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)

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include L2,

{\text{L2}}_1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hbox {L2}_1$$\end{document}

, 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

{\text{H3N3-2}}_{\sigma }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\hbox {H3N3-2}_\sigma $$\end{document}

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.

Keywords

Exponential Caputo derivative / Caputo-Hadamard derivative / H3N3 formula / Truncation error / 26A33

References

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Shanghai University