On the generalized LU-fuzzy derivative and fuzzy differential equations

Abstract

The generalized differentiability of a fuzzy-number-valued function of a real variable, as recently introduced by Bede and Gal (Fuzzy Sets and Systems, vol. 151, 2005), can be expressed by first defining a generalized Hukuhara difference and using it for the differentiability; to do so, the basic elements are the lower and upper functions which characterize the level-cuts of the fuzzy quantities i.e. functions that are monotonic over [0,1]. Using this fact, we present a (parametric) representation of fuzzy numbers and its application to the solution of fuzzy differential (initial value) equations (FDE). The representation uses a finite decomposition of the membership interval [0,1] and models the level-cuts of fuzzy numbers and fuzzy functions to obtain the formulation of a fuzzy differential equation y'=f(x,y) in terms of a set of ordinary (non fuzzy) differential equations, defined by the lower and upper components of the fuzzy-valued function f(x,y). From a computational view, the resulting ODE's can be analyzed and solved by standard methods of numerical analysis.