Wavelet Method For Sloving Differential Equations

Wavelet analysis is a new promising mathematical branch.This new analytical method is the result of decades of assiduous research in the field of harmonic analysis. At present,wavelet analysis has become a popular and widely used subject in the areas of scientific research and engineering projects.In the mathematical field,wavelet analysis is a powerful tool for numerical analysis, which can be used to solve the differential and integral equation directly and effectively. It can also be well applied to the linear problem and nonlinear problem.Based on the research into various differential equations using wavelet,a series of results are presented in this thesis.The whole thesis is divided into four chapters and the main issues discussed in each chapter as follows:In Chapter 1,the basic knowledge and general theories of wavelet analysis and differential equations are illustrated,the applications of wavelet analysis in the differential equations are talked about and the feasibility of using orthogonal wavelet to solve differential equations is proposed.In Chapter 2,the numerical methods of the differential equations are investigated. They are discussed in three aspects:numerical methods for the initial-value problems and boundary-value problems of constant-coefficient differential equations;difference methods for partial differential equations;finite element,methods for partial differential equations.The basic ideas of constructing numerical methods are explained by means of some classical,common and effective numerical methods.In Chapter 3,orthogonal wavelet bases mainly deals with Haar wavelet defined by orthogonal multi-analysis,Sine-cosine wavelet and CAS wavelet constructed by sine and cosine functions,as well as Legendre wavelt and Chebyshev wavelet constructed by orthogonal polynomial.The properties of these wavelets are discussed and the corresponding operational matrices are established.Based on one dimensional CAS orthogonal wavelet,two dimensional CAS orthogonal wavelet is constructed using the tensor product,and its properties and operational matrices are also discussed.In Chapter 4,the differential equation for finite-length beam,constant coefficients equation for convection-diffusion,and control differential equation for the bending problem of lamellas on the elastic groundsill are solved by many types of orthogonal wavelet bases.As the advantage of the property of localization of wavelet,the solutions to some problems could be identified precisely by wavelet analysis.Moreover,with the increase of dimensions and speeding up of convergence,the differential equations could be solved more effectively.This has overcome many calculative obstacles caused by the large amount of calculations and the slow speed of convergence in the process of solving differential problems using many classic methods.