The aim of this paper is to presents a parallel processor technique for solving eigenvalue problem for ordinary differential equations using artificial neural networks. The proposed network is trained by back propagation with different training algorithms quasi-Newton, Levenberg-Marquardt, and Bayesian Regulation. The next objective of this paper was to compare the performance of aforementioned algorithms with regard to predicting ability. 1. Introduction These days every process is automated. A lot of mathematical procedures have been automated. There is a strong need of software that solves differential equations (DEs) as many problems in science and engineering are reduced to differential equations through the process of mathematical modeling. Although model equations based on physical laws can be constructed, analytical tools are frequently inadequate for the purpose of obtaining their closed form solution and usually numerical methods must be resorted to. The application of neural networks for solving differential equations can be regarded as a mesh-free numerical method. It has been proved that feed forward neural networks with one hidden layer are capable of universal approximation, for problems of interpolation and approximation of scattered data. 2. Related Work Neural networks have found application in many disciplines: neurosciences, mathematics, statistics, physics, computer science, and engineering. In the context of the numerical solution of differential equations, high-order derivatives are undesirable in general because they can introduce large approximation error. The use of higher order conventional Lagrange polynomials does not guarantee to yield a better quality (smoothness) of approximation. Many methods have been developed so far for solving differential equations; some of them produce a solution in the form of an array that contains the value of the solution at a selected group of points [1]. Others use basis functions to represent the solution in analytic form and transform the original problem usually to a system of algebraic equations [2]. Most of the previous study in solving differential equations using artificial neural network (Ann) is restricted to the case of solving the systems of algebraic equations which result from the discretisation of the domain [3]. Most of the previous works in solving differential equations using neural networks is restricted to the case of solving the linear systems of algebraic equations which result from the discretisation of the domain. The minimization of the networks energy function
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