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.

References in corpus (1)

• Reaction Front in an A+B -> C Reaction-Subdiffusion Process

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