الفهرس 
Introduction
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Abstract
The primary objective of the thesis is to outline a general framework for tackling linear twodimensional timedependant diffusion problems in rectangular domains using the waveletGalerkin method. The trial (test) functions for the twodimensional Galerkin method are obtained via tensor product of translated versions of dilated Daubechies scaling function. An estimate to the coefficient of the twodimensional waveletGalerkin expansion using these trial functions is given. An iterative scheme is deduced for the case of constant coefficients problems then it is generalized for the case of variable coefficients. Constructing this scheme is based on connection coefficients, the Kronecker product of matrices, and the CrankNicolson scheme. The discretization technique is applied to enforce boundary conditions to the obtained system of equations. The validity of the proposed method is verified by applying it to timedependent diffusion problems with different types of nonhomogeneity terms and boundary conditions.The thesis is organized as follows. Chapter one gives a historical view of the subject of wavelets, a summary of the methods used in solving linear partial differential equations, the problem statement, a literature review, and the work objective. Chapter two begins with a discussion of the concept of scale followed by the basic concepts of multiresolution analysis in one and two dimensions. Finally, the basic properties of Daubechies wavelets are illustrated. Chapter three is divided into four sections. The first section is devoted to discuss the main concepts of the Galerkin method. The second section is the waveletGalerkin section in which an estimate to the waveletGalerkin expansion coefficient is given and the definitions of the needed connection coefficients are introduced. In sections three and four, the iterative scheme for solving the specified problem is deduced for constant and variable coefficients cases, respectively. In chapter four the structure of the matrices arising in the application of the method on the chosen domain and the figures showing the results of applying the method to some numerical experiments are displayed.The variable time step technique is used to overcome a stability problem appears when much smaller time step is used. Chapter five is devoted to present some conclusion remarks and the future work. The thesis includes three appendices. The first appendix presents some features of Hilbert and Sobolev spaces that are used inside the thesis. The second appendix lists the main properties of the Kronecker product and the definitions of matrix norm and condition number. The last appendix is devoted to present the algorithm used for twodimensional interpolation and the approximation properties of polynomial interpolation using Chebyshev nodes. The thesis ends with the references used in the search.