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Recovering the Initial Distribution for a Time-Fractional Diffusion Equation

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Abstract

We consider the inverse problem of reconstructing the initial condition of a one-dimensional time-fractional diffusion equation from measurements collected at a single interior location over a finite time-interval. The method relies on the eigenfunction expansion of the forward solution in conjunction with a Tikhonov regularization scheme to control the instability inherent in the problem. We show that the inverse problem has a unique solution provided exact data is given, and prove stability results regarding the regularized solution. Numerical realization of the method and illustrations using a finite-element discretization are given at the end of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Yarmouk University, Irbid, 21163, Jordan

    Mohammad F. Al-Jamal

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  1. Mohammad F. Al-Jamal
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Correspondence to Mohammad F. Al-Jamal.

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Al-Jamal, M.F. Recovering the Initial Distribution for a Time-Fractional Diffusion Equation. Acta Appl Math 149, 87–99 (2017). https://doi.org/10.1007/s10440-016-0088-8

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  • DOI: https://doi.org/10.1007/s10440-016-0088-8

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