Non-Standard Finite Difference Schemes for Solving Variable-Order Fractional Differential Equations

A non-standard finite difference (NSFD) methodology of Mickens is a popular method for the solution of differential equations. In this paper, we discusses how we can generalize NSFD schemes for solving variable-order fractional problems. The variable-order fractional derivatives are described in the Riemann-Liouville and Grunwald-Letinkov sense. Special attention is given to the Grunwald-Letinkov definition which is used to approximate the variable-order fractional derivatives. Some applications of the variable-order fractional in viscous-viscoelasticity oscillator model and chaotic financial system are included to demonstrate the validity and applicability of the proposed technique.