Granular Differentiability of Fuzzy-Number-Valued Functions

Abstract

In this paper, using the concept of horizontal membership functions, a new definition of fuzzy derivative called granular derivative is proposed based on granular difference. Moreover, a new definition of fuzzy integral called granular integral is defined, and its relation with the granular derivative is given. A new definition of a metric—granular metric—on the space of type-1 fuzzy numbers, and a concept of continuous fuzzy functions are also presented. Restrictions associated to previous approaches—Hukuhara differentiability, strongly generalized Hukuhara differentiability, generalized Hukuhara differentiability, generalized differentiability, Zadeh's extension principle, and fuzzy differential inclusions—dealing with fuzzy differential equations (FDEs) are expressed. It is shown that the proposed approach does not have the drawbacks of the previous approaches. It is also demonstrated how this approach enables researchers to solve FDEs more conveniently than ever before. Moreover, we showed that this approach does not necessitate that the diameter of the fuzzy function be monotonic. It is also proved that the result of each of the four basic operations on fuzzy numbers introduced based on the proposed approach leads to a fuzzy number. Moreover, the condition for the existence of the granular derivative of a fuzzy function is provided by a theorem. Additionally, by two examples, it is shown that the existence of the granular derivative of a fuzzy function does not imply the existence of the generalized Hukuhara differentiability of the fuzzy function, and vice versa. The terms doubling property and unnatural behavior in modeling phenomenon are also introduced. Furthermore, using some examples, the paper proceeds to elaborate on the efficiency and effectiveness of the proposed approach. Moreover, as an application of the proposed approach, the response of Boeing 747 to impulsive elevator input is obtained in the presence of uncertain initial conditions and parameters.