分数阶偏微分方程的近似算法研究

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.

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。

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。..本项目主要做了如下研究工作：用同伦摄动方法研究了时间分数阶KdV方程的近似解析解；研究了n维时间分数阶反应扩散方程的数值解问题；研究了非线性矩阵方程组的Hermitian正定解；研究了一类时间分数阶偏微分方程的同伦分析Sumudu变换解法；研究了非线性时间分数阶微分方程的exp(-Φ(ξ))解法；研究了分数阶扩散波动方程的有限差分/局部间断有限元解法；用全离散局部间断有限元方法研究了时间分数阶反应扩散方程的数值解；构造了一种无条件稳定的完全离散局部间断Galerkin方法用来求解一类具有多项时间分数阶的扩散方程；研究了Cattaneo方程的近似数值解。..本项目的研究内容将会为工程应用提供有力的参考，同时丰富微分方程理论成果，为我们进一步研究高效数值算法提供必要的研究经验、方法和思路。