Numerical Finite Difference Scheme For A Two-Dimensional Time-Fractional Semi Linear Partial Differential Equations

Abstract

The fundamental tools for modelling neural dynamics are time-fractional partial differential equations. In order to solve a two-dimensional, time-fractional semilinear parabolic equation under Dirichlet boundary conditions, this paper introduces the Crank-Nicolson (C.N.) finite difference methodology. The proposed scheme's consistency, stability, and convergence are also thoroughly investigated. Two numerical experiments are conducted to support the theoretical results. The effectiveness of the method is carefully assessed and analysed in terms of absolute mistakes, order of accuracy, and computational time.  The outcomes show that, while being conditionally stable, the suggested scheme may be used successfully with a high rate of convergence to calculate numerical solutions for the issue at hand.