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.

Key words: Exponential Caputo derivative, Caputo-Hadamard derivative, H3N3 formula, Truncation error