Wavelet In The Numerical Solution Of Partial Differential Equations

Wavelet analysis, which is used to solve lots of practical problems, is the development and perfection of the Fourier analysis. It inherits the advantages of the Fourier analysis, also makes up for its short. Wavelets have good localization properties and multi-resolution analysis, which made them to be widely used in scientific research and engineering applications in various fields. There is of great significance to research its theory and practical application. Wavelet analysis, as a new technology for time-frequency analysis, has been a hot field for research. Especially, recent years, wavelet which has many good natures applied in the numerical solution of partial differential equations has been greatly concerned to.1. Presented an overview of wavelet analysis and the basic theoretical knowledge of wavelet analysis, which contained its definition, CWT, DWT, orthogonal wavelet and MRA. Gave a review of wavelet applied and development in the numerical solution of the partial differential equation, especially, the Haar wavelet integral method based integral matrix applied for equations, systems and etc.2. Made a study of the Haar wavelet and its integral matrix, researched into the Haar integral method of a class of evolution equations concluding Burgers equation and Fisher equation extend to the numerical simulation of Fisher equation, which had general initial and boundary conditions, Figures and Tables verified the simplicity and viability of this method.3. Based on the study of the Haar integral method for the parabolic evolution equation with initial and boundary value conditions, discussed to apply this method to the numerical solution of the one-dimensional second order hyperbolic telegraph equation, present the Haar integral method for telegraph equation, charts showed the method is feasible and simple. Simultaneously, made further improvement of the method, the Legendre wavelet integral method and the Chebyshev wavelet integral method are given for the telegraph equation. Results showed these two methods are better than the Haar approximation, and also, orthogonal wavelets derived from orthogonal polynomials played an important role in engineering.