الفهرس 
Introduction
The stiffness matrixOnly 14 pages are availabe for public view
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Abstract
Wavelets offer a comprehensive mathematical power and provide a good application potential in many interesting physical problems. The vast attention that wavelets received has resulted, in the last years, in the development of both the mathematical theories and applications. Nowadays, the use of wavelets covers many areas, from engineering and signal processing to physics and pure mathematics. The objective of the thesis is to extend the use of wavelet-Galerkin method to obtain the approximate solution of coupled systems of partial differential equations in rectangular domains. The wavelet-Galerkin method is applied to a general coupled nonhomogeneous system of partial differential equations with constant coefficients, then it is generalized for coupled systems with variable coefficients. A recursive scheme is deduced for the two cases. Constructing this scheme is based on connection coefficients and Crank-Nicolson scheme. The validity of the proposed method is verified through it’s application to different coupled systems of partial differential equations. The thesis is organized as follows: Chapter one gives a brief historical view of wavelets, a summary of some methods used in solving coupled systems of partial differential equations, coupled partial differential equations in literature, solution methods sited in literature and Finally the work objective. Chapter two begins with the concept of scaling then the basic concepts of multiresolution analysis and the chapter ends by a discussion of the basic properties of Daubechies wavelets. Chapter three is divided into four sections, the first one presents the main concept of the Galerkin method and the second highlights the wavelet -Galerkin method. The third section shows the application of wavelet-Galerkin method to a general coupled nonhomogeneous system of partial differential equations with constant coefficients. The field variables in the original system are approximated by the j level wavelet series and the Galerkin discretization scheme is then used. The connection coefficients are used in the evaluation of terms occurred in the application of the wavelet-Galerkin procedure. Two cases are considered, the first when the nonhomogeneous term is variable separable and the second when it is not separable. The analysis ended up with a recursive linear system of equations, the solution of which yields an approximation to the solution of the coupled system of PDEs. In the fourth section, the wavelet-Galerkin method is applied to a general coupled nonhomogeneous system with variable coefficients for the two cases, when the nonhomogeneous term is separable and when it is not. Chapter four presents the structure of the matrices arising in the application of above method. Some numerical experiments are carried out. Chapter five is devoted to present some conclusion remarks and suggestions for the future work.