New Results on Multiple Solutions for Nth-Order Fuzzy Differential Equations under Generalized Differentiability

We firstly present a generalized concept of higher-order differentiability for fuzzy functions. Then we interpret Nth-order fuzzy differential equations using this concept. We introduce new definitions of solution to fuzzy differential equations. Some examples are provided for which both the new solutions and the former ones to the fuzzy initial value problems are presented and compared. We present an example of a linear second-order fuzzy differential equation with initial conditions having four different solutions

Improved determination of the weights in a clustering approach based on a weighted dissimilarity measure between fuzzy data

In the literature, a popular dissimilarity measure between each pair of fuzzy data is defined as a weighted sum of the squared Euclidean distances among the centers and the spreads. The flexible weights introduced by D'Urso and Giordani in [Computational Statistics gf Data Analysis 50 (2006) 1496-1523] may be helpful in certain situations as shown in that paper. The flexible weights are obtained by the minimization algorithm, and a larger weight is given to the center distance than the spread one.In the present study, the weights are proposed to be fixed and equal to 0.5, which can contribute to higher performances of clustering methods in some cases. For instance, where the distance among the spreads is longer than that among the centers, i.e., the centers do not play a relevant role while the spreads play the predominant role, it is clear that a clustering method that takes the advantage of equal weights would outperform the other. Indeed, when there is no external condition, choosing the weights fixed and equal to 0.5 would be the best strategy.To deeply investigate our claim, we provide a wide simulation study to show whether and under what circumstances the flexible weights are (not) helpful. The results furnished by numerical experiments on both simulated and benchmark datasets demonstrate the merit of using the fixed and equal to 0.5 weights suggested in the present paper

A Numerical Method for Fuzzy Differential Equations and Hybrid Fuzzy Differential Equations

Numerical algorithms for solving first-order fuzzy differential equations and hybrid fuzzy differential equations have been investigated. Sufficient conditions for stability and convergence of the proposed algorithms are given, and their applicability is illustrated with some examples

On the linear fuzzy model associated with Caputo–Fabrizio operator

Abstract In this paper, we introduce the fuzzy Caputo–Fabrizio operator under generalized Hukuhara differentiability concept. In this setting, we study the linear fuzzy fractional initial value problems and present the general form of their solutions. Some examples are given to illustrate our results