الفهرس 
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Abstract
The main purpose of this thesis is to present and use the Adomians decomposition method (ADM) and the numerical quadrature techniques in solving linear and nonlinear integral equations and systems of such equations. ADM is a powerful, very effective and reliable device for solving deterministic and stochastic equations as well as systems of such equations. In this work, ADM is applied to solve a class of integral equations. Some convergence theorems are proved. It is used to solve some different problems which illustrate the reliability of the method. we compare ADM with the numerical quadrature methods. It is found that the ADM requires less computational efforts to obtain an accurate solutio but takes more CPU time to obtain the same accuracy. The thesis is organized in five chapters as follows: Chapter one gives a brief introduction to the concept of integral equations, its main classifications, the well known integral equations, some special kinds of kernels and some forms of systems of integral equations. Also, a general historical introduction for the ADM is presented with some references. A simple introduction to Nystrom (quadrature) method is shown with its principle rules. In chapter two, ADM is presented and applied to linear and nonlinear Volterra integral equations and linear and nonlinear Fredholm integral equations. The solution algorithm and proof of existence and uniqueness are discussed for each type. It is found that the ADM is equivalent to the successive approximation method (Neumann series) in the linear case. A special case of nonlinearity is studied. Some examples are solved on each type to illustrate the method. In chapter three, ADM is presented and applied to various types of systems of integral equations, linear and nonlinear system of Volterra integral equations and system of linear and nonlinear Fredholm integral equations. The solution algorithm and proof of existence and uniqueness are discussed for each system. Some examples are solved on each type to illustrate the effectiveness of the method. In chapter 4 Nystrom (quadrature) method is shown with its principle rules, composite Simpson?s rule, composite Boole?s rule, and the generalized Gaussian rule. They are used to solve the nonlinear Fredholm integral equations and systems of such equations numerically. A number of examples are solved using the three numerical rules to distinguish the best which gives the most accurate results. Chapter five completes the thesis by introducing some conclusion remarks on the obtained results in the preceding chapters. Also suggestions for future work are stated. At the end, two appendices are added. The first summarize some preliminary definitions and important theorems for functional analysis. The second displays the Adomian?s polynomials with some famous examples.