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 $δ_0^{(α)}=1-ζ(α-1),\; δ_n^{(α)}=(n-1)^{1 -α}-n^{1-α},$ $$ δ_1^{(α)}=2^{1-α}-2+2ζ(α-1),\; δ_2^{(α)}=1-2^{2-α}+3^{1-α}-ζ(α-1),$$ $$δ_k^{(α)}=(k-1)^{1-α}-2k^{1-a}+(k+1)^{1-α},\quad (k=3\cdots,n-1),$$ where $y$ is a differentiable function which satisfies $y'(0)=0$. The numerical solutions of the fractional relaxation and the time-fractional subdiffusion equations are discussed.