… In section 4, we determine the second order approximation (4) for the Caputo derivative
by modifying the first three coefficients of (1) with values of the Riemann zeta function. … In
section 4 we use the approximation for the fractional integral (6), to determine the second
order approximations for the Caputo fractional derivative … In section 4, we present a proof
for the second order approximation (4) of the Caputo derivative. …

When $0<\alpha< 1$, the approximation for the Caputo derivative $$ y^{(\alpha)}(x)=\frac
{1}{\Gamma (2-\alpha) h^\alpha}\sum_ {k= 0}^ n\sigma_k^{(\alpha)} y (x-kh)+ O\bigl (h^{2-
\alpha}\bigr), $$ where $\sigma_0^{(\alpha)}= 1,\sigma_n^{(\alpha)}=(n-1)^{1-a}-n^{1-a} $
and $$\sigma_k^{(\alpha)}=(k-1)^{1-\alpha}-2k^{1-a}+(k+ 1)^{1-\alpha},\quad (k= 1..., n-1),
$$ has accuracy $ O\bigl (h^{2-\alpha}\bigr) $. We use the expansion of $\sum_ {k= 0}^
nk^\alpha $ to determine an approximation for the fractional integral of order $2-\alpha $ and …