变阶分数阶微分方程相关问题的定性研究

Variable-order fractional differential equations, as a novel mathematical modeling tool, has attracted much attention of experts and scholars both at home and abroad, and the study of its qualitative theory is a hot research area and difficulty at home and abroad in recent years. Due to fractional derivative of this kind of equations contains variable exponents, some classical results and analysis methods of constant-order fractional differential equations can not be generalized directly to study variable-order fractional differential equations. In this project, we intend to research the extremum principles of variable-order fractional differential equations and the existence, uniqueness, stability of solution, multiple solutions for initial and boundary value problem of several kinds of variable-order fractional differential equations. This subject has important theoretical significance and broad application prospects for the development and improvement of variable-order fractional calculus.

变阶分数阶微分方程作为一种新颖的数学建模工具，倍受国内外专家学者的关注，其定性理论的研究是近几年兴起的国内外研究热点及难点。由于此类方程中分数阶导数含有变指数的部分，有关常阶分数阶微分方程的经典结果和经典分析方法不能自然地推广到变阶微分方程的研究当中。本项目拟研究变阶分数阶微分方程极值原理以及几类变阶分数阶微分方程初、边值问题解的存在性、唯一性、多解性及稳定性等动力学问题，这对发展和完善变阶分数阶微积分理论具有重要的理论意义和广泛的应用前景。