Solving Singular Perturbation Problems With Initial and Boundary Conditions By Using Modified Neuro System

Abstract

The aim of this paper is to design a neural network for solving the singular perturbation problems by using neural networks. The modified neuro system using a polynomial of second degree is to replace each component in the training set. The foundation of this approach is to swap off each x in the input vector training set       (x_j ) ⃗=(x_1  , x_2  , …, x_n )  , x_j∈[a,b]  , the polynomial will be as  ξ(x) =λ/( 2 )(x^2+x+1), λ∈(a ,b). The appropriate value is determined within a certain range, which has a significant impact on the accuracy of the solution. The numerical results show that the modified neuro system method is better and more accurate than usual artificial neural network method, the main reason for this point is connected with the chosen value of  . Finally, a method of updating the neural network is clarified by the numerical results of some examples that are compared to the usual artificial neural network method and through which the accuracy of the solution and the rapidity of convergence is proved.