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In this paper, we present a new method for solving some certain differential systems in the artificial neural networks field. The analytic and approximate solutions are given with series form in the spaces W[a,b] and H[a,b]. The method used in this thesis has several advantages; first, it is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical, physical, and engineering problems; second, it is accurate, need less effort to achieve the results, and is developed especially for the nonlinear cases; third, in the proposed method, it is possible to pick any point in the interval of integration and as well the approximate solutions will be applicable; fourth, the method does not require discretization of the variables, and it is not effected by computation round off errors and one is not faced with necessity of large computer memory and time. Results presented in this thesisshow potentiality, generality, and superiority of our method as compared with the Range Kutta method.

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