upper and lower solutions technique along with the use of a monotone integral operator. Our results show that the technique we developed is another good method for approaching traveling wave solutions. In addition, we suggest that this method can be applied to other reaction-diffusion equations that model a wide range of biological, physical, and chemical phenomena.