关于分数阶偏泛函微分方程基本理论的研究

Fractional differential equations possess more advantages than integer-order differential equations in modelling some memory processes, hereditary properties and heterogenerous materials.The essential laws of viscoelastic systems, dielectric polarization systems, electric conduction in biological systems,fractal dynamics and synchronization of chaos,etc,can be described properly by fractional differential systems.The theory of fractional ordinary differential equations, fractional partial differential equations and fractional functional differential equations have gained good developments.Studying of fractional partial functional differential equations extensively and deeply is inevitable tendency.At present, however,investigation of the theory of fractional partial functional differential equations has just started.This project aims to deal with the fundamental theory such as the existence,the uniqueness and the continuous dependence,etc,of fractional partial functional differential equations with initial problems,boundary value problems and the related problems in Banach space.The results of investigation of this project will enrich and develop the theory of parital differential equations,functional differential equations and fractional differential equations.

分数阶微分方程在建模一些具有记忆过程、遗传性质以及异质材料时比整数阶微分方程模型更具优势。粘弹性系统、电解质极化系统、生物系统中的电导、分形动力学、混沌的同步等都可以用分数阶微分系统来描述其本质规律。分数阶常微分方程、分数阶偏微分方程、分数阶泛函微分方程理论已经得到了很好的发展，广泛而深入地研究分数阶偏泛函微分方程理论是必然的趋势。然而，目前对分数阶偏泛函微分方程理论的研究刚刚起步。本项目拟研究线性、非线性、常系数、变系数等分数阶偏泛函微分方程初值问题和边值问题解的存在性、唯一性、稳定性等基本理论和Banach空间中的相关问题。本项目的研究结果将丰富和发展偏微分方程理论、泛函微分方程理论和分数阶微分方程理论。

分数阶微分方程在模拟记忆、遗传等过程时比整数阶方程更具有优越性。目前，对分数阶微分方程理论的研究是数学领域和控制领域内的热点研究课题。本项目主要研究了分数阶偏泛函微分方程初值问题解的存在性、有界性和稳定性、分数阶反应扩散方程初边值问题弱解的存在性、唯一性和时滞分数阶偏泛函微分方程初值问题全局适度解的存在性等相关问题；同时研究了分数阶微分系统状态解的稳定性、能控性和同步等相关问题。建立了时间分数阶非齐次反应扩散方程的齐次化原理，给出了相应初值问题解的表示和解的稳定性判别准则； 建立了时间和空间都具有分数阶导数的反应扩散方程初边值问题弱解的存在唯一性定理； 建立了具有时滞的分数阶偏泛函微分方程初边值问题全局适度解的存在性定理；建立了分数阶微分系统状态解稳定性、能控性和同步的若干判别准则等。项目的研究结果丰富和发展了分数阶微分方程理论、偏微分方程理论和泛函微分方程理论。