Generalized differentiability of fuzzy-valued functions
TL;DR: Using novel generalizations of the Hukuhara difference for fuzzy sets, new generalized differentiability concepts for fuzzy valued functions are introduced and studied.
About: This article is published in Fuzzy Sets and Systems.The article was published on 2013-11-01 and is currently open access. It has received 497 citations till now. The article focuses on the topics: Fuzzy number & Fuzzy set.
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245 citations
TL;DR: A generalization of the Hukuhara difference for closed intervals on the real line is used to develop a theory of the fractional calculus for interval-valued functions.
166 citations
TL;DR: The related theorems and properties of fuzzy Caputo fractional differential equation FCFDE under the Generalized Hukuhara differentiability are proved in detail and the method is illustrated by solving some examples.
Abstract: In this paper the fuzzy Caputo fractional differential equation FCFDE under the Generalized Hukuhara differentiability is introduced. Also the existence and uniqueness of the solution for a class of fuzzy Caputo fractional differential equation with initial value is studied, and an analytical solution of FCFDE is obtained. The related theorems and properties are proved in detail and the method is illustrated by solving some examples.
142 citations
TL;DR: It is 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, and the condition for the existence of the granular derivative of a fuzzy function is provided by a theorem.
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.
120 citations
Cites background or methods from "Generalized differentiability of fu..."
...It is easy to see that w̃ = ũ gH ṽ does not exist [23]....
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...Definition 6: [23] The fuzzy g-derivative of the fuzzy function f̃ : (a, b) ⊆ R → E1 at the point x0 ∈ (a, b) is defined as...
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...Theorem 2: [23] Let f̃ : (a, b) ⊆ R → E1 be such that [f̃(x)] = [f(x), f μ (x)]....
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...Following the definition of gH-derivative, to reduce gHderivative restrictions, another derivative—called generalized derivative (g-derivative)—was presented in which generalized difference (g-difference) was used that is more general compared to the others [23]....
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...Definition 5: [23] Let f̃ : (a, b) ⊆ R → E1 be a fuzzy function and x0 ∈ (a, b)....
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TL;DR: The existence and uniqueness results of solutions for FFFDEs under Caputo generalized Hukuhara differentiability are studied and the solution to fuzzy fractional functional initial value problem by a modified Adams–Bashforth–Moulton method is presented.
114 citations
References
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TL;DR: F fuzzy-set-valued mappings of a real variable whose values are normal, convex, upper semicontinuous and compactly supported fuzzy sets in Rn are studied and the existence and uniqueness theorem for a solution to a fuzzy differential equation is given.
1,475 citations
TL;DR: This paper shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in [4] to define differentiation and integration of fuzzy-valued functions in ways that parallel closely the corresponding definitions for real differentiation and Integration.
1,273 citations
"Generalized differentiability of fu..." refers background in this paper
...The "nested" property is the basis for the LU representation (L for lower, U for upper) (see [14], [42])....
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..., [12], [18], [14]....
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TL;DR: The Radstrom embedding theorem is generalized and is used to define the concept of the differential of a fuzzy function.
985 citations
TL;DR: generalized concepts of differentiability (of any order n@?N), which solves this shortcoming of fuzzy number differentiability, are introduced and some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c@?g(x).
911 citations
TL;DR: Generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated and the extension principle and the use of extremal solutions of deterministic initial value problems are applied.
812 citations