Abstract
We give a new definition of fuzzy fractional derivative called fuzzy conformable fractional derivative. Using this definition, we prove some results and we introduce new definition of generalized fuzzy conformable fractional derivative.
Highlights
Fuzzy set theory is a powerful tool for modeling uncertainty and for processing vague or subjective information in mathematical models. eir main directions of development have been diversed, and its applications have been varied [1,2,3,4].e derivative for fuzzy valued mappings was developed by Puri and Ralescu [5], which generalized and extended the concept of Hukuhara differentiability for set-valued mappings to the class of fuzzy mappings
We have investigated generalized fuzzy conformable fractional differentiability. e conformable q-differentiability introduced here is a very general differentiability concept, being practically applicable, and we can calculate by the fuzzy conformable derivative of the product of two functions (Tq(f · g)) because all fractional derivatives do not satisfy the known formula Tq(f · g) Tq(f)g + fTq(g)
E disadvantage of fuzzy generalized conformable differentiability of a function seems to be that a simple fuzzy differential equation (y(q) + y 0, 0 < q ≤ 1, y(0) y0 ∈ RF) has not got a unique solution, so it may have several solutions. e advantage of the existence of these solutions is that we can choose the solution that reflects better the behaviour of the modelled real-world system
Summary
Fuzzy set theory is a powerful tool for modeling uncertainty and for processing vague or subjective information in mathematical models. eir main directions of development have been diversed, and its applications have been varied [1,2,3,4]. Fuzzy set theory is a powerful tool for modeling uncertainty and for processing vague or subjective information in mathematical models. E derivative for fuzzy valued mappings was developed by Puri and Ralescu [5], which generalized and extended the concept of Hukuhara differentiability for set-valued mappings to the class of fuzzy mappings. Using the H-derivative, Kaleva [6] started to develop a theory for FDE. In [7], a new well-behaved simple fractional derivative called “the conformable fractional derivative” depending just on the basic limit definition of the derivative, namely, for a function f(0, ∞) ⟶ R the (conformable) fractional derivative of order 0 < q ≤ 1 of f at t > 0 was defined by f t + εt1− q − f(t). Is defined the fractional derivative at 0 as (Tqf)(0) limt⟶0+ (Tqf)(t). E aim of this paper is to study and generalize the fuzzy conformable fractional derivative Is defined the fractional derivative at 0 as (Tqf)(0) limt⟶0+ (Tqf)(t). e aim of this paper is to study and generalize the fuzzy conformable fractional derivative
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