Solving Higher-Order Fractional Differential Equations by the Fuzzy Generalized Conformable Derivatives

In this paper, the generalized concept of conformable fractional derivatives of order q ∈ (n, n + 1] for fuzzy functions is introduced. We presented the definition and proved properties and theorems of these derivatives. $e fuzzy conformable fractional differential equations and the properties of the fuzzy solution are investigated, developed, and proved. Some examples are provided for both the new solutions.


Introduction
A closed-form solution for nonlinear fractional differential equations (FDEs) plays a significant role in understanding the qualitative as well as quantitative features of complex physical phenomena.
e nonlinear FDEs appear in different sciences and engineering problems such as control theory, signal processing, finance, electricity, mechanics, plasma physics, stochastic dynamical system, economics, and electrochemistry [1][2][3][4][5][6][7].A fuzzy fractional differentiation and fuzzy integration operators have different kinds of definitions that we can mention, the fuzzy Riemann-Liouville definition [8,9], the fuzzy Caputo definition [9,10], and so on.Lately, Khalid et al. [11] introduced a new simple definition of the fractional derivative named the conformable fractional derivative, which can redress shortcomings of the other definitions, and this new definition satisfies formulas of derivative of product and quotient of two functions [12,13].Harir et al. [14] introduced the fuzzy generalized conformable fractional derivative, which generalized and extended the concept of Hukuhara differentiability for set-valued mappings to the class of fuzzy mapping [15,16].
Our objective of this article is to present a generalized concept of conformable fractional derivative of order q ∈ (n, n + 1], n ∈ N for fuzzy functions.
en, we have investigated in more detail some new properties of these derivatives and we have proved some useful related theorems.We interpret fuzzy conformable fractional differential equations using this concept.We introduce new definitions of solutions.Two examples are provided.

Preliminaries
Let us denote by R F � u: R ⟶ [0, 1] { } the class of fuzzy subsets of the real axis satisfying the following properties [13,17]: (i) u is normal, i.e., there exists an x 0 ∈ R such that u(x 0 ) � 1. (ii) u is fuzzy convex, i.e., for x, y ∈ R and 0 < λ ≤ 1, (1) . By P K (R), we denote the family of all nonempty compact convex subsets of R and define the addition and scalar multiplication in P K (R) as usual.
Lemma 1 (see [18]).Let u, v: R ⟶ [0, 1] be the fuzzy sets. en, e following arithmetic operations on fuzzy numbers are well known and frequently used below [17]

⎧ ⎨ ⎩
(3) 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.
Theorem 2 (see [19]) for general a, b ∈ R, the above property does not hold.(iv) For any λ ∈ R and any u, v ∈ R F , we have λ (5) 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) [17]: (7) for all u, v, w ∈ R F and λ ∈ R.
Let (A k ) be a sequence in P K (R) converging to A. en, eorem in [17] gives us an expression for the limit.
Let I � (0, a) ⊂ R be an interval.We denote by C(I, R F ) the space of all continuous fuzzy functions on I which is a complete metric space with respect to the metric

Generalized Fuzzy Conformable Fractional Derivatives
Definition 2 (see [14]).Let F: I ⟶ R F be a fuzzy function.q th order "fuzzy conformable fractional derivative" of F is defined by for all t > 0, q ∈ (0, 1).If F is q-differentiable in some I, and lim t⟶0 + T q (F)(t) exists, then and the limits exist (in the metric d).
Remark 1 (see [14]).From the definition, it directly follows that if F is q-differentiable, then the multivalued mapping F α is q-differentiable for all α ∈ [0, 1] and Here, T q F α is denoted as the conformable fractional derivative of F α of order q. e converse result does not hold, since the existence of Hukuhara differences We consider the following definition [14].

