The Wavelet Mathod For Partial Differential Equation

As is well known, many problems in science 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, such as the ill condition matrix by FEM and a lot of integrals needed by the BIE. But when wavelets are used, we will not meet the disadvantages mentioned above. Usually applying wavelet to the PDE, we need to solving such problems as follows (1) choose a suitable wavelet basis according to the problem we want to deal with (2) design or modify a good algorithm in order to make use of the characters of the wavelet we choose, etc.In this dissertation, based on the research of wavelet theory, the following problems are discussed:1.A new conception. wavelet operator group, is presented, which is the generalization of the wavelet transformation. Having discussed the properties, we prove that the continuous wavelet transformation belongs to the wavelet operator group. We come to the conclusion that the physical wavelet is the conjugate of the wavelet operator group.2.The substitution is made of the interval wavelet basis for the compact one in order to produce the basis of Sobolev space with the lower condition number, which will improve the speed and the accuracy of the linear system made by wavelet Galerkin method. Also the algorithms based on the local or global correction are presented and their properties are discussed in detail. The numerical results on two point boundary problems test and verify the algorithms.3.Two ways are discussed to produce solution spaces when the BIE is used with the basis of period wavelet. Based on these, the convergence and the error estimation are given. The MRA with more then one-wavelet function is presented and the multi-wavelet basis is produced by unitary matrix. This basis is used successfully in scattering problem of sonic wave and nonlinear Hanimersterin integral equation.4.The optimal approximation property is shown when the cubic spline interval wavelet is used in the interpolation with some special points. This property ensures the identical of the interpolation by use of scale basis and of wavelet basis. The interpolation error is also discussed. A new algorithm is presented in which two one-order differential matrixes are used to substitute for the two-order one. This way produces less vibration for the numerical solution of Burgers equation when the coefficient is somewhat larger.