H3N3 Approximate Formulae for Typical Fractional Derivatives

The existing numerical approximation formulae for two kinds of typical fractional derivatives—the exponential Caputo and Caputo-Hadamard derivatives both of order \(\alpha \in (1,2)\) $\alpha \in (1,2)$ include L2, \(\hbox {L2}_1\) ${\text{L2}}_1$ , H2N2, but their convergence orders are all less than 2. To obtain a higher accuracy convergence order, we construct H3N3 approximation formulae based on the H2N2 formulae of these two kinds of derivatives and the \(\hbox {H3N3-2}_\sigma \) ${\text{H3N3-2}}_{\sigma }$ formula of the Caputo derivative, determine their truncation errors, and show the coefficients’ properties. Simultaneously, we display the numerical examples which support the theoretical analysis.