A Second Order Approximation for the Caputo Fractional Derivative

Abstract

When $0<α<1$, the approximation for the Caputo derivative $$y^{(α)}(x) = \frac{1}{Γ(2-α)h^α}\sum_{k=0}^n σ_k^{(α)} y(x-kh)+O\bigl(h^{2-α}\bigr),$$ where $σ_0^{(α)} = 1, σ_n^{(α)} = (n-1)^{1-a}-n^{1-a}$ and $$σ_k^{(α)} = (k-1)^{1-α}-2k^{1-a}+(k+1)^{1-α},\quad (k=1...,n-1),$$ has accuracy $O\bigl(h^{2-α}\bigr)$. We use the expansion of $\sum_{k=0}^n k^α$ to determine an approximation for the fractional integral of order $2-α$ and the second order approximation for the Caputo derivative $$y^{(α)}(x) = \frac{1}{Γ(2-α)h^α}\sum_{k=0}^n δ_k^{(α)} y(x-kh)+O\bigl(h^{2}\bigr),$$ where $δ_k^{(α)} = σ_k^{(α)}$ for $2\leq k\leq n$, $$δ_0^{(α)} = σ_0^{(α)}-ζ(α-1), δ_1^{(α)} = σ_1^{(α)}+2ζ(α-1),δ_2^{(α)} = σ_2^{(α)}-ζ(α-1),$$ and $ζ(s)$ is the Riemann zeta function. The numerical solutions of the fractional relaxation and subdiffusion equations are computed.