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To Solve Laplace Equations with Periodic Quasi Wavelet Boundary Element Method

Author: HongJiFang
Tutor: RanQiWen
School: Harbin Institute of Technology
Course: Applied Mathematics
Keywords: boundary element method boundary element integral wavelet periodic quasi wavelet
CLC: O241.82
Type: Master's thesis
Year: 2009
Downloads: 14
Quote: 0
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Abstract


Boundary element method is an effective method to solve partial differential equation. Boundary element method naturalizes the differential equation in solve region to boundary and then disperses boundary to solve. The main virtue of boundary element method is the reduce of dimensions, thus equations and dates the question needed is decreased so the solving work is simplified, that is advantage to solve high dimension equations.Wavelet theory as a new mathematical branche starts from the study of S. Mallat and Y. Meyer in 1980s, that is the construction of wavelet basis and has developed rapidly ever since. Wavelet transform overcomes defects of traditional Fourier transform,which is a breakthrough progress after Fourier analysis. Wavelet has well localized property both in time and frequency domains, creates an wavelet upsurge in application fields. It possess abundant theory and is applied widely, such as signal processing, image analysis. It is a powerful method and tools in application fields. It gives new idea to related fields and is concerned by more and more math research workers, because of its own smoothness and local compact support properties, can solve numerical computing integral and differential equations better.Since was introduced in 90s last century, wavelet boundary element method is always hot spots studied by scholars home and aboard. Wavelet BEM possess high iteration efficiency and preconditioning matrix simple, thus concerned by numerous scholars as a fast BEM solving method.The paper is divided to four chapters. The first chapter mainly relates background of selected subjects, history and present situations of wavelet and boundary element method researched and study work of this paper. The second chapter relates basic wavelet analysis theory including definition and property of wavelet, wavelet transform, multi resolution analysis and scale function as well as periodic quasi wavelet theory. The third chapter introduces basis theory of boundary element, summary of method of weight residual and variational method and deduction process of boundary integral equation. The fourth chapter is the main work of this paper, its mainly studied numerical computing of two dimensions Laplace equation using quasi periodic wavelet boundary element method. First using boundary element method coverts equation to be solved into boundary integral equation, then spread boundary integral equations by quasi wavelet basis turns to corresponding algebraic equations and solve it getting approximately solutions. In the process of solving equations do matrix transform using wavelet matrix and use multi scale method solving new algebraic equations reduce calculated amount, discuss convergence, complexity, error analysis of this arithmetic.

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CLC: > Mathematical sciences and chemical > Mathematics > Computational Mathematics > Numerical Analysis > The numerical solution of differential equations, integral equations > Numerical Solution of Partial Differential Equations
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