الفهرس 
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Abstract
The primary objective of this thesis is to introduce a general framework for tackling some classes of linear and nonlinear partial differential equations using the wavelet-Galerkin method. Wavelet-Galerkin method for solving these classes of equations depends mainly on the computation of the unbounded connection coefficients. So, accurate algorithms for computing the required connection coefficients are introduced. Then, a wavelet-Galerkin technique is introduced for solving some nonlinear partial differential equations. Also, a comparison between finite element method based on Petrov-Galerkin and wavelet-Galerkin method is given by considering convection-diffusion model. Finally, the Rothe-wavelet-Galerkin method is introduced for solving both linear parabolic partial differential equations of order 2s and nonlinear diffusion-reaction equations. The error analysis for both cases is also studied. The thesis is organized as follows: The introductory chapter is devoted to give a simple introduction to partial differential equations including revision of some methods of solution that most frequently used in solving them. A historical preview of the wavelets is also given. Then short highlights of the thesis together with the aim of the thesis are presented. Chapter two gives a literature review and some basic concepts of wavelets. Then numerical computations of scaling and wavelet functions, derivatives of the scaling function and moments of the scaling function are reviewed. Also, an accurate algorithms for computing the unbounded two- and three-term connection coefficients are introduced. This is followed by introducing the concept of periodized wavelets and algorithm for evaluating scaling function expansion. Chapter three begins with presenting some basic concepts of nonlinear partial differential equations. An efficient algorithm for solving a class of nonlinear partial differential equations is introduced. Finally, some numerical experiments are studied. Chapter four is devoted to the numerical solution of convection-diffusion equations by finite element based on Petrov-Galerkin approach and wavelet-Galerkin methods. The two methods are compared by applying them to certain numerical experiment. In chapter five the Rothe-wavelet technique for solving a class of both linear and nonlinear parabolic partial differential equations is introduced. A full error analysis and numerical experiment for each case are given. Chapter six is devoted to present the conclusion and the suggested future work. The thesis includes three appendices. The first appendix is devoted to the treatment of the different kinds of boundary conditions. The second appendix presents the Newton’s method for solving system of nonlinear algebraic equations. The third one gives a summary of some basic definitions and theorems of functional analysis that is needed in the thesis. The thesis ends with the list of references used in the research. Keywords: Partial differential equations, nonlinear partial differential equations, wavelet-Galerkin methods, Rothe-wavelet technique.