变量阶常微分方程问题解的研究

In this project, by means of some analysis techniques, we will consider the qualitative theory (existence, uniqueness and stability of solutions ) and the numerical solutions to some class of intial value and boundary vlaue problems for differential equations of variable-order. Variable-order operators are derivatives and integrals whose order are the functions of certain variables. The well-known fractional derivatives and integrals are the special case of variable-order operators, that is, their order are constants. In recent years, many scholars at home and abroad, based on the transformation between fractional derivatives and integrals, using the nonlinear functional analysis, considered the qualitative theory of solutions to problems for fractional differential equations. But, in most cases, variable-order derivatives and integrals do not have this transformation nature. Without this transformtaion, the nonlinear functional analysis method will not be used very well to discuss the qualitative theory of solutions to problems for variable-order differential equations, as those done for fractional differantial equations problems. Recently, variable-order operators and differential equations of variable-order have appeared frequently in Engineering applications.However, their extensive applications urgently call for systematic studies on the qualitative theory (existence, uniqueness and stability of solutions) and the numerical solutions of initial value and boundary value problems of these ordinary differential equations.

本项目中，我们将应用一定的分析技巧论证一类变量阶微分方程的初值问题和边值问题的解的定性理论（解的存在性、唯一性和稳定性）及其数值解。变量阶微积分算子是阶数为某些变量的函数的导数和积分，人们常说的分数阶导数和积分，即阶数为常数的导数和积分，是变量阶微积分算子的特殊情形。近年来，国内外诸多学者基于分数阶导数和积分所具有的能够相互转换的性质，运用非线性泛函分析的方法论证了分数阶微分方程问题解的定性理论。但是，一般情况下，变量阶导数和积分不具有能够相互转换的性质，这种转换性质的缺失，使我们无法如同分数阶微分方程问题一样运用非线性泛函分析的方法讨论变量阶微分方程问题解的定性理论。然而，近些年来，变量阶微分算子及其相应的微分方程在工程等领域的广泛应用迫切需要对这类常微分方程的初值问题、边值问题的解的定性理论和数值解进行系统的研究。

变量阶算子是比常数阶分数阶算子（分数阶导数和积分）更一般的一种分数阶算子，是对整数阶导数和积分的一种推广。变量阶算子及其相应的微分方程近年来越来越频繁地出现于几乎所有的研究领域和工程应用之中，出现了些许讨论一些变量阶微分方程的应用模型和数值解的论文，但是有关变量阶微分方程解的存在性等问题的结果却很少。本项目基于整数阶和常数分数阶微分方程解的存在性、唯一性的相关结论探讨了变量阶（及其它的特殊情形-常数阶）微分方程解的存在性和唯一性及其相关应用。我们探讨了变量阶算子的基本性质，通过一些分析技巧讨论了某类变量阶微分方程解的存在性和唯一性，讨论结果以论文形式呈现，我们在国内外一些期刊上发表论文二十余篇，其中被SCI所收录十五篇左右。