Simulation and evaluation of interval-valued fuzzy linear Fredholm integral equations with interval-valued fuzzy neural network

In the present paper, using generalizations of the fuzzy integral equations for interval-valued fuzzy sets, we introduce and study new generalized interval-valued fuzzy linear Fredholm integral equation concepts. The work of this paper is an expansion of the research of real fuzzy linear Fredholm integral equations. In this paper interval-valued fuzzy neural network (IVFNN) can be trained with crisp and interval-valued fuzzy data. In this paper, a novel hybrid method based on IVFNN and Newton–Cotes methods with positive coefficient for the solution of interval-valued fuzzy linear Fredholm integral equations (IVFLFIEs) of the second kind is presented. Within this paper the fuzzy neural network model is used to obtain an estimate for the fuzzy parameters in a statistical sense. Then a simple algorithm from the cost function of the interval-valued fuzzy neural network is proposed, in order to find the approximate parameters. We propose a learning algorithm from the cost function for adjusting of interval-valued fuzzy weights. Here neural network is considered as a part of a larger field called neural computing or soft computing. Finally, we illustrate our approach by some numerical examples.