Numerical Approximations for Fractional Differential Equations

Abstract

The Grünwald and shifted Grünwald 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 Grünwald and shifted Grünwald 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 Grünwald 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.