2
Applied Computational Intelligence and Soft Computing Definition 3. Let F: I ⟶ R F be a fuzzy function and q ∈ (0, 1].One says, F is q (1) -differentiable at point t > 0 if there exists an element T q (F)(t) ∈ R F such that for all ε > 0 sufficiently near to 0, there exist F(t + εt 1− q )⊖F(t), F(t)⊖F(t − εt 1− q ) and the limits (in the metric d) F is q (2) -differentiable at t > 0 if for all ε < 0 sufficiently near to 0, there exist If F is q (n) -differentiable at t > 0, we denote its q-derivatives, for n � 1, 2. Definition 4. Let F: I ⟶ R F be a fuzzy function and now we introduce definitions and theorems for q ∈ (n, n + 1] for some natural number n.For the sake of convenience, we concentrate on q ∈ (1, 2] case be n-differentiable at t, where t > 0. en the fuzzy conformable fractional derivative of f of order q is defined by where q ∈ (n, n + 1] and [q] is the smallest integer greater than or equal to q and the limits exist (in the metric d).
exists on I and it is (m)-differentiable on I. e second derivatives of F are denoted by D (2)  n, m F(t) for n, m � 1, 2.
Proof.We present the details only for n � m � 1, since the other case is analogous.Let h � εt 2− q in Definition 4, then ε � t q− 2 h.erefore, if ε > 0 and α ∈ [0, 1], we have Dividing by ε, we have and passing to the limit lim Similarly, we obtain Applied Computational Intelligence and Soft Computing and passing to the limit and ε � t q− 2 h gives □ Theorem 5. Let 0 < q, p < 1, and F be a fuzzy function 2 times differentiable on an open real interval I, then the fuzzy conformable derivative obeys to the following where n, m � 1, 2.
Proof.Let n, m � 1, 2. By using eorem 7 and eorem 8 in [14], we have the following relation; On the other hand, It follows that for all n, m � 1, 2. e proof is complete.

□
Lemma 2. Let 0 < q < 1 and F be a fuzzy function 2 times differentiable on an open real interval I, then the fuzzy conformable derivative obeys the following where n, m � 1, 2.
Proof.Let n, m � 1, 2. By using eorems 7 and 8 in [14], we have the following relation: On the other hand, It follows that for all n, m � 1, 2. e proof is complete.
□ Theorem 6.Let F: I ⟶ R F , 1 < q ≤ 2 and p � q − 1. en the fuzzy conformable fractional derivative of order q, where D 1 n F(t) exists, is defined by where n, m � 1, 2.
Proof.We present the details only for the case n � m � 1, since the other case is analogous.By using eorem 8 in [14] and eorem 2.2 in [20], we have and of eorem 4, it is that Applied Computational Intelligence and Soft Computing q-differentiable (conformable fractional derivatives of order q ∈ (0, 1]) of F is denoted by T q (n,m) F(t) for n, m � 1, 2.
Theorem 7. Let 1 < q ≤ 2 and p � q − 1. en the fuzzy conformable fractional derivative of F order q, where D 1  1 Proof.We present the details only for n � m � 1, since the other cases are analogous.If ε > 0 and α ∈ [0, 1], we have Dividing by ε, we have Dividing by ε, we have Similarly, we obtain and passing to the limit, we have and using eorem 6 gives the theorem.

Fuzzy Conformable Fractional Differential Equations of Order q
In this section, we study the fuzzy conformable fractional differential equations of order q ∈ (1, 2], p ∈ (0, 1]: where a, b > 0, σ 0 , σ 1 ∈ R F , and g(t) is a continuous fuzzy function on some interval I.We give the following definition for the solutions of (41).

Conclusion
By using the concept of conformable generalized derivative and its extension to fractional derivatives of order q ∈ (0, 2], we show that we have several possibilities or types to define fractional derivatives of order q ∈ (0, 2] of fuzzy-numbervalued functions.en, we propose a new method to solve fuzzy fractional differential equations based on the selection of conformable derivative types covering all former solutions.With these ideas, the selection of conformable derivative type in each step of deprivation plays a crucial role.
For future research, we will solve the fractional fuzzy conformable partial differential equations [22,23] by using the proposed method.