Fuzzy Conformable Fractional Differential Equations

In this study, fuzzy conformable fractional differential equations are investigated. We study conformable fractional differentiability, and we define fractional integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy fractional differential equation by using the concept of conformable differentiability. 'is concept is based on the enlargement of the class of differentiable fuzzy mappings; for this, we consider the lateral Hukuhara derivatives of order q ∈ (0, 1].


Introduction
Fractional calculus is generalization of differentiation and integration to an arbitrary order. e derivative for fuzzy-valued mappings was developed by [1] that generalized and extended the concept of Hukuhara differentiability (H-derivative) for setvalued mappings to the class of fuzzy mappings. Subsequently, using the H-derivative [2,3] started to develop a theory for FDE. e concept of the fuzzy fractional derivative was introduced by [4] and developed by [5][6][7][8][9][10][11], but these researchers tried to put a definition of a fuzzy fractional derivative. Most of them used an integral from the fuzzy fractional derivative, two of which are the most popular ones, Riemann-Liouville definition and Caputo definition [12][13][14]. All definitions above satisfy the property that the fuzzy fractional derivative is linear. is is the only property inherited from the first fuzzy derivative by all of the definitions. However, the following are some of the setbacks of the other definitions [15]. e fuzzy conformable derivative may facilitate some computations: (i) It satisfies all concepts and rules of an ordinary derivative such as quotient, product, and chain rules while the other fractional definitions fail to meet these rules (ii) It can be extended to solve exactly and numerically fractional differential equations and systems easily and efficiently And it was introduced and developed in [16,17]. e objective of this study is to present some results for fuzzy conformable differentiability and fuzzy fractional integrability of such functions; we study the fuzzy fractional differential equations (FFDEs) by using this derivative and give an existence and uniqueness theorem for a solution of FFDEs.

Preliminaries
{ } the class of fuzzy subsets of the real axis satisfying the following properties: (1) { u(x) ≥ α}; then, from (i) to (iv), it follows that the α-level set the family of all nonempty compact convex subsets of R and define the addition and scalar multiplication in P K (R) as usual.
Theorem 1 (see [7]). If u ∈ R F , then is a nondecreasing sequence which converges to α, and then, ; α ∈ (0, 1] is a family of closed real intervals verifying (i) and (ii), then A α defined a fuzzy number u ∈ R F such that [u] α � A α for 0 < α ≤ 1 and Lemma 1 (see [18] e following arithmetic operations on fuzzy numbers are well known and frequently used below. If u, v ∈ R F , then Definition 1 (see [19,20]). Let u, v ∈ R F . If there exists w ∈ R F such as u � v + w, then w is called the H-difference of u, v, and it is denoted as u⊖v.
Definition 2 (see [21]). Let we denote Define d: where d H is the Hausdorff metric.
It is well known that (R F , d) is a complete metric space. We list the following properties of d(u, v): for all u, v, w ∈ R F and λ ∈ R.
Let (A k ) be a sequence in P K (R) converging to A. en, theorem in [2] gives us an expression for the limit.

Fuzzy Conformable Fractional
Differentiability. Now, we present our new definition, which is the simplest and most natural and efficient definition of fractional derivative of order q ∈ (0, 1]. Definition 3 (see [17]). Let F: (0, a) ⟶ R F be a fuzzy function, and q th order fuzzy conformable fractional derivative of F is defined by Hence, If F is q-differentiable in some (0, a) and lim t⟶0 + F (q) (t) exists, then and the limits (in the metric d).
where T q F α is denoted from the conformable fractional derivative of F α of order q.
Proof. For each s ∈ [t 1 , t 2 ], there exists δ(s) > 0 such that the H-differences F(s + εs 1− q )⊖F(s) and F(s)⊖F(s − εs 1− q ) exist for all 0 ≤ ε < δ(s). en, we can find a finite sequence t 1 � s 1 < s 2 < · · · < s n � t 2 such that the family for some A 1 , A 2 , λ i ∈ R F . Hence, Proof. Let t, t + t 1− q ε ∈ (0, a) with ε > 0. en, by properties of equation (7) and the triangle inequality, we have where ε is so small that the H-difference F(t + t 1− q ε)⊖F(t) exists. By the differentiability, the right-hand side goes to zero as ε ⟶ 0 + , and hence, F is right continuous. e left continuity is proved similarly.
e proof is similar to the proof of eorem 8 case (i) in [17] and is omitted.
and we get that is, the H-difference By similar reasoning, we get that there exist u 2 (t, εt 1− q ) and v 2 (t, εt 1− q ) such that and so that is, the H-difference We observe that Finally, by multiplying (21) and (24) with 1/ε and passing to limit with lim ε⟶0 + , we get that F + G is q-differentiable and T q (F + G)(t) � T q F(t) + T q G(t). e case (ii) is similar to the previous one. . ((0, a), R) ∩ L 1 ((0, a), R) for all α ∈ [0, 1] and let International Journal of Differential Equations for α n ∈ [0, 1] and i � 1, 2. Obviously, g i is integrable on (0, a). erefore, if α n ↑α, then by Lebesque's dominated convergence theorem, we have From eorem 1, the proof is complete. ((0, a), R F ) define the fuzzy fractional integral for q ∈ (0, 1], where the integral t 0 (f α i /x 1− q )(x)dx for i � 1, 2 is the usual Riemann improper integral. Also, the following properties are obvious. Lemma 3. Let q ∈ (0, 1] and F, G: (0, a) ⟶ R F be fractional integrable and λ ∈ R. en, e proof is similar to the proof of eorem 4.3 cases (i) and (ii) in [2] and is omitted.

Fuzzy Comformable Fractional Differential Equations
We study the fuzzy initial value problem T q x(t) � F(t, x(t)), q ∈ (0, 1], where F: (0, a) × R F ⟶ R F is the continuous fuzzy mapping, and x 0 is the fuzzy number. From eorems 5, 8, and 9, it immediately follows.

Conclusion
In this study, for developing and proving some results for fuzzy conformable differentiability and fuzzy fractional integrability of such functions, we provided existence and uniqueness solutions to fuzzy fractional problems for order q ∈ (0, 1] FFDEs, which is interpreted by using the generalized conformable fractional derivatives concept. For future research, we will solve the fractional fuzzy conformable partial differential equations [22,23] and a class of linear differential dynamical systems [24] by using the proposed method.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.