Numerical Solution of Fuzzy Differential Equations with Z-numbers Using Bernstein Neural Networks

Abstract

The uncertain nonlinear systems can be modeled with fuzzy equations or fuzzy differential equations (FDEs) by incorporating the fuzzy set theory. The solutions of them are applied to analyze many engineering problems. However, it is very difficult to obtain solutions of FDEs.In this paper, the solutions of FDEs are approximated by two types of Bernstein neural networks. Here, the uncertainties are in the sense of Z-numbers. Initially the FDE is transformed into four ordinary differential equations (ODEs) with Hukuhara differentiability. Then neural models are constructed with the structure of ODEs. With modified back propagation method for Z-number variables, the neural networks are trained. The theory analysis and simulation results show that these new models, Bernstein neural networks, are effective to estimate the solutions of FDEs based on Z-numbers.

The Shooting Method for Solving Second Order Fuzzy Two-Point Boundary Value Problems

“…Existence and uniqueness of numerical solutions was considered by Fard et al [5], while in Fatullayev et al [6], the authors considered initial value solvers and Gaussian iteration to solve the systems involved. Solving FDE using Bernstein neural network was done by Jafaeri et al [9] while Adams and Nystrorm methods and predictor-corrector methods for solving FDEs can be found in Khastan and Ivaz [12] and Khastan and Nieto [13] respectively. Euler method was applied for solving initial value problem for FDEs in Palligkinis et al [17] and Runge-Kutta methods and numerical simulation using general linear method and B-series was done in Rabiei et al [18] and Rabiei et al [19], respectively.…”

Fuzzy Differential Equations for Modeling and Control of Fuzzy Systems

“…In recent days, many methods involving uncertainties have used fuzzy numbers [1][2][3][4][5][6][7][8], where the uncertainties of the system are represented by fuzzy coefficients. Fuzzy method is a highly favorable tool for uncertain nonlinear system modeling.…”

The Optimization of Marine Diesel Engine Rotational Speed Control Process by Fuzzy Logic Control Based on Particle Swarm Optimization Algorithm

“…On the other hand, the study of fuzzy logic theory has also conducted through some articles. Jafari et al, had modeled the uncertain nonlinear systems along with fuzzy differential equations (FDEs) by fuzzy set [9]. Moreover, the determination of uncertainty property through Z-number of fuzzy equation has also been investigated by Jafari et al [10].…”