本项目计划将算子半群理论中的一些方法与分数次演算自身的特点相结合，藉以研究分数阶发展方程中的几个核心问题。我们将主要研究以下几方面问题：算子分数幂的定义域特征，特别是那些可以生成某种分数次预解算子族但不能生成算子半群的偏微分算子分数幂的定义域；UMD空间上分数次预解算子族的平方根分解问题；基于Mittag-Leffler函数特性的相应于分数次预解算子族的代数同态；分数阶发展方程关于空间和时间的离散化逼近；不适定分数阶发展方程解的正则化方法；分数阶微分方程和随机过程的关系。

In this project we plan to study some core topics on fractional evolution equations via the methods from the theory of semigroups of operators and the properties of fractional calculus. We will concentrate on the following aspects: characterizing the domain of the fractional powers of the operators, especially those partial differential operators that can generate some kind of fractional resolvent families but not a semigroup of operators; square root reduction for fractional resolvent families on UMD spaces; constructing the algebraic homomorphism for fractional resolvent families based on the properties of Mittag-Leffler functions; creating the theory of discrete approximations in space and in time for fractional evolution equations; investigation and development of algorithms for solution of ill-posed fractional evolution equations involving regularization methods; establishing the relations between some stochastic processes and fractional differential equations.