International Journal of Differential Equations

Volume 2021, Issue 1 6655450

Research Article

Open Access

Fuzzy Conformable Fractional Differential Equations

Atimad Harir,

Corresponding Author

Atimad Harir

[email protected]

orcid.org/0000-0003-3603-9268

Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, P.O. Box 523, Beni Mellal 23000, Morocco universitesms.com

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Said Melliani,

Said Melliani

Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, P.O. Box 523, Beni Mellal 23000, Morocco universitesms.com

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Lalla Saadia Chadli,

Lalla Saadia Chadli

Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, P.O. Box 523, Beni Mellal 23000, Morocco universitesms.com

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Atimad Harir,

Corresponding Author

Atimad Harir

[email protected]

orcid.org/0000-0003-3603-9268

Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, P.O. Box 523, Beni Mellal 23000, Morocco universitesms.com

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Said Melliani,

Said Melliani

Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, P.O. Box 523, Beni Mellal 23000, Morocco universitesms.com

Search for more papers by this author

Lalla Saadia Chadli,

Lalla Saadia Chadli

Laboratory of Applied Mathematics and Scientific Computing, Sultan Moulay Slimane University, P.O. Box 523, Beni Mellal 23000, Morocco universitesms.com

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First published: 04 February 2021

https://doi.org/10.1155/2021/6655450

Citations: 5

Academic Editor: Peiguang Wang

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Abstract

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. This 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].

1. Introduction

Fractional calculus is generalization of differentiation and integration to an arbitrary order. The derivative for fuzzy-valued mappings was developed by [1] that generalized and extended the concept of Hukuhara differentiability (H-derivative) for set-valued mappings to the class of fuzzy mappings. Subsequently, using the H-derivative [2, 3] started to develop a theory for FDE. The concept of the fuzzy fractional derivative was introduced by [4] and developed by [5–11], 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 [12–14]. All definitions above satisfy the property that the fuzzy fractional derivative is linear. This 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]. The 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]. The 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.

2. Preliminaries

Let us denote by ℝ_ℱ = {u : ℝ⟶[0,1]} the class of fuzzy subsets of the real axis satisfying the following properties:

(i)
u is normal, i.e, there exists x₀ ∈ ℝ such that u(x₀) = 1
(ii)
u is fuzzy convex, i.e, for x, y ∈ ℝ and 0 < λ ≤ 1,
(1)
(iii)
u is upper semicontinuous
(iv)
[u]⁰ = cl{x ∈ ℝ|u(x) > 0} is compact

Then, ℝ_ℱ is called the space of fuzzy numbers. Obviously, ℝ ⊂ ℝ_ℱ. For 0 < α ≤ 1, denote [u]^α = {x ∈ ℝ|u(x) ≥ α}; then, from (i) to (iv), it follows that the α-level set [u]^α ∈ P_K(ℝ) for all 0 ≤ α ≤ 1 is a closed bounded interval which is denoted by . 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.

Theorem 1 (see [7].)If u ∈ ℝ_ℱ, then

(i)
[u]^α ∈ P_K(ℝ) for all 0 ≤ α ≤ 1
(ii)
for all 0 ≤ α₁ ≤ α₂ ≤ 1
(iii)
{α_k} ⊂ [0,1] is a nondecreasing sequence which converges to α, and then,

(2)

Conversely, if 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].)Let u, v : ℝ_ℱ⟶[0,1] be the fuzzy sets. Then, u = v if and only if [u]^α = [v]^α for all α ∈ [0,1].

The following arithmetic operations on fuzzy numbers are well known and frequently used below. If u, v ∈ ℝ_ℱ, then

(3)

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

(4)

Define d : ℝ_ℱ × ℝ_ℱ⟶ℝ₊ ∪ {0} by the equation

(5)

where d_H is the Hausdorff metric.

(6)

It is well known that (ℝ_ℱ, d) is a complete metric space. We list the following properties of d(u, v):

(7)

for all u, v, w ∈ ℝ_ℱ and λ ∈ ℝ.

Let (A_k) be a sequence in P_K(ℝ) converging to A. Then, theorem in [2] gives us an expression for the limit.

Theorem 2 (see[2]). If d(A_k, A)⟶0 as k⟶∞, then

(8)

3. Fuzzy Conformable Fractional Differentiability and Fuzzy Fractional Integral

3.1. 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

(9)

for all t > 0, q ∈ (0,1). Let F^(q)(t) stands for T_q(F)(t). Hence,

(10)

If F is q-differentiable in some (0, a) and exists, then

(11)

and the limits (in the metric d).

Remark 1. 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

(12)

where T_qF_α is denoted from the conformable fractional derivative of F_α of order q.

Theorem 3 (see[17]). Let F : (0, a)⟶ℝ_ℱ be q-differentiable. Denote . Then, and are q-differentiable and

(13)

Theorem 4. Let F : (0, a)⟶ℝ_ℱ is q-differentiable on (0, a). If t₁, t₂ ∈ (0, a) with t₁ ≤ t₂, then there exists λ ∈ ℝ_ℱ such that F(t₂) = F(t₁) + λ.

Proof. For each s ∈ [t₁, t₂], 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). Then, we can find a finite sequence t₁ = s₁ < s₂ < ⋯<s_n = t₂ such that the family covers [t₁, t₂] and . Pick , such that s_i < x_i < s_i+1. Then,

(14)

for some A₁, A₂, λ_i ∈ ℝ_ℱ. Hence,

(15)

Theorem 5. If F : (0, a)⟶ℝ_ℱ is q-differentiable, then it is continuous.

Proof. Let t, t + t^1−qε ∈ (0, a) with ε > 0. Then, by properties of equation (7) and the triangle inequality, we have

(16)

where ε is so small that the H-difference F(t + t^1−qε)⊖F(t) exists. By the differentiability, the right-hand side goes to zero as ε⟶0⁺, and hence, F is right continuous. The left continuity is proved similarly.

Theorem 6. Let q ∈ (0,1]. If F is differentiable and F is q-differentiable, then

(17)

The proof is similar to the proof of Theorem 8 case (i) in [17] and is omitted.

Theorem 7. Let q ∈ (0,1], and if F, G : (0, a)⟶ℝ_ℱ are q-differentiable and λ ∈ ℝ, then

T_q(F + G)(t) = T_q(F) + T_q(G) and
T_q(λF)(t) = λT_q(F)(t)

Proof. Since F is q-differentiable, it follows that F(t + εt^1−q)⊖F(t) exists, i.e., there exists u₁(t, εt^1−q) such that

(18)

Analogously, since G is q-differentiable, there exists v₁(t, εt^1−q) such that

(19)

and we get

(20)

that is, the H-difference

(21)

By similar reasoning, we get that there exist u₂(t, εt^1−q) and v₂(t, εt^1−q) such that

(22)

and so

(23)

that is, the H-difference

(24)

We observe that

(25)

Finally, by multiplying (21) and (24) with 1/ε and passing to limit with , we get that F + G is q-differentiable and T_q(F + G)(t) = T_qF(t) + T_qG(t). The case (ii) is similar to the previous one.

3.2. Fuzzy Fractional Integral

Let q ∈ (0,1] and F : (0, a)⟶ℝ_ℱ be such that

for all t ∈ (0, a) and α ∈ [0,1]. Suppose that

for all α ∈ [0,1] and let

(26)

Lemma 2. The family {A_α; α ∈ [0,1]}, given by equation (26), defined a fuzzy number F ∈ ℝ_ℱ such that [F]^α = A_α.

Proof. For α < β, we have and . It follows A_α⊇A_β. Since , we have

(27)

for α_n ∈ [0,1] and i = 1,2. Obviously, g_i is integrable on (0, a). Therefore, if α_n↑α, then by Lebesque’s dominated convergence theorem, we have

(28)

From Theorem 1, the proof is complete.

Definition 4. Let F ∈ C((0, a), ℝ_ℱ)∩L¹((0, a), ℝ_ℱ) define the fuzzy fractional integral for q ∈ (0,1],

(29)

by`

(30)

where the integral

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. Then,

(i)
I_qλF(t) = λI_qF(t)
(ii)
I_q(F + G)(t) = I_qF(t) + I_qG(t)

Proof. The proof is similar to the proof of Theorem 4.3 cases (i) and (ii) in [2] and is omitted.

Theorem 8. T_qI_q(F)(t) = F(t), for t ≥ 0, where F is any continuous function in the domain of I_q

Proof. Since F is continuous, then I_q(F)(t) is clearly q-differentiable because

(31)

and t^q−1F(t) is continuous for all t ∈ (0, a); then, by Theorem 5.6 in [2] and Theorem 6, the fractional integral is q-differentiable. Hence,

(32)

Theorem 9. Let q ∈ (0,1] and F be q-differentiable in (0, a), and assume that the conformable derivative F^(q) is integrable over (0, a). Then, for each s ∈ (0, a), we have

(33)

Proof. Let q ∈ (0,1] and α ∈ [0,1] be fixed. We shall prove that

(34)

where

is the Hukuhara conformable fractional derivative of F_α; then, using Theorems 3 and 6 gives us the following equation.

(35)

By equation (29), we have

(36)

So,

(37)

where

is the Hukuhara derivative of F_α; equation (37) is also true for a fuzzy mapping F : (0, a)⟶ℝ_ℱ. The equality (34) now follows Theorem 5.7 in [2].

4. Fuzzy Comformable Fractional Differential Equations

We study the fuzzy initial value problem

(38)

where F : (0, a) × ℝ_ℱ⟶ℝ_ℱ is the continuous fuzzy mapping, and x₀ is the fuzzy number. From Theorems 5, 8, and 9, it immediately follows.

Theorem 10. A mapping x : (0, a)⟶ℝ_ℱ is a solution to problem (38) if and only if it is continuous and satisfies the integral equation:

(39)

for all t ∈ (0, a) and q ∈ (0,1].

Theorem 11. Let F : (0, a) × ℝ_ℱ⟶ℝ_ℱ be continuous, and assume that there exists k > 0 such that

(40)

for all t ∈ (0, a), x, y ∈ ℝ_ℱ. Then, problem (38) has a unique solution on (0, a).

Proof. If in problem (38) we consider the conformable derivative x^(q) for all q ∈ (0,1] Theorem 3, then from Theorem 6.1 in [2] and using Definition 4 and Lemma 1,(0, a) we can prove that there exists an unique solution on (0, a), and the proof is now complete.

Remark 2. In [15], it is observed that if we fuzzify the equivalent ordinary differential equation x^(q) + x = 0, then we will get fuzzy differential equations (the equation was fuzzified by adding a forcing term σ(t) in the right-hand side). That is, if we consider fuzzy differential equation x^(q) + x = σ(t) with the same initial condition x(t₀) = x₀, we get the result.

Consider the following linear fractional equation:

(41)

where σ ∈ C((0, a) × ℝ_ℱ). Denote

, and

Theorem 12. Equation (41) has a unique solution in (0, a), and for given initial x₀ ∈ ℝ_ℱ, it is given by

(42)

Proof. Equation (41) can be written, levelwise, as

(43)

for every α ∈ [0,1], so that

(44)

Thus, for t ∈ (0, a),

(45)

and, therefore, it can be deduced that

(46)

This proves that, for α ∈ [0,1],

(47)

So,

(48)

5. 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.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Open Research

Data Availability

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

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Fuzzy Conformable Fractional Differential Equations

Abstract

1. Introduction

2. Preliminaries

3. Fuzzy Conformable Fractional Differentiability and Fuzzy Fractional Integral

3.1. Fuzzy Conformable Fractional Differentiability

3.2. Fuzzy Fractional Integral

4. Fuzzy Comformable Fractional Differential Equations

5. Conclusion

Conflicts of Interest

Open Research

Data Availability

References

Citing Literature

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Fuzzy Conformable Fractional Differential Equations

Abstract

1. Introduction

2. Preliminaries

3. Fuzzy Conformable Fractional Differentiability and Fuzzy Fractional Integral

3.1. Fuzzy Conformable Fractional Differentiability

3.2. Fuzzy Fractional Integral

4. Fuzzy Comformable Fractional Differential Equations

5. Conclusion

Conflicts of Interest

Open Research

Data Availability

References

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