Three-Point Compact Approximation for the Caputo Fractional Derivative

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}{\Gamma(2-\alpha)h^\alpha}\sum_{k=0}^{n} \delta_k^{(\alpha)} y_{n-k}=\dfrac{13}{12}y^{(\alpha)}n-\dfrac{1}{6}y^{(\alpha)}{n-1}+\dfrac{1}{12}y^{(\alpha)}_{n-2}+O\left(h^{3-\alpha}\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 $y^{\mathrm{\prime }}(0)=0$. The numerical solutions of the fractional relaxation and the time-fractional subdiffusion equations are discussed.