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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

Department of Cybernetics, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia
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Mathematics 2019, 7(12), 1138; https://doi.org/10.3390/math7121138
Received: 24 October 2019 / Revised: 14 November 2019 / Accepted: 19 November 2019 / Published: 21 November 2019
(This article belongs to the Special Issue Inverse and Ill-Posed Problems)
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. View Full-Text
Keywords: inverse problem; source reconstruction; final overdetermination; subdiffusion; tempered subdiffusion; fractional wave equation; generalized fractional derivative; Atangana–Baleanu derivative inverse problem; source reconstruction; final overdetermination; subdiffusion; tempered subdiffusion; fractional wave equation; generalized fractional derivative; Atangana–Baleanu derivative
MDPI and ACS Style

Kinash, N.; Janno, J. 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 2019, 7, 1138.

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