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

Authors: 

Tran Thi Khieu, Tra Quoc Khanh*

Source title: 
Numerical Algorithms, 89: 1743-1767, 2022 (ISI)
Academic year of acceptance: 
2021-2022
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.