Design Feed Forward Neural Network to Solve Singular Boundary Value Problems

The aim of this paper is to design feed forward neural network for solving second-order singular boundary value problems in ordinary differential equations. The neural networks use the principle of back propagation with different training algorithms such as quasi-Newton, Levenberg-Marquardt, and Bayesian Regulation. Two examples are considered to show that effectiveness of using the network techniques for solving this type of equations. The convergence properties of the technique and accuracy of the interpolation technique are considered. 1. Introduction The study of solving differential equations using artificial neural network (Ann) was initiated by Agatonovic-Kustrin and Beresford in [1]. Lagaris et al. in [2] employed two networks, a multilayer perceptron and a radial basis function network, to solve partial differential equations (PDE) with boundary conditions defined on boundaries with the case of complex boundary geometry. Tawfiq [3] proposed a radial basis function neural network (RBFNN) and Hopfield neural network (unsupervised training network). Neural networks have been employed before to solve boundary and initial value problems. Malek and Shekari Beidokhti [4] reported a novel hybrid method based on optimization techniques and neural networks methods for the solution of high order ODE which used three-layered perceptron network. Akca et al. [5] discussed different approaches of using wavelets in the solution of boundary value problems (BVP) for ODE, also introduced convenient wavelet representations for the derivatives for certain functions, and discussed wavelet network algorithm. Mc Fall [6] presented multilayer perceptron networks to solve BVP of PDE for arbitrary irregular domain where he used logsig. transfer function in hidden layer and pure line in output layer and used gradient decent training algorithm; also, he used RBFNN for solving this problem and compared between them. Junaid et al. [7] used Ann with genetic training algorithm and log sigmoid function for solving first-order ODE. Abdul Samath et al. [8] suggested the solution of the matrix Riccati differential equation (MRDE) for nonlinear singular system using Ann. Ibraheem and Khalaf [9] proposed shooting neural networks algorithm for solving two-point second-order BVP in ODEs which reduced the equation to the system of two equations of first order. Hoda and Nagla [10] described a numerical solution with neural networks for solving PDE, with mixed boundary conditions. Majidzadeh [11] suggested a new approach for reducing the inverse problem for a domain to an equivalent