We can not list the applications or the fields that use the anomalous sub-diffusion equations due to their wide area, one of these important applications are in the chemical reactions when a single substance tends to move from an area of high concentration to an area of low concentration until the concentration is equal across space. The mathematical model that describes these physical-chemical phenomena is called the reaction subdiffusion equation. In our study, we try to solve the 2D variable order version of these equations (2DVORSE) (linear and nonlinear) using an accurate numerical technique which is the weighted average finite difference method (WAFDM). We will study the stability of the resulting scheme using the fractional version of the John von Neumann stability analysis procedure. An accurate specific stability condition that is valid for some parameters in the resulting schemes is derived and checked. At the end of the study, we present some numerical examples to demonstrate the accuracy of the proposed technique.

In this study, we propose to derive an accurate numerical procedure to solve the mathematical model which describes the electrical R-L circuit composed of resistors and inductors driven by a voltage of current source, which is the fractional-order model for the electrical RL-circuit. Our study depends on the spectral collocation method via the useful properties of the Chebyshev polynomials of the third-kind. Some theorems about the convergence analysis are given. The study concludes by comparing the resulting approximate solutions of the proposed model with the exact solution in the classical case. Illustrative graphical and numerical analysis of the derived results are also included in this study.

This paper present an accurate numerical algorithm to solve the space fractional-order Fisher’s equation where the derivative operator is described in the Caputo derivative sense. In the presented discretization process, first we use the compact nite difference (CFD) to occur a semi-discrete in time derivative, and implement the Chebyshev spectral collocation method (CSCM) of the third-kind to discretize the spatial fractional derivative. The presented method converts the studied problem to be a system of algebraic equations which can be easily solved. To study the convergence and stability analysis, some theorems are given with their profs. A numerical simulation is given to test the accuracy and the applicability of our presented algorithm.