Numerical Approximations for Fractional Differential Equations

TL;DR

This paper develops higher-order numerical approximation formulas for fractional derivatives, improving accuracy for solving fractional differential equations and sub-diffusion models.

Contribution

It introduces second and third order approximations for Gr"unwald formulas using weighted averages, enhancing the accuracy of numerical solutions for fractional differential equations.

Findings

Second and third order approximations achieved.

Implicit difference schemes with second order accuracy.

Enhanced numerical solutions for fractional equations.

Abstract

The Gr\"unwald and shifted Gr\"unwald formulas for the function $y(x)-y(b)$ are first order approximations for the Caputo fractional derivative of the function $y(x)$ with lower limit at the point $b$. We obtain second and third order approximations for the Gr\"unwald and shifted Gr\"unwald formulas with weighted averages of Caputo derivatives when sufficient number of derivatives of the function $y(x) $ are equal to zero at $b$, using the estimate for the error of the shifted Gr\"unwald formulas. We use the approximations to determine implicit difference approximations for the sub-diffusion equation which have second order accuracy with respect to the space and time variables, and second and third order numerical approximations for ordinary fractional differential equations.