Generalized Non-Standard Finite Difference Method for Fractional PDEs on Non-Uniform Grids

This paper proposes a novel Generalized Non-Standard Finite Difference (GNSFD) scheme for the numerical solution of a class of fractional partial differential equations (FrPDEs). The formulation of the method is grounded in optimization and leverages the fractional Taylor series (FrTS) expansion associated with Caputo fractional derivatives (FrDs). To discretize the time derivatives, the non-trivial denominator functions are utilized. The theoretical analysis establishes the consistency, stability, and convergence of the proposed scheme. Results are compared against existing methods to substantiate the accuracy and computational efficiency of the approach.