分数阶偏微分方程的不变流形

Motivated by the huge success of the applications of the abstract fractional equations in many areas of science and engineering and compare with the classic(ordinary or partial) differential equations theory, there have many unsolved questions in this filed. The project intends to explore the dynamical behavior of fractional partial differential equation and relate problem, main concern the existence of invarianat manifold for fractional ordinary and partial differential equation with finite and infinite dimention case. Such that, we can extend the present result of existence stable manifolds for planar fractional differential equations. Also we intends to compare the dynamical behavior of fractional differential equations. When the fractional derivative of time tend to one, what is the relationship of invariant manifolds between fractioanl differential equations and general deterministic differential equations. Partiallarly, the new pheomena of dynamical for fractional differential equations will be explored.

目前分数阶微分方程越来越成功运用科学与工程的许多领域，与经典整数阶微分方程(常微或者偏微)已有的结果相比较，分数阶微分方程有着许多问题有待解决。本课题目标为研究由分数阶偏微分方程所生成的系统的动力学性质以及相关问题，主要研究有限维与无穷维分数阶常微以及偏微分方程系统的不变流形的存在性，从而对目前已有的二维分数阶微分方程稳定流形存在性结果做出相应的推广，比较分数阶确定性微分方程与一般确定性微分方程不变流形之间的关系。特别的，考察分数阶微分方程在动力系统中带来的新现象和新问题。

鉴于目前分数阶微分方程越来越成功运用科学与工程的许多领域，同时与经典整数阶微分方程(常微或者偏微)已有的结果相比较，分数阶微分方程有着许多问题有待解决。目标为研究由分数阶偏微分方程以及随机微分方程所生成的系统的动力学性质以及相关问题，主要研究了有限维与无穷维分数阶常微以及随机偏微分方程系统的不变流形的存在性。已经取得的结果有：研究了一类带线性噪声随机微分方程不变流行的存在性并取得相应的结果，同时考虑了一类随机偏微分方程区域扰动问题，并得到相应的收敛结果。利用已有结果与方法处理一类分数阶微分方程相应的问题。