非线性分析方法与奇异摄动理论在力学中的若干应用

Nonlinear analysis is a particularly important topic from the theoretical as well as the applied point of view.The methods in nonlinear analysis can be applied in many kinds of nonlinear differential equations, integral equations and other kinds of equations. These methods play important roles in many other fields, for example computational mathematics and control theory. Singular perturbation theory is mainly a kind of effective theories and methods to study the asymptotic solutions of differential equations, and now has become a powerful tool to deal with the nonlinear problems.. This project mainly uses the methods in nonlinear analysis and the singular perturbation theory to research some problems in mechanics. First of all, some problems from elastic beams will be discussed. To do this, we will first study the calculation of topological degree under the lattice structure and the number of fixed points in the theory of global structure. And then, six fundamental types of beam equations will be processed in a unified framework and some more comprehensive results of the existence and the number of positive solutions are expected to be obtained. Secondly, with the aid of the singular perturbation theory, we will investigate the singular perturbation of bending problem of elastic thick plate on nonlinear foundation including two parameters and the stability analysis of nonlinear vibration of large deflection plate.

非线性分析是现代数学中一个既有深刻理论意义又有广泛应用价值的研究方向。非线性分析方法可以应用于各种非线性微分方程、积分方程和其他类型的方程以及计算数学、控制理论等许多领域。奇异摄动理论主要是研究微分方程渐近解的一种有效的理论和方法，现在已经成为处理非线性问题的一种强有力的工具。. 本课题主要利用非线性分析方法和奇异摄动理论去研究力学中的若干问题。首先，利用格结构理论下拓扑度的计算方法以及全局结构理论下不动点个数问题的研究去处理弹性梁问题。主要是将六种梁方程类型纳入一个统一的框架下处理，期望对正解的存在性、个数等有更加深入的讨论。其次，利用奇异摄动理论去研究含双参数非线性地基上弹性厚板弯曲问题的奇异摄动，以及板的大挠度非线性振动稳定性分析。