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.

本文的目的是研究恢复扩散方程的历史分布的问题，其中扩散算子由局部和非局部类型的耦合描述。这个问题主要出现在许多现实世界的环境中，包括生物种群动态，其中种群竞争资源并通过布朗和列维过程的组合进行扩散。我们首先设计一个典型的例子来说明问题的不适定性质。一个分数然后提出过滤方法来实现问题的可靠逼近。仔细分析了所提出方法的稳定性和收敛性。在有限差分法和快速傅里叶变换的支持下，通过四个数值例子来验证理论结果，包括不适定性和正则化效果。数值结果与理论分析一致。