分数阶偏微分方程的近似算法研究

项目名称： 分数阶偏微分方程的近似算法研究

项目编号： No.11461072

项目类型： 地区科学基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 张新东

作者单位： 新疆师范大学

项目金额： 40万元

中文摘要： 分数阶微分方程是经典微分方程自然的数学推广，具有深刻的物理背景和丰富的理论内涵，在物理、生物、化学等多个学科领域具有广泛的应用。由于分数阶算子的特殊性质，使得分数阶微分方程的解析解很难求得，为此，研究分数阶微分方程的数值方法就显得很有必要。本项目主要研究分数阶Tricomi-型方程和扩散方程的近似数值算法。研究内容包括：局部间断有限元方法(LDG) 在求解分数阶偏微分方程数值解中的应用，此方法易于处理复杂边界问题，具有灵活处理间断问题的能力同时便于并行算法的实现；有限元方法(FEM)在求解不规则区域中分数阶微分方程的应用，此方法对区域的形状适应性强，同时便于通用程序的编写。目的是想实现分数阶Tricomi-型方程和扩散方程的LDG方法的并行计算及FEM方法的高维通用程序的编写。

中文关键词： 近似算法；有限元方法；分数阶偏微分方程；稳定性分析；误差分析

英文摘要： Fractional differential equation (FDE) is the natural mathematical promotion of classic calculus differential equation. FDE has profound physical background and rich theoretical connotation, which has been successfully applied to problems in physical, biological, chemical, and other disciplines. Due to the special properties of the fractional order operator, which makes the analytical solution of the FDE is difficult to be calculated, therefore, the numerical method for FDE is very necessary. In the project, we will study the numerical algorithm of fractional Tricomi-type equation by Local discontinuous Galerkin method (shortly LDG) and the fractional diffusion equation by Finite element method (shortly FEM). The research content mainly includes: we will study the applications of the LDG for FDE. It is easy to deal with the problem with complex boundary by LDG method, which has the ability of dealing with the discontinuous problems, and also can be used for parallel algorithm; we also study the application of FEM for FDE in the irregular area. The method can be used for any shape of the area, at the same time it is easy for the writing of the general program.Our purpose is to achieve parallel algorithm of LDG method for fractional Tricomi-type equation and the general program of FEM for diffusion equation in high-dimensional case.

英文关键词： Approximate algorithm;Finite element method;Fractional partial differential equation;Stability analysis;Error analysis