The boundary knot method for the solution of two-dimensional advection reaction-diffusion and Brusselator equations Mehdi Dehghan and Vahid Mohammadi Department of Applied Mathematics, Amirkabir University of Technology, Tehran, Iran Abstract Purpose – This study aims to apply a numerical meshless method, namely, the boundary knot method (BKM) combined with the meshless analog equation method (MAEM) in space and use a semi-implicit scheme in time for finding a new numerical solution of the advection–reaction–diffusion and reaction–diffusion systems in two-dimensional spaces, which arise in biology. Design/methodology/approach – First, the BKM is applied to approximate the spatial variables of the studied mathematical models. Then, this study derives fully discrete scheme of the studied models using a semi-implicit scheme based on Crank–Nicolson idea, which gives a linear system of algebraic equations with a non-square matrix per time step that is solved by the singular value decomposition. The proposed approach approximates the solution of a given partial differential equation using particular and homogeneous solutions and without considering the fundamental solutions of the proposed equations. Findings – This study reports some numerical simulations for showing the ability of the presented technique in solving the studied mathematical models arising in biology. The obtained results by the developed numerical scheme are in good agreement with the results reported in the literature. Besides, a simulation of the proposed model is done on buttery shape domain in two-dimensional space. Originality/value – This study develops the BKM combined with MAEM for solving the coupled systems of (advection) reaction–diffusion equations in two-dimensional spaces. Besides, it does not need the fundamental solution of the mathematical models studied here, which omits any difficulties. Keywords Singular value decomposition, Boundary knot method, Meshless analog equation method, Multiquadric radial function, The advection–reaction–diffusion and reaction–diffusion equations, Mathematical biology Paper type Research paper 1. Introduction 1.1 Two-dimensional nonlinear advection–reaction–diffusion equation As mentioned in Sarra (2012), the advection–reaction–diffusion and reaction–diffusion equations arise in various scientific disciplines such as biology (Sarra, 2012). For example, chemical reactions, population dynamics, flame propagation, the evolution of concentrations in environmental, biological processes and the study of chemical and biological systems (Islam et al., 2010; Mohammadi et al., 2014; Sarra, 2012). To understand the biological growth, Alan M Turing, a British mathematician, showed that a simple model of coupled reaction–diffusion equations could give rise to spatial Special thanks go to four reviewers for carefully reading this paper and putting useful comments and suggestions, which have improved the quality of this paper. Boundary knot method Received 2 October 2019 Revised 3 March 2020 Accepted 30 March 2020 International Journal of Numerical Methods for Heat & Fluid Flow © Emerald Publishing Limited 0961-5539 DOI 10.1108/HFF-10-2019-0731 The current issue and full text archive of this journal is available on Emerald Insight at: https://www.emerald.com/insight/0961-5539.htm