两类分数阶微分方程有效数值计算方法研究

The fractional differential equations (FDEs) are able to fully capture the behavior of interesting phenomena than the differential equations with integer order in many applications （e. g. biological, geological, financial）, therefore it is a most impartent and valuable scientific research topic.The difficulties for numerical approximating FDEs are due to the nonlocal of the fractional differen- tial operator, which led to large amount of calculation and with low order. In this proposal, we aim to construct effectual and higher order numerical methods to approximate fractional integro-differential equations and multiterm time-space fractional partial differential equations. The main ideas of this proposal are as follows. (1) The collocation methods for fractional integro-differential equations with weakly kernels are considered for super convergence. We will choose special collocation points to construe the high order collocation methods and analysis the stability and error of this method. Then, we will give some numerical results to validate the effective and convergence of this method. (2) A finite difference and finite element methods for the multiterm time-space fractional partial differen- tial equations are considered, which may be effectual and convergent with high order. We will show algerithem of this method, proof the stability and convergence of this high order method. Then, we will approximate some numerical examples to check the convergence by the numerical results. These problems in this proposal are the cutting-edge topics, which have some certain research foundation.

分数阶微分方程在许多领域（如生物、地质、金融等）比通常的整数阶微分方程能更深刻准确地描述某些内在的特性，在实际中有很广阔的研究前景，其相关的研究工作具有重要的科学价值。分数阶微分方程数值算法研究的困难在于，分数阶微分算子的非局部性质引起的计算过程中的计算量大、计算复杂、收敛阶低等。本项目的主要目的在于构造高效的计算方法来近似求解分数阶积微分方程及分数阶偏微分方程。主要研究内容：（1）具有弱奇核的分数阶积微分方程的配置方法，分析其超收敛现象，选择特殊配置节点构造快速、稳定的数值算法，并进行误差分析，给出数值实验，验证方法的可行性及收敛性结果；（2）复合型时间空间分数阶偏微分方程的有限差分和有限元方法，给出实用的高阶差分方法和有限元方法格式，证明稳定性，进而研究算法收敛阶，给出数值例子验证收敛结果。本项目拟研究的问题都具有一定的研究基础，并且是非常有价值的。

本项目对分数阶微分方程的数值算法和算法的效率进行了分析，主要针对复合型时间空间分数阶偏微分方程的有限差分方法和有限元方法的分析，以及对分数阶积微分方程的配置方法的分析。同时还研究了二维四阶抛物型偏微分方程的差分方法和基于七维带边流形与带挠率Dirac算子相关的KKW定理等相关问题。课题研究取得了部分理论成果和实验数据。