Banach空间上抽象分数阶微分方程的逼近

In the past twenty years, the theory of fractional calculus gained fast development. The notion of α-times resolvent family was introduced by Bajlekova to study the Cauchy problem of fractional order. She was the first researcher who systematically studied abstract fractional Cauchy problem by using operator method. After then, the theory of fractional Cauchy problem on Banach spaces has been developed broadly.. This project is devoted to study approximation of fractional Cauchy problem on Banach spaces. Normally, we considered approximation when both original and approximated equations are on the same space. The case when approximation spaces are different from the original is much complicated. We systematically consider this kind of approximation in this project which consists of three parts:. In the first part, we will consider the full discretization of fractional resolvent family and the solution of fractional differential equation by Crank-Nicholson difference;. In the second part, we will study the full discretization of the solution of semilinear fractional Cauchy problem;. In the third part, we will study the approximation of the solution of another semilinear fractional Cauchy problem which is more ordinary and widespread.

过去二十年中, 分数阶微积分理论取得了长足发展. 在2001年, Bajlekova在她的博士论文中提出了α-次预解算子族的概念, 并利用其来研究分数阶Cauchy问题. 是她首次用算子方法研究了抽象分数阶Cauchy问题. 故在此之后, Banach空间上分数阶Cauchy问题理论取得了长足进展.. 本项目研究Banach空间上分数阶Cauchy问题的逼近. 一般在讨论逼近时, 初始方程和逼近方程都是在同一个空间上进行的. 本项目旨在对初始方程和逼近方程在不同空间的情形进行系统研究. 主要包含三个部分:. (一).用Crank-Nicholson差分来研究分数阶预解族及分数阶微分方程解的全离散化逼近; . (二).研究半线性分数阶Cauchy问题解的全离散逼近; . (三).研究一类更为普遍的半线性Cauchy问题解的离散逼近.

本项目对Banach空间上抽象分数阶Cauchy问题的逼近进行了研究, 主要分为两个部分: . 首先, 我们根据Caputo导数的定义, 通过将积分区间分段并在每段上取近似而得到有限分数阶差分, 并用其来逼近Caputo导数. 我们去掉了对分数阶微分方程解的限制, 分别用隐式差分和显式差分这两种方法, 来处理时间分数阶微分方程的全离散化逼近, 研究其收敛性, 并得出其收敛阶.. 其次, 我们采用有限差分和投影法, 研究了当算子A生成紧解析分数阶预解族且函数f充分光滑时时, 半线性分数阶Cauchy问题的解的半离散化逼近.