Three-Point Compact Approximation for the Caputo Fractional Derivative

Abstract

In this paper we derive the fourth-order asymptotic expansions of the trapezoidal approximation for the fractional integral and the $L1$ approximation for the Caputo derivative. We use the expansion of the $L1$ approximation to obtain the three point compact approximation for the Caputo derivative \begin{equation*} \dfrac{1}{Γ(2-α)h^α}\sum_{k=0}^{n} δ_k^{(α)} y_{n-k}=\dfrac{13}{12}y^{(α)}_n-\dfrac{1}{6}y^{(α)}_{n-1}+\dfrac{1}{12}y^{(α)}_{n-2}+O\left(h^{3-α}\right), \end{equation*} with weights ${\delta }_0^{(\alpha )}=1-\zeta (\alpha -1),{\delta }_n^{(\alpha )}=(n-1{)}^{1-\alpha }-n^{1-\alpha },$ $${\delta }_1^{(\alpha )}=2^{1-\alpha }-2+2\zeta (\alpha -1),{\delta }_2^{(\alpha )}=1-2^{2-\alpha }+3^{1-\alpha }-\zeta (\alpha -1),$$ $${\delta }_k^{(\alpha )}=(k-1{)}^{1-\alpha }-2k^{1-a}+(k+1{)}^{1-\alpha },(k=3\cdots ,n-1),$$ where $y$ is a differentiable function which satisfies $y^{\prime }(0)=0$. The numerical solutions of the fractional relaxation and the time-fractional subdiffusion equations are discussed.

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• An analysis of the L1 Scheme for the subdiffusion equation with nonsmooth data
• Two Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data
• A Second Order Approximation for the Caputo Fractional Derivative
• A Matlab toolbox for fractional relaxation-oscillation equations
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Cited by in corpus (1)

• Higher-Order Numerical Solutions of the Fractional Relaxation-Oscillation Equation using Fractional Integration