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Fractional filter method for recovering the historical distribution for diffusion equations with coupling operator of local and nonlocal type

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Abstract

The purpose of this paper is to investigate the problem of recovering the historical distribution for diffusion equations in which the diffusion operators are described by the coupling of local and nonlocal type. The problem essentially arises in many real-world circumstances including the biological population dynamic where a population competes for the resources and diffuses by a combination of the Brownian and Lévy processes. We first design a typical example to illustrate the ill-posed nature of the problem. A fractional filter method is then proposed to achieve reliable approximations of the problem. The stability and convergence of the proposed method are gingerly analyzed. Four numerical examples, with the support from the finite difference method and the fast Fourier transform, are implemented to validate the theoretical results including the ill-posedness and the effect of regularization. The numerical results agree with the theoretical analysis.

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Acknowledgements

The authors would like to thank Professor Hoang-Hung Vo for introducing the coupling operator and the problem. The authors are also very grateful to the anonymous referees for the very careful reading and helpful suggestions which led to the improvement of the original manuscript.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Science, Vietnam National University Ho Chi Minh City (VNU-HCM), 227 Nguyen Van Cu street, District 5, Ho Chi Minh City, Vietnam

    Tran Thi Khieu

  2. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Tra Quoc Khanh

  3. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Tra Quoc Khanh

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  1. Tran Thi Khieu
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Correspondence to Tra Quoc Khanh.

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Khieu, T.T., Khanh, T.Q. Fractional filter method for recovering the historical distribution for diffusion equations with coupling operator of local and nonlocal type. Numer Algor 89, 1743–1767 (2022). https://doi.org/10.1007/s11075-021-01171-0

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  • DOI: https://doi.org/10.1007/s11075-021-01171-0

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