Study On The Wavelet Precise Integration Method Of Partial Differential Equations

As is well know, many problems in engineering can be eventually presented in the form of partial differential equation(PDE). While the methods now we using to analysis and Compute PDE have their own advantages, they also have some shortcomings. In recent years, With the development of wavelet theory, wavelet analysis can solve many different 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. Many numerical value performance superior new algorithm is mergered by integrating the wavelet method and the traditional numerical value method. Moreover local characteristic of function is well analyzed by the wavelet method, therefore the wavelet method is extremely adapt to the numerical solution of the nonlinear PDE. The wavelet collocation method is representative among those wavelet numerical value methods ,which is a method of combining the high precision of the global method with the stability of the local method. In this thesis, a new method which is used to solve linear and nonlinear diffusion equation is developed by combining the wavelet collocation method with the precise integration method. The spatial domain of PDE is discreted by interval quasi-shannon wavelet collocation method, the precise integration method is used to the time domain. The following problems is proposed and solved.1. For linear Parabolic partial differential equation, the interval quasi-shannon scaling function is selected as base function ,the spatial domain was discreted by collocation method, so the system of ordinary differential equation is built. Then a high efficient method, that is the precise integration method is used to solve the system. Accordingly a effective numerical method for solving partial differential equations is obtained. The numerical results demonstrates the effect of our method. 2. The interval quasi-shannon wavelet is selected for the nonlinear PDE, the spatial domain is discreted by collocation method and the system of nonlinear ordinary differential equation is built. According to the thought of precise integration method, two disposal method of the nonlinear equations is proposed,(l) The nonlinear term of the system is disposed linearization by Taylor expanding within an inergral interval, then the linear equations which is obtained is solved by the precise integration method. (2)A homogenization precise integration method for solving the system of nonlinear ODE is obtained by Taylor expanding the nonlinear term of the system within an adequate inergral interval and by introducing homogeneous technique, which is unnecessary to compute the inverse matrixes. Good results are obtained by solving Burgers equation and MKDV equation.3. A high accuracy algorithm for numerical solution of partial differential equation is obtained by combining wavelet precise integration method with energy conservation property , which respective solves hyperbolic partial differential equations and parabolic partial differential equation. Firstly, generalized energy function and generalized energy integral corresponding to the PDE is obtained by definition, according to energy conservation theorem, energy conservation formula corresponding to the PDE is obtained. Then the spatial variable of energy conservation formula is discreted by wavelet collocation method, So the system of ordinary differential equations about time variable are built. Finally, the system of ordinary differential equations are solved by precise integration method. The numerical results show that the proposed method possesses the proterties of higher numerical stability and high precision.