الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
Numerical and analytical methods for solving Fredholm integral equation, Volterra integral equation and Fredholm-Volterra integral equation have drawn much attention in recent years.
Adomian decomposition method (ADM), the modified decomposition method, the variational iteration method, the direct computation method, the successive approximations method, the method of regularization and the homotopy perturbation method are examples of the analytical methods for solving integral equation. For concrete problems, sometimes we can’t obtain exact solutions using analytical methods. In this case we determine the solution in a series form that may converge to exact solution if such a solution exists. Other series may not give exact solution, and in this case the obtained series can be used for numerical purposes. The more terms that we determine the higher accuracy level that we can achieve.
Collocation method, Galerkin method, Nystrom method and iteration method are most used numerical methods for solving integral equations. But, all these numerical methods reduce the solution of the nonlinear integral equation to the solution of a nonlinear system of algebraic equations. The iteration methods, for example Newton’s method, for solving such cumbersome nonlinear system are usually sensitive to the selection of initial guess. In this thesis we overcome this obstacle and solve nonlinear integral equations using a discrete version of Adomain decomposition method. This proposed method will be called discrete Adomain decomposition method (DADM).
The thesis is organized in five chapters. Chapter (1) presents a brief introduction to integral equations, classification of Integral equations and classification of integro-differential equations. Then, some mathematical preliminaries are introduced. Finally, the chapter is concluded by a literature review of mathematical researches in integral equations.
In chapter (2), Adomian decomposition method and discrete Adomain decomposition method are applied to solve linear and nonlinear Fredholm integral equation. Then, the convergence of the technique is discussed and the error is estimated. Finally, to verify the theoretical results, some numerical examples are presented.
In chapter (3), Adomian decomposition method and discrete Adomain decomposition method are applied to solve linear and nonlinear Volterra integral equation. Then, the convergence of the technique is discussed and the error is estimated. Finally, to verify the theoretical results, some numerical examples are presented.
In chapter (4), Adomian decomposition method and discrete Adomain decomposition method are applied to solve two dimensional nonlinear Fredholm-Volterra integral equations. Then, the convergence of the technique is discussed and the error is estimated. The theoretical results are verified using some numerical examples. Finally, chapter (5) contains the conclusion and suggested future work.