H3N3 Approximate Formulae for Typical Fractional Derivatives

Abstract

The existing numerical approximation formulae for two kinds of typical fractional derivatives—the exponential Caputo and Caputo-Hadamard derivatives both of order $\alpha \in (1,2)$ include L2, $\hbox {L2}_1$, 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 $ 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.