分数阶泊松方程的建模和计算

Fractional calculus plays an efficient role in characterizing long-range interactions and history dependence. It is well-known that fractional calculus has important and broad applications in physics, chemistry, mechanics, biology, computer science, and engineering. Although there are lots of research topics in studying fractional calculus, the present program mainly focuses on the steady state problems with long-range interactions, that is, it emphasizes modeling and computing of fractional Poisson equations. In details, the contents of this program are planned as follows. (1) By using fractional calculus, the anomalous diffusion in Lévy process will be characterized. And the corresponding fractional Poisson equation models will be established. (2) Numerical methods for the derived equation models shall be established. The high order finite difference methods are going to be built up for one-dimensional fractional Poisson equation. In order to avoid computing complexity induced by the nonlocality of the fractional Laplacians, Monte Carlo methods will be used for fractional Poisson equations in two and three space dimensions. Through these studies, the applicants wish to deeply develop modeling, analyzing and computing of fractional Poisson equations in order to provide solid theoretical fundaments and strongly technical support for real applications.

分数阶微积分是刻画长距离相互作用和历史依赖的有效工具。现已知道，它在多个学科领域如物理、化学、力学、生物、计算机科学、工程等有着重要和广泛的应用。虽然分数阶微积分有众多的研究课题，但本项目重点关注具有长距离相互作用的稳态问题，着重研究分数阶泊松方程的建模和计算。具体研究内容如下：(1) 运用分数阶微积分的有关知识，刻画列维过程中的反常扩散现象，建立分数阶泊松方程。(2) 对建立的分数阶泊松方程模型进行数值计算。对一维模型而言，运用高阶差分方法进行求解。对二、三维模型而言，为了克服因分数阶拉普拉斯算子非局部性带来的计算复杂度的困难，拟采用蒙特卡罗方法进行求解。通过本项目的研究，推动分数阶泊松方程建模、分析和计算的深入发展，为现实应用奠定坚实的理论基础和提供强有力的技术支持。