Numerical Solution Of Differential Equation Based On Wavelet Theory

As a new mathematics embranchment,wavelet started at the S.Mllat and Y.Meyer's work, which was to construct wavelet base in a general way (multiresolution analysis) in the middle and latter half of the 1980s.Later the wavelet gets a drastic development. In application, it raises an upsurge, such as signal processing,image analysis,singular Check,marginal analysis,numerical solution of differential equations and so on. This paper studies the knowledge of wavelet theory and its applications in numerical solution of differential equation systemically, the content includes the following aspects:Chapter 1 summarizes the wavelet analysis's development course and its application in numerical solution of differential equation.Chapter 2 particularly analyzes the basic wavelet theory and algorithm which refer to this paper, such as multiresolution analysis and mallat algorithm.Chapter 3 gives a detailed introduction to Daubechies wavelets, meanwhile introduces periodic Daubechies wavelet and some basic theory and knowledge for numerical solution of differential equation which are based on wavelet. All of these are ready for chapter 4.In chapter 4, firstly we use Wavelet-Galerkin method to solve one-dimension Helmholtz equation which has periodic boundary condition;secondly Wavelet-Galerkin method combines with Euler method are used to solve one-dimension heat equation with periodic initial and boundary condition;finally we propose Wavelet Optimized Finite Difference method(this method works by using wavelets to generate an irregular grid which is then exploited for the finite difference method.) to solve nonlinear Burgers equation whose boundary and initial condition have periodicity, meanwhile make a compare with Mol method. It turns out that this method has a potential to solve nonlinear problem.Some numerical experiments prove that through numerical methods based on wavelet theory to solve differential equations can not only obtain numerical solutions which have high precision (validated by the Helmholtz equation which has real solution) and deal with large problem effectively (validated by the heat equation which has real solution), but also make a very good numerical simulation of nonlinear problems whose solution is singular (validated by the nonlinear Burgers equation), at the same time have higher efficiency than some other methods(eg. Mol) to solve this kind of problems, all of these sufficiently show the superiority of algorithms based on wavelet theory.