Numerical Solution of The Two-dimensional Diffusion Logistic Model (Crank-Nicolson )

Abstract

The paper presents a detailed numerical analysis of the nonlinear fourth-order fractional reaction-diffusion equation using the compact difference method. The introduction of the fourth-order fractional derivative adds additional complexity to the equation, making its analytical solution challenging. Therefore, a numerical approach becomes necessary to understand the behavior of the equation and obtain approximate solutions. The compact difference method, known for its accuracy and efficiency in solving differential equations, is used to discretize the spatial and temporal derivatives of the equation. The fractional derivatives are approximated using suitable fractional difference operators. The resulting system is solved iteratively using appropriate numerical techniques. The study delves into a reaction-diffusion model utilized in brain gliomas, incorporating two different diffusion functions. In order to achieve a thorough comprehension, the analysis is broadened to encompass various types of tissue environments. Diverse scenarios are scrutinized, with the diffusion coefficient staying consistent to depict a uniform tissue environment. Furthermore, instances where the diffusion coefficient changes spatially are explored, bringing heterogeneity into the model. This spatial diversity accommodates the differing characteristics of distinct regions within the brain. Following this, the examination is expanded to include heterogeneous tissue environments in two dimensions.