A Second Order Approximation for the Caputo Fractional Derivative

Abstract

When $0<\alpha <1$, the approximation for the Caputo derivative $$y^{(\alpha )}(x)=\frac{1}{\Gamma (2-\alpha )h^{\alpha }}\sum _{k=0}^n{\sigma }_k^{(\alpha )}y(x-kh)+O\left(h^{2-\alpha }\right),$$ where ${\sigma }_0^{(\alpha )}=1,{\sigma }_n^{(\alpha )}=(n-1{)}^{1-a}-n^{1-a}$ and $${\sigma }_k^{(\alpha )}=(k-1{)}^{1-\alpha }-2k^{1-a}+(k+1{)}^{1-\alpha },(k=\mathrm{1...},n-1),$$ has accuracy $O\left(h^{2-\alpha }\right)$. We use the expansion of ${\sum }_{k=0}^n{k}^{\alpha }$ to determine an approximation for the fractional integral of order $2-\alpha$ and the second order approximation for the Caputo derivative $$y^{(\alpha )}(x)=\frac{1}{\Gamma (2-\alpha )h^{\alpha }}\sum _{k=0}^n{\delta }_k^{(\alpha )}y(x-kh)+O\left(h^2\right),$$ where ${\delta }_k^{(\alpha )}={\sigma }_k^{(\alpha )}$ for $2\le k\le n$, $${\delta }_0^{(\alpha )}={\sigma }_0^{(\alpha )}-\zeta (\alpha -1),{\delta }_1^{(\alpha )}={\sigma }_1^{(\alpha )}+2\zeta (\alpha -1),{\delta }_2^{(\alpha )}={\sigma }_2^{(\alpha )}-\zeta (\alpha -1),$$ and $\zeta (s)$ is the Riemann zeta function. The numerical solutions of the fractional relaxation and subdiffusion equations are computed.

References in corpus (3)

• Numerical Approximations for Fractional Differential Equations
• Two Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data
• A Matlab toolbox for fractional relaxation-oscillation equations

Cited by in corpus (6)

• Approximations for the Caputo derivative (II)
• Three-Point Compact Approximation for the Caputo Fractional Derivative
• Higher-Order Numerical Solutions of the Fractional Relaxation-Oscillation Equation using Fractional Integration
• A New Method for Numerical Solution of the Fractional Relaxation and Subdiffusion Equations Using Fractional Taylor Polynomials
• Numerical solutions of ordinary fractional differential equations with singularities
• Fractional-compact numerical algorithms for Riesz spatial fractional reaction-dispersion equations