Neuro-fuzzy system for solving fuzzy singular perturbation problems

In this paper we have introduced a new method, by using fully fuzzy neural network (FINN) to find the numerical solution for solving fuzzy singular perturbation problems (FSPPS) for ordinary differential equations. It uses the Levenberg-Marquardt training machine algorithm (TrianLM) for calculating the Fuzzy neural network parameters and the sigmoid function of the hidden units is hyperbolic tangent activation function, that is: Gamma(x) = e(x) -e(-x)/e(x) +e(-x). The fuzzy trial solution of FSPPs is written as a sum of two parts. The first part satisfies the fuzzy conditions, it contains no fuzzy adjustable parameters. The second part involves a feed-forward fuzzy neural network containing fuzzy adjustable parameters. In comparison with existing numerical methods, shows upp that the use of fuzzy neural networks provides solutions with good generalization and high accuracy. The proposed method is illustrated by several examples.