Solutions of higher order linear fuzzy differential equations with interactive fuzzy values

Abstract

In this study, we consider higher order linear differential equations with additional conditions (initial and/or boundary) given by interactive fuzzy numbers. The concept of interactivity arises from the notion of a joint possibility distribution (J). The proposed method for solving fuzzy differential equations is based on an extension of the classical solution via sup-J extension, which is a generalization of Zadeh's extension principle. We prove that under certain conditions, the solution via Zadeh's extension principle is equal to the convex hull of the solutions produced by the sup-J extension. We also show that the solutions based on the Fréchet derivatives of fuzzy functions coincide with the solutions obtained via the sup-J extension. All of the results are illustrated based on a 3rd order fuzzy boundary value problem.