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

Abstract

An inverse source problem for non-smooth multiterm time Caputo fractional diffusion with fractional order designed as β0 < β1 < ⋯ < βM < 1 is the case of study in a bounded Lipschitz domain in \(\mathbb {R}^{d}\). 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 Grönwall’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.

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Funding

K. Van Bockstal is supported by a postdoctoral fellowship of the Research Foundation - Flanders (106016/12P2919N).

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Correspondence to K. Van Bockstal.

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

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Hendy, A.S., Van Bockstal, K. A solely time-dependent source reconstruction in a multiterm time-fractional order diffusion equation with non-smooth solutions. Numer Algor (2021). https://doi.org/10.1007/s11075-021-01210-w

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Keywords

  • Inverse source problem
  • Multiterm fractional diffusion
  • Graded meshes
  • Non-uniform Rothe’s method
  • Prior estimates
  • Convergence

Mathematics Subject Classification (2010)

  • 35A15
  • 35R11
  • 47G20
  • 65M12