The Numerical Solution Of Several Types Of Nonlinear Dynamical Problems Based On Wavelets Analysis

The study of solving partial differential equations has played an important role in the development of mechanics. At present, there are few methods to solve partial differential equations. One of the primary methods to solve them is numerical method, and the other is Galerkin method, which change partial differential equations into normal differential equations, and then solve them with the traditional method. Wavelets analysis can solve many difficult problems that Fourier analysis can not solve, which has become a new bench developing rapidly in mathematical field. It is an innovation of the tools and methods for research recently, and becomes the focus of many subjects. Wavelets analysis has well local property, so it is very good for the numerical solution of nonlinear partial equations. The numerical method based on quasi wavelets is a new numerical method that has not only the high accuracy of global methods but also the flexibility of local methods. The numerical method based on quasi wavelets is introduced to study the numerical solution of some typical partial differential equations.The whole paper consists of six chapters. In the first chapter, the actuality of wavelet analysis, nonlinear dynamics and their foreground have been introduced as well as the main task and the main work of the thesis. In the second chapter, the theory of wavelets analysis is introduced as the base of the numerical method based on quasi wavelets. In the third chapter, the numerical methods based on quasi wavelets and discrete singular convolution have been studied. It provides that the discrete singular convolution is a numerical method based on quasi wavelets in the fact, because the same discrete algorithm can be gotten from the theory of quasi wavelets and discrete singular convolution respectively. In the fourth chapter, a new method of boundary treatment for the method of quasi wavelets is constructed, and the numerical solution based on quasi wavelets of diffusion problems has been studied. In fifth chapter, two typical nonlinear dynamic partial differential equations, MKdV equation and Klein-Gordon equation, have been studied to test the reliability and efficiency of the numerical method based on quasi wavelets for solving the nonlinear dynamic partial differential equations of this kind. In the sixth chapter, the dynamic responses of the nonlinear elastic pole and panel under periodic load have been studied. The numerical method based on quasi wavelets has been used in the computation of dynamic responses, the computation gotten by which agree well with the results gotten by perturbation method and Galerkin method. In the latter problem, it provides the numerical method based on quasi wavelets can analyze chaos for that it gives the possibility in the nonlinear elastic panel through the Poicare mapping, phase plane and time history. On the contrary, it provides the rationality of Galerkin truncation. In the last, the summary of this paper and the prospect of the application of wavelets analysis in mechanics are given.