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

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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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 (2021). https://doi.org/10.1007/s11075-021-01171-0

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Keywords

  • Nonlocal diffusion equation
  • Coupling operator
  • Ill-posed problem
  • Filter regularization

Mathematics Subject Classification (2010)

  • 65N20
  • 35R25
  • 47J06
  • 26A33