时间变量为分数阶导数的抛物型偏微分方程动力学行为及其应用研究

Fractional differential equations have been widely applied to many engineering and scientific disciplines such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, economics, signal and image transmission, fitting of experimental data, and so forth. Fractional derivatives provide an excellent mathematical tool for the description of memory and hereditary properties of various materials and processes. In consequence, the subject of fractional order differential equations is gaining much importance and attention. There are a large number of good results involving in the dynamic behaviors of fractional ordinary differential systems. However, there is the relatively scarce results of the study on the dynamic behaviors of fractional partial differential systems. Therefore, we mainly study the dynamic behaviors such as existence, uniqueness, multiplicity, stability, attraction and oscillation of positive solutions, almost periodic solutions, travelling wave solutions and blow-up solutions for fractional order parabolic partial differential equations with impulses and time delays. As applications, we will investigate the above mentioned dynamic properties for the fractional order reaction-diffusion neural network systems, some fractional reaction-diffusion ecological dynamics models and fractional reaction-diffusion models of population dynamics.

分数阶微分方程被广泛应用于许多工程和科学领域，例如物理学、化学、空气动力学、复杂介质电动力学、聚合物流变学、反馈放大器的波德分析、电容器理论、电气电路、电子分析化学、生物学、控制论、经济学、信号和图像传输过程以及实验数据拟合等领域。对于描述和刻画具有记忆和遗传性质的各种材料和工艺，分数阶微分是一个非常好的数学工具。因此，分数阶微分方程受到了很多关注和重视。关于分数阶常微分系统动力学行为的研究已经获得许多很好的成果。但是，对分数阶偏微分方程动力学行为的研究结果目前相对较少。所以，我们主要研究具有时间延迟项和脉冲项的分数阶抛物型偏微分方程动力学行为，譬如正解、概周期解、行波解、爆破解的存在唯一性、多解性、稳定性、吸引性和振荡性。作为应用，我们将对分数阶反应扩散神经网络系统、某些分数阶反应扩散生态动力学模型和分数阶反应扩散人口动力学模型的上述动力学性质进行研究。

在物理学、化学、空气动力学、复杂介质电动力学、聚合物流变学、反馈放大器的波德分析、电容器理论、电气电路、电子分析化学、生物学、控制论、经济学、信号和图像传输过程以及实验数据拟合等科学和工程领域，分数阶微分方程都有非常广泛的应用。对于描述和刻画具有记忆和遗传特性的现象和过程，分数阶微积分具有很大的优越性。依托本项目，一方面我们研究了一些具有脉冲效应或延迟效应非线性分数阶微分方程或耦合方程组多点边值问题，获得了系统解的存在性和多解性的重要且新颖的充分判据。另一方面我们也研究了几类脉冲或时滞Lotka–Volterra生态系统和Gilpin–Ayala生态系统，获得正周期解和正概周期解的存在性以及全局指数稳定性。所得结果易于验证且对生态系统的保护和开发利用具有一定的指导作用。