Traveling wave solutions for systems of nonlinear advection-diffusion-reaction equations with delay and variable coefficients

Document Type : Research Paper


1 Institute for Systems Science and KZN e-Skills CoLab, Durban University of Technology, South Africa

2 DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa

3 Department of Applied Mathematics and School for Data Science and Computational Thinking, Stellenbosch University, South Africa

4 National Institute for Theoretical and Computational Sciences (NITheCS), South Africa


This paper introduces the methods for constructing the exact solutions of systems of nonlinear Advection-Diffusion-Reaction (ADR) equations with delay and variable coefficients. ADR systems of equations are coupled models which can be used to describe a set of interacting processes. Precepts are given for reducing such systems of equations to simpler systems of delayed ordinary differential equations by using modified methods of functional constraints. New exact solutions are presented in the form of traveling wave solutions. Exact solutions are prescribed to particular nonlinear ADR systems of equations for illustration. Significant arbitrary functions are present in the solutions which justify the suitability of the solutions for solving various modelling problems, validating the potency of numeric, asymptotic, and approximate analytical methods. The range of applicability of the results in this paper is universal as the results involve variable coefficients and delay.


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Volume 14, Issue 5
May 2023
Pages 337-348
  • Receive Date: 06 November 2022
  • Revise Date: 20 December 2022
  • Accept Date: 29 December 2022
  • First Publish Date: 03 January 2023