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A solely time-dependent source reconstruction in a multiterm time-fractional order diffusion equation with non-smooth solutions

A. S. Hendy and Karel Van Bockstal (UGent)
(2022) NUMERICAL ALGORITHMS. 90(2). p.809-832
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Abstract
An inverse source problem for non-smooth multiterm time Caputo fractional diffusion with fractional order designed as beta(0) < beta(1) < ... < beta(M) < 1 is the case of study in a bounded Lipschitz domain in Double-struck capital R-d. The missing solely time-dependent source function is reconstructed from an additional integral measurement. The existence, uniqueness and regularity of a weak solution for the inverse source problem is investigated. We design a numerical algorithm based on Rothe's method over graded meshes, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the multiterm time Caputo fractional subdiffusion problem is that the solution possibly lacks the smoothness near the initial time, although it would be smooth away from t = 0. In this contribution, we will establish an extension of Gronwall's inequalities for multiterm fractional operators. This extension will be crucial for showing the existence of a unique solution to the inverse problem. The theoretical obtained results are supported by some numerical experiments.
Keywords
Applied Mathematics, Inverse source problem, Multiterm fractional diffusion, Graded meshes, Non-uniform Rothe's method, Prior estimates, Convergence, BOUNDARY-VALUE-PROBLEMS, INVERSE SOURCE PROBLEM, ERROR ANALYSIS, GRADED MESHES, SOURCE-TERM, TRANSPORT

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Citation

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MLA
Hendy, A. S., and Karel Van Bockstal. “A Solely Time-Dependent Source Reconstruction in a Multiterm Time-Fractional Order Diffusion Equation with Non-Smooth Solutions.” NUMERICAL ALGORITHMS, vol. 90, no. 2, 2022, pp. 809–32, doi:10.1007/s11075-021-01210-w.
APA
Hendy, A. S., & Van Bockstal, K. (2022). A solely time-dependent source reconstruction in a multiterm time-fractional order diffusion equation with non-smooth solutions. NUMERICAL ALGORITHMS, 90(2), 809–832. https://doi.org/10.1007/s11075-021-01210-w
Chicago author-date
Hendy, A. S., and Karel Van Bockstal. 2022. “A Solely Time-Dependent Source Reconstruction in a Multiterm Time-Fractional Order Diffusion Equation with Non-Smooth Solutions.” NUMERICAL ALGORITHMS 90 (2): 809–32. https://doi.org/10.1007/s11075-021-01210-w.
Chicago author-date (all authors)
Hendy, A. S., and Karel Van Bockstal. 2022. “A Solely Time-Dependent Source Reconstruction in a Multiterm Time-Fractional Order Diffusion Equation with Non-Smooth Solutions.” NUMERICAL ALGORITHMS 90 (2): 809–832. doi:10.1007/s11075-021-01210-w.
Vancouver
1.
Hendy AS, Van Bockstal K. A solely time-dependent source reconstruction in a multiterm time-fractional order diffusion equation with non-smooth solutions. NUMERICAL ALGORITHMS. 2022;90(2):809–32.
IEEE
[1]
A. S. Hendy and K. Van Bockstal, “A solely time-dependent source reconstruction in a multiterm time-fractional order diffusion equation with non-smooth solutions,” NUMERICAL ALGORITHMS, vol. 90, no. 2, pp. 809–832, 2022.
@article{8747792,
  abstract     = {{An inverse source problem for non-smooth multiterm time Caputo fractional diffusion with fractional order designed as beta(0) < beta(1) < ... < beta(M) < 1 is the case of study in a bounded Lipschitz domain in Double-struck capital R-d. The missing solely time-dependent source function is reconstructed from an additional integral measurement. The existence, uniqueness and regularity of a weak solution for the inverse source problem is investigated. We design a numerical algorithm based on Rothe's method over graded meshes, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the multiterm time Caputo fractional subdiffusion problem is that the solution possibly lacks the smoothness near the initial time, although it would be smooth away from t = 0. In this contribution, we will establish an extension of Gronwall's inequalities for multiterm fractional operators. This extension will be crucial for showing the existence of a unique solution to the inverse problem. The theoretical obtained results are supported by some numerical experiments.}},
  author       = {{Hendy, A. S. and Van Bockstal, Karel}},
  issn         = {{1017-1398}},
  journal      = {{NUMERICAL ALGORITHMS}},
  keywords     = {{Applied Mathematics,Inverse source problem,Multiterm fractional diffusion,Graded meshes,Non-uniform Rothe's method,Prior estimates,Convergence,BOUNDARY-VALUE-PROBLEMS,INVERSE SOURCE PROBLEM,ERROR ANALYSIS,GRADED MESHES,SOURCE-TERM,TRANSPORT}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{809--832}},
  title        = {{A solely time-dependent source reconstruction in a multiterm time-fractional order diffusion equation with non-smooth solutions}},
  url          = {{http://doi.org/10.1007/s11075-021-01210-w}},
  volume       = {{90}},
  year         = {{2022}},
}
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