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On a reconstruction of a solely time-dependent source in a time-fractional diffusion equation with non-smooth solutions

A. S. Hendy and Karel Van Bockstal (UGent)
(2022) JOURNAL OF SCIENTIFIC COMPUTING. 90(1).
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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 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.
Keywords
Computational Theory and Mathematics, General Engineering, Theoretical Computer Science, Software, Applied Mathematics, Computational Mathematics, Numerical Analysis, Inverse source problem, Reconstruction, Fractional diffusion, Uniform and nonuniform (graded) meshes, Prior estimates, Convergence, INVERSE SOURCE PROBLEM, FINITE-DIFFERENCE METHOD, BOUNDARY-VALUE-PROBLEMS, SOURCE-TERM, REGULARIZATION, IDENTIFICATION

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Citation

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MLA
Hendy, A. S., and Karel Van Bockstal. “On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-Smooth Solutions.” JOURNAL OF SCIENTIFIC COMPUTING, vol. 90, no. 1, 2022, doi:10.1007/s10915-021-01704-8.
APA
Hendy, A. S., & Van Bockstal, K. (2022). On a reconstruction of a solely time-dependent source in a time-fractional diffusion equation with non-smooth solutions. JOURNAL OF SCIENTIFIC COMPUTING, 90(1). https://doi.org/10.1007/s10915-021-01704-8
Chicago author-date
Hendy, A. S., and Karel Van Bockstal. 2022. “On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-Smooth Solutions.” JOURNAL OF SCIENTIFIC COMPUTING 90 (1). https://doi.org/10.1007/s10915-021-01704-8.
Chicago author-date (all authors)
Hendy, A. S., and Karel Van Bockstal. 2022. “On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-Smooth Solutions.” JOURNAL OF SCIENTIFIC COMPUTING 90 (1). doi:10.1007/s10915-021-01704-8.
Vancouver
1.
Hendy AS, Van Bockstal K. On a reconstruction of a solely time-dependent source in a time-fractional diffusion equation with non-smooth solutions. JOURNAL OF SCIENTIFIC COMPUTING. 2022;90(1).
IEEE
[1]
A. S. Hendy and K. Van Bockstal, “On a reconstruction of a solely time-dependent source in a time-fractional diffusion equation with non-smooth solutions,” JOURNAL OF SCIENTIFIC COMPUTING, vol. 90, no. 1, 2022.
@article{8733484,
  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 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.}},
  articleno    = {{41}},
  author       = {{Hendy, A. S. and Van Bockstal, Karel}},
  issn         = {{0885-7474}},
  journal      = {{JOURNAL OF SCIENTIFIC COMPUTING}},
  keywords     = {{Computational Theory and Mathematics,General Engineering,Theoretical Computer Science,Software,Applied Mathematics,Computational Mathematics,Numerical Analysis,Inverse source problem,Reconstruction,Fractional diffusion,Uniform and nonuniform (graded) meshes,Prior estimates,Convergence,INVERSE SOURCE PROBLEM,FINITE-DIFFERENCE METHOD,BOUNDARY-VALUE-PROBLEMS,SOURCE-TERM,REGULARIZATION,IDENTIFICATION}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{33}},
  title        = {{On a reconstruction of a solely time-dependent source in a time-fractional diffusion equation with non-smooth solutions}},
  url          = {{http://doi.org/10.1007/s10915-021-01704-8}},
  volume       = {{90}},
  year         = {{2022}},
}
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