Some Nonlinear Partial Differential Equations Of The Wavelet Methods

Wavelet method is a new numerical method, and it is developed in recent years. Wavelet function has many compelling advantages: orthogonality, arbitrary-order differentiability, and a good time-frequency localization.At present it is quite few to its research. But because wavelet analysis has the very strong flexibility in partial time-frequency analysis, moreover its fast algorithm is very convenient to analyze and solve problems. Therefore, this method shows its unique advantages and strong vitality from the beginning. Different wavelets have different characteristics. We can choose appropriate wavelet depending on different problem. For example, Shannon wavelet can solve effectively the problem of singular integral problem; Haar wavelet can make Equation (group) of the stiffness matrix into a sparse matrix or similar sparse matrix; Daubechies wavelets with compact support and so on, thus greatly reducing the computation, so the extensive application of wavelet method has become a trend.This paper is divided into five chapters.Chapter 1 introduces the history of the development of the wavelet method, partial differential equations some of the solution, as well as wavelet method for solving partial differential equations in applications, and gives the wavelet method for solving partial differential equations in the main results achieved.Chapter 2 introduces the Daubechies wavelet, Daubechies wavelet character detailedly and gives the scaling function and wavelet function of the demand law and differential operator orthogonal wavelet representation.Chapter 3 introduces the wavelet Galerkin method, and application of wavelet Galerkin method for a class of nonlinear partial differential equations Noumann Boundary Problem and Dirichlet boundary value problem, first be a discrete time variable, the establishment of differential format, and then for each fixed time layer using wavelet Galerkin methods, the linear equations or algebraic equations, and thus obtain the numerical solution of the original problem.Chapter 4 sets up a discrete form of time for a class of time fractional problem by wavelet Galerkin method, and gives the format of stability and convergence proof of it.Chapter 5 paper first introduces the history of the development Haar wavelet and the nature and advantages (as opposed to Wolfowitz Transform Laplace transform and the convergence rate), and then introduced the band on the Haar wavelet, and use it Rosenau-Burgers equations were solved, and then give a basis for"Haar wavelet has a simple expression"characteristic solution of partial differential equations approach.