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Fractional Calculus and Time-Fractional Differential Equations: Revisit and Construction of a Theory

Author

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  • Masahiro Yamamoto

    (Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan
    Academy of Romanian Scientists, Ilfov, nr. 3, 062217 Bucuresti, Romania
    Accademia Peloritana dei Pericolanti, Palazzo Università, Piazza S. Pugliatti 1, 98122 Messina, Italy
    Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., 117198 Moscow, Russia)

Abstract

For fractional derivatives and time-fractional differential equations, we construct a framework on the basis of operator theory in fractional Sobolev spaces. Our framework provides a feasible extension of the classical Caputo and the Riemann–Liouville derivatives within Sobolev spaces of fractional orders, including negative ones. Our approach enables a unified treatment for fractional calculus and time-fractional differential equations. We formulate initial value problems for fractional ordinary differential equations and initial boundary value problems for fractional partial differential equations to prove well-posedness and other properties.

Suggested Citation

  • Masahiro Yamamoto, 2022. "Fractional Calculus and Time-Fractional Differential Equations: Revisit and Construction of a Theory," Mathematics, MDPI, vol. 10(5), pages 1-55, February.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:5:p:698-:d:756679
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    References listed on IDEAS

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    1. Li, Zhiyuan & Liu, Yikan & Yamamoto, Masahiro, 2015. "Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 381-397.
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    Cited by:

    1. Yuri Luchko, 2023. "Fractional Integrals and Derivatives: “True” versus “False”," Mathematics, MDPI, vol. 11(13), pages 1-2, July.

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