Nonstandard Finite Difference Schemes for Solving Systems of two Linear Fractional Differential Equations

the case of real eigenvalues

Abstract

This paper provides non-standard finite difference methods for solving a Caputo-type fractional linear system with two equations with real eigenvalues. The linear system's real eigenvalues are classified into two types: distinct and repeated eigenvalues. The scenario of repeated eigenvalues is classified into two categories based on whether the dimension of the corresponding eigenspace is one or two. For each of the three scenarios, we obtained the exact solution and developed the numerator and denominator functions for the nonstandard finite difference scheme. Each of the three proposed numerical scheme's convergence has been established by proving consistency and stability. We showed that each of the proposed techniques is unconditionally stable when the system's eigenvalues are negative. Three examples were used to demonstrate the performance of the proposed methods.