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Identifying the source function for time fractional diffusion with non-local in time conditions

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Abstract

The diffusion equation has many applications in fields such as physics, environment, and fluid mechanics. In this paper, we consider the problem of identifying an unknown source for a time-fractional diffusion equation in a general bounded domain from the nonlocal integral condition. The problem is non-well-posed in the sense of Hadamard, i.e, if the problem has only one solution, the solution will not depend continuously on the input data. To get a stable solution and approximation, we need to offer the regularization methods. The first contribution to the paper is to provide a regularized solution using the modified fractional Landweber method. Two choices are proposed including a priori and a posteriori parameter choice rules, to estimate the convergence rate of the regularized methods. The second new contribution is to use truncation to give an estimate of \(L^p\) for the convergence rate.

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Authors and Affiliations

  1. Division of Applied Mathematics, Thu Dau Mot University, Thu Dau Mot, Binh Duong Province, Vietnam

    Nguyen Hoang Luc & Le Dinh Long

  2. Department of Mathematics, Cankaya University, Ankara, Turkey

    Dumitru Baleanu

  3. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

    Dumitru Baleanu

  4. Institute of Space Sciences, Magurele-Bucharest, Romania

    Dumitru Baleanu

  5. Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., MSC 172, Kingsville, TX, USA

    Ravi P. Agarwal

Authors
  1. Nguyen Hoang Luc
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  2. Dumitru Baleanu
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  3. Ravi P. Agarwal
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  4. Le Dinh Long
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Correspondence to Le Dinh Long.

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Communicated by José Tenreiro Machado.

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Luc, N.H., Baleanu, D., Agarwal, R.P. et al. Identifying the source function for time fractional diffusion with non-local in time conditions. Comp. Appl. Math. 40, 159 (2021). https://doi.org/10.1007/s40314-021-01538-y

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  • DOI: https://doi.org/10.1007/s40314-021-01538-y

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