Some Fundamental Results on Fuzzy Conformable Differential Calculus

In this paper, we combine fuzzy calculus, and conformable calculus to introduce the fuzzy conformable calculus. We define the fuzzy conformable derivative of order 2Ψ and generalize it to derivatives of order pΨ. Several properties on difference, product, sum, and addition of two fuzzy-valued functions are provided which are used in the solution of the fuzzy conformable differential equations. Also, examples in each case are given to illustrate the utility of our results.


Introduction
In the real world, the data sometimes cannot be collected precisely. For example, the water level of a river cannot be measured in an exact manner because of fluctuation. Similarly, the temperature in a room also cannot be measured precisely because of a similar reason. Therefore, the fuzzy numbers are used to deal with non-precise quantities possessing vagueness. Thus a more appropriate way to describe the water level is to take the water level " around 25 meters" as a fuzzy number.
The concept of the fuzzy sets was originally introduced by Zadeh as an extension of the classical set [1]. The basic arithmetic structure of the fuzzy numbers was later developed by Mizumoto and Tanaka [2], Dubois and Prade [3]. Fuzzy calculus is a basic structure in fuzzy mathematics. The concept of the fuzzy derivative was first introduced by Chang, and Zadeh ([4], [5]). H-derivative [6] was the starting point of the topic of 1. These definitions, however, are non-local, which makes them unsuitable for investigating properties related to local scaling or fractional differentiability. 2. Riemann Liouville's derivative does not fulfill D Ψ (1) = 0. 3. For Caputo's derivative, we have to assume that the function is differentiable.
Otherwise, we cannot apply this definition.
In short, all fractional derivatives are deficient in some mathematical properties like product rule, chain rule, and quotient rule. Therefore, the solution of differential equations is not easy to obtain using these definitions.
Recently, [14] introduced a new definition of fractional derivative called conformable derivative. This new definition is different from other fractional derivatives and similar to the classical definition of the derivative. It depends on the limit definition of the derivative of a function. So this definition seems to be a natural extension of the ordinary derivative. [15] defined left and right conformable derivative, conformable power series, and conformable Laplace transform. This theory has attracted many researchers to work within and so many new concepts are introduced in conformable fractional calculus. For more research works, we refer the interested reader to ( [16], [17], [18]) and the references therein. Modeling real-life problems with conformable derivatives ( [19], [20]). Other fractional derivatives do not have geometrical interpretation but conformable derivative has [21]. For relevant concepts on conformable calculus, see for example ([22], [23], [24]).
A combination of these theories would be a ground-breaking strategy and can model better real-world phenomena. Recently, few attempts have been made in this area such as [25]. The authors introduced a fuzzy conformable derivative of order Ψ. Now, we have extended their work to the fuzzy conformable derivative of order 2Ψ, and also provide a generalization to the fuzzy conformable derivative of order pΨ. Also, we have proved relevant results on the fuzzy conformable derivative of addition, subtraction, multiplication, and H-difference of two the fuzzy-valued functions.
We organized this paper as follows: Section 2 includes the basic concepts on the fuzzy and conformable differential calculus. In Section 3, we have investigated the fuzzy conformable derivative of order Ψ, Section 4 contains the fuzzy conformable derivative of order 2Ψ while in Section 5, we have provided the generalization of the fuzzy conformable derivative up to order pΨ. In the end, a summary of our results is presented. We use R Φ to denote the space of all the real fuzzy numbers. Definition 2.3. For 0 γ < 1, γ-cuts for a fuzzy number η is defined as

Example 2.4.
A triangular fuzzy number η, denoted by an ordered triple (a, b, c) , with the condition a b c. The γ-cuts associated with the triangular fuzzy number η are Proposition 2.6. Arithmetic operations on the space of the fuzzy numbers are generalized from that of real intervals.
Then addition on space of the fuzzy numbers by γ-cuts is defined as

Scalar multiplication is defined as
Subtraction on the space of fuzzy numbers is

Proposition 3.5. For a fuzzy-valued function
Proof. By definition of H-conformable differentiability, we have Using the definition of H-conformable differentiability, and parametric form of level sets, we obtain This gives us our result in the form Thus proved.
Remark 3.6. But the converse of the above proposition is not true. i.e. The H-conformable differentiability of Φ does not guarantee the existence of conformable differentiability of Φ * and Φ * . For example, a fuzzy-valued function is conformable Hdifferentiable, but the level-sets are not conformable differentiable for π 2 , π . Remark 3.7. A conformable H-differentiable function Φ may not be differentiable with the H-derivative.
Remark 3.8. Conformable H-derivative of a fuzzy-valued function has some disadvantages such as

1.
It does not gives a bounded solution.

For a fuzzy-valued function
where η is a fuzzy number and ψ is a real-valued conformable differentiable function, conformable H-derivative does not exist when ψ Ψ (v) < 0. So we use strongly generalized conformable differentiability which is a generalization of conformable H-differentiability.
If a strongly generalized Ψ-derivative of a fuzzy-valued function Φ exists, then we say that Φ is strongly generalized Ψ-differentiable on (a, b) .

Definition 3.10.
We say that Φ is strongly generalized differentiable of type (Ψ-1) if Φ is differentiable in the first case, and Φ is strongly generalized differentiable of type (Ψ-2) if Φ is differentiable in the second case.
Example 3.11. Define a fuzzy-valued function Φ as where η is a fuzzy number, and ψ :
Since Φ is differentiable of type (Ψ-1), therefore we have Similarly, Now by dividing both sides by 1 θ , and taking lim θ→0 on both sides of equation (3.5), and equation (3.6), we obtain our required result. Theorem 3.13. Relation between Ψ-strongly generalized differentiable fuzzy-valued function and strongly generalized differentiable fuzzy-valued function is 1. If Φ is strongly generalized differentiable of type (Ψ-1), then we have

2.
If Φ is strongly generalized differentiable of type (Ψ-2), then we have Proof. Proof for case (2) is provided here. Case (1) can be easily proved similarly.
Put θv 1−Ψ = λ, we have the above expression in the form So we have Thus our required result is obtained.
Theorem 3.14. If a fuzzy-valued function Φ is Ψ-strongly generalized differentiable at any point v, then Φ is continuous at v.
Proof. The proof is straight-forward so left for the reader.
1. Strongly generalized conformable derivative for fuzzy-valued function Φ does not obey index law for any non-negative constants Ψ, ∆ ∈ (0, 1) , that is However, this deficiency can be removed by taking ∆ = 1.

2.
Strongly generalized conformable derivative for fuzzy-valued function Φ does not obey commutative law for any non-negative constants Ψ, ∆ ∈ (0, 1) , that is This result also holds with p 2.
3. Triangular inequality for strongly generalized conformable derivative also does not hold.
Lemma 3.15. ( [29]) Let Φ, and ψ : (0, ∞) → R Φ have strongly generalized Ψ-differentiability of the same case at any point v > 0, then Φ + ψ is strongly generalized Ψ-differentiable, and we have Theorem 3.16. Consider fuzzy-valued functions Φ, and ψ are strongly generalized differentiable of a different type. If Φ(v) ψ(v) exists, then Φ ψ is strongly generalized differentiable, and its derivative is Proof. We take the case when Φ is differentiable of type (1), and ψ is differentiable of type (2). Then there exist fuzzy numbers η 1 , and η 2 such that and we get So we have, Hence we obtain So we have our required result.
Theorem 3.17. If Φ is Ψ-differentiable real-valued function, and ψ is strongly generalized Ψdifferentiable fuzzy-valued function, then we have the following possibilities.
, and Φ ψ satisfies condition (B), then Φ ψ is also differentiable, and we have , and Φ ψ satisfies the condition (A), then Φ ψ is also differentiable, and , and Φ ψ satisfies the condition (B), then Φ ψ is also differentiable, and we have Proof. We provide proof of case (1) only, other cases can be proved in a similar pattern. As Φ is a continuous fuzzy-valued function, therefore we have Φ(v), Φ v − θv 1−Ψ , Φ v + θv 1−Ψ with the same sign. As ψ is differentiable of type (Ψ-2), so there exists a fuzzy number η 1 (v, θ, Ψ), and υ 1 (v, θ, Ψ) such that Also Now, multiply both sides with −1 θ , and taking lim θ→0 , we obtain our required result.
So our required result is obtained.
where η is a fuzzy number, and if ψ is differentiable, then Φ is also differentiable, and its fuzzy conformable derivative is given as
Since ψ is differentiable of type (Ψ-2). So there exist fuzzy numbers υ 1 (v, θ, Ψ), and Also (Φ + ψ) satisfies the condition (B), then the H-differences ψ Then we have Similarly Now by multiplying both sides by 1 θ , and taking limit both sides, we obtain our required result.

Fuzzy Conformable Derivative of Order 2Ψ
Definition 4.1. A fuzzy-valued function Φ is strongly generalized differentiable of order 2Ψ if (∃) a fuzzy number Φ 2Ψ (v) and , and Φ Ψ (v 0 ) both are fuzzy conformable differentiable of the same type, then case (1) of the above definition can be written in the form 2 If Φ (v 0 ), and Φ Ψ (v 0 ) are fuzzy conformable differentiable of a different type, then case (2) of the above definition can be written in the form Theorem 4.2. Relation between the strongly generalized conformable differentiable fuzzy-valued function of order 2Ψ, and the strongly generalized differentiable fuzzy-valued function of second order is 1 If Φ Ψ is the conformable strongly generalized differentiable fuzzy-valued function of the type (Ψ-1), then we have 2 If Φ Ψ is the conformable strongly generalized differentiable fuzzy-valued function of the type (Ψ-2), then we have Proof. We prove case (1) here. Case (2) can be proved in a similar pattern easily.
Since we have Put θv 1−Ψ = λ, we obtain We prove the case (1) here. Case (2) can be proved in a similar pattern easily. Since we have Put θv 1−Ψ = λ, we obtain Then we have This gives us our required result in the form 2 If Φ, and ψ Ψ are differentiable of type (Ψ-1), and Φ Ψ , ψ are differentiable of type (Ψ-2). If H-differences Φ(v) ψ(v), and Φ Ψ (v) ψ Ψ (v) exists, then Φ(v) ψ(v) is strongly generalized differentiable of 2Ψ, and we have 3 If Φ is differentiable of type (Ψ-2), and Φ Ψ , ψ, and ψ Ψ are differentiable of type (Ψ- is strongly generalized differentiable of order 2Ψ, and we have is strongly generalized differentiable of order 2Ψ, and we have Proof. We provide proof of the case (1) only. Other cases can be proved similarly.
, and therefore we get Thus we obtain our required result.

Example 4.4. Consider strongly generalized differentiable fuzzy-valued functions Φ,
and ψ of order 2Ψ defined as is differentiable of type (Ψ-2). So applying case (1) of the above theorem, we obtain

If we have the situation where
, then applying case (3) of the above theorem, we obtain , