Applications of the Nonstandard Finite Difference Method to a Fractional Model Explaining Diabetes Mellitus and Its Complications

Abstract

This work examines a mathematical model of diabetes mellitus and its consequences in a population using fractional differential equations. It attempts to solve the problem using a nonstandard way because standard finite difference numerical methods can result in numerical instabilities. The nonstandard finite difference scheme (NSFDS), which satisfies dynamical consistency, is the recommended nonstandard method for discretising the model. To demonstrate the stability of the model at the equilibrium points, analyses of both discrete and continuous models are performed.Stability analysis is carried out at the discretised models equilibrium point using the Schur-Cohn criterion. Consequently, the models asymptotically stable state is demonstrated. Furthermore, by contrasting the stability for various step sizes with conventional techniques like Finite Difference Scheme (FDS), the benefits of the NSFDS are shown. The NSFDS has been shown to converge at bigger step sizes. Furthermore, a graphical comparison is shown between the numerical findings acquired by the NSFDS and the FDS. It is noted that the NSFDS is accurate.