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On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

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Abstract

An inverse source problem for a non-automonous time fractional diffusion equation of order \((0<\beta <1)\) is considered in a bounded Lipschitz domain in \(\mathbb {R}^d\). The missing solely time-dependent source is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is studied. We design two numerical algorithms based on Rothe’s method over uniform and graded grids, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the fractional subdiffusion problem is that the solution lacks the smoothness near the initial time, although it would be smooth away from \(t = 0\). Rothe’s method on a uniform grid addresses the existence of a such a solution (non-smooth with \(t^\gamma \) term where \(1>\gamma > \beta \)) under low regularity assumptions, whilst Rothe’s method over graded grids has the advantage to cope better with the behaviour at \(t=0\) (also here \(t^\beta \) is included in the class of admissible solutions) for the considered problems. The theoretical obtained results are supported by numerical experiments and stay valid in case of smooth solutions to the problem.

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Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Code Availability Statement

The authors used the open-source computing platform FEniCS for computations.

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Acknowledgements

The authors are grateful to the handling editor and the anonymous referees for their constructive feedback and helpful suggestions, which highly improved the paper. The authors would also like to thank Professor Vladimir G. Pimenov of Ural Federal University and Professor Marián Slodička of Ghent University, for their generosity and guidance, which has always been so valuable to them.

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  1. A. S. Hendy

    Present address: Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg, Russia, 620002

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  1. Department of Mathematics, Faculty of Science, Benha University, Benha, 13511, Egypt

    A. S. Hendy

  2. Research Group NaM2, Department of Electronics and Information systems, Ghent University, Krijgslaan 281, 9000, Ghent, Belgium

    K. Van Bockstal

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Correspondence to K. Van Bockstal.

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A. S. Hendy wishes to acknowledge the support of the RSF grant, project 22-21-00075. K. Van Bockstal is supported by a postdoctoral fellowship of the Research Foundation - Flanders (106016/12P2919N).

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Hendy, A.S., Van Bockstal, K. On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions. J Sci Comput 90, 41 (2022). https://doi.org/10.1007/s10915-021-01704-8

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