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An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations

Author

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  • Nataliia Kinash

    (Department of Cybernetics, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia)

  • Jaan Janno

    (Department of Cybernetics, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia)

Abstract

In this article, we consider two inverse problems with a generalized fractional derivative. The first problem, IP1, is to reconstruct the function u based on its value and the value of its fractional derivative in the neighborhood of the final time. We prove the uniqueness of the solution to this problem. Afterwards, we investigate the IP2, which is to reconstruct a source term in an equation that generalizes fractional diffusion and wave equations, given measurements in a neighborhood of final time. The source to be determined depends on time and all space variables. The uniqueness is proved based on the results for IP1. Finally, we derive the explicit solution formulas to the IP1 and IP2 for some particular cases of the generalized fractional derivative.

Suggested Citation

  • Nataliia Kinash & Jaan Janno, 2019. "An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations," Mathematics, MDPI, vol. 7(12), pages 1-16, November.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:12:p:1138-:d:289627
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    References listed on IDEAS

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    1. Atangana, Abdon, 2016. "On the new fractional derivative and application to nonlinear Fisher’s reaction–diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 948-956.
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    Cited by:

    1. Tarasov, Vasily E., 2023. "Nonlocal statistical mechanics: General fractional Liouville equations and their solutions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    2. Mohammed Al-Refai & Yuri Luchko, 2023. "The General Fractional Integrals and Derivatives on a Finite Interval," Mathematics, MDPI, vol. 11(4), pages 1-13, February.

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