Analysis of symmetries, conservation laws, and exact solutions of (1+1) reaction‐diffusion equation

This research work provides an investigation of the (1+1) reaction-diffusion equation, which models population dynamics with spatially varying growth rates represented by z(x) using Lie point symmetries analysis. Our methodology involves categorising this equation into three distinct types based on the constraints imposed on the spatially dependent growth rate during the solution of the Lie group determining equations. For each category, we systematically derive the corresponding conservation laws associated with the identified symmetries. Additionally, we develop exact solutions for each type, offering a widespread understanding of the population dynamics modelled by the equation. We pay special attention to scale-invariant solutions, which are explored using the global invariants of the one-parameter group. This in-depth investigation not only enhances our theoretical understanding of reaction-diffusion processes in heterogeneous environments but also highlights the utility of symmetry methods in solving complex differential equations.