Study On Wavelet-Galerkin Method Of Partial Differential Equation

Partial differential equation, growing with the birth and development of calculus, is a subject of long history. Its importance in application is quickly showed since it appeared. There are many traditional numerical methods to solve partial differential equation, such as finite element method and finite difference method. These methods have achieved great success in solving partial differential equation, however, there are disadvantage to solve the paradoxical perturbation equation of small parameter. The methods to solve partial differential equation have been developed rapidly since the birth of wavelet named mathematic microscope. The Wavelet-Galerkin method to solve partial differential equation is studied in this thesis.The main contents of the thesis can be summarizd as follows:(1) The basic theory of wavelet ,autocorrelation function of Daubechies scale function and its multi-analysis are introduced. Some related property and conclusions of the autocorrelation function are demonstrated.The Wavelet-Galerkin method and its merits in solving partial differential equation are presented.(2) Autocorrelation function of Daubechies scale function is chosen as base function, nonlinear partial differential equation is solved by Wavelet-Galerkin method.For autocorrelation function of Daubechies scale function has interpolation property which makes the solution is just the function value at the equinoxes, the transformation from wavelet space into physical space is avoided , so the amount of calculation is reduced and the efficiency is increased. The numerical example demonstrates this method is feasible and effective .(3) Autocorrelation function of Daubechies scale function is further studied.It is dealt with by using Lagrange polynomials, and the result is regarded as base function using Wavelet-Galerkin method to solve linear and nonlinear partial differential equation. Compared with the approach in chapter 3, the numerical examples indicate that this method not only increases accuracy of the numerical solution but also reduces the error near the boundaries.