Numerical Simulations For The Variable Order Two-Dimensional Reaction Sub-Diffusion Equation: Linear And Nonlinear

The applications and the fields that use the anomalous sub-diffusion equations cannot be easily listed due to their wide area. Sure, one of the main physical reasons for using and researching fractional order diffusion equations is to explain anomalous diffusion that occurs in transport processes through complex and/or disordered structures, such as fractal media. One of the important applications is their use in chemical reactions, where a single material continues to shift from a high concentration area to a low concentration area until the concentration across the space is equal. The mathematical model that describes these physical-chemical phenomena is called the reaction sub-diffusion equation. In our study, we try to solve the 2D variable order version of these equations (2DVORSE) (linear and nonlinear) by using an accurate numerical technique which is the variable weighted average finite difference method (WAFDM). We will analyze the stability of the resulting scheme by using a modified suitable version of the John Von Neumann procedure. Specific stability conditions that occur or some parameters in the resulting schemes are derived and checked. At the end of the study, numerical examples are simulated to check the stability and the accuracy of the proposed technique.