Numerical Solution Of Partial Differential Equations Based On Wavelet Collocation Method

Wavelet analysis is an extension and development of Fourier analysis, since thebirth and the development of its so far, the theoretical framework constantly improveand mature. Compared with the Fourier analysis, wavelet analysis has moreadvantage,which in many areas has been fully reflected. From the purely mathematicaltheory to many application areas, such as astrology, economics, oceanography,seismology, wavelet analysis plays their respective role. In the field of mathematics,wavelet analysis can solve many problems, especially in the numerical solution ofpartial differential equations.This paper mainly describes and discuss the Haar wavelet collocation method andthe Shannon wavelet collocation method for solving partial differential equations in themechanics: wave equation and convection-diffusion equation, and give the methods forthe numerical solution of two equations. Finally the approximation degree of thewavelet solution, which is deduced by two wavelet collocation methods, is analyzed.The specific content of the work is as follows:1. Introduced generation and development of wavelet analysis; the review ofwavelet’s application in partial differential equations; the basic wavelet knowledgewhich the paper involves, including: the concept of the wavelet; two kinds of wavelettransform; multi-resolution analysis; orthogonal wavelet.2. Introduced definition and properties of Haar wavelet. Integral operator matrix ofthe corresponding numerical solution of partial differential equations is obtained byHaar wavelet. Via these integral operator matrices and their properties, wave equationwas decomposed into algebraic equations, and then the equations were solved by thepreconditioning technique. Finally the approximation form of solution was obtained.Examples illustrated the effectiveness of this method, and the approximation results,which are obtained by Haar wavelet collocation method, are investigated.3. Introduced the definition and properties of Shannon scaling function and thewavelet function, and the basis functions which is derived by Shannon scaling functioninL2R. With base functions and their properties, the convection-diffusion equation was decomposed into one-dimensional equations. Then approximation solution wasobtained by the Runge-Kutta-Gill method, as well as the preconditioning technique.Finally examples illustrated the effectiveness of this method, and the approximationresults, which are obtained by Shannon wavelet collocation method, are investigated.