Invariant Subspace and Lie Symmetry Analysis, Exact Solutions and Conservation Laws of a Nonlinear Reaction-Diffusion Murray Equation Arising in Mathematical Biology

Abstract

Reaction-diffusion type equations are seen as models of pattern formation in biology and chemistry. The concept of Lie symmetry and invariant subspace (ISM) methods play a vital role in the study of partial differential equations (PDEs). Lie symmetry method helps to derive point symmetries, symmetry algebra and exact solution by reducing the PDEs to and ordinary differential equation (ODEs), while the invariant subspace method determines an invariant subspace and construct exact solutions of the PDEs by also reducing the PDEs to ODEs. In this article, the two methods are applied to derive the exact solutions of a nonlinear reaction-diffusion murray equation appearing in mathematical biology. Several kinds of solutions of the model are presented, including topological, singular and exponential function solutions. We classify the conservation laws (Cls) of the model using the multipliers approach. The paper conclude by giving a comprehensive physical interpretations and comparative study of the results showing the molecular nature of the acquired solutions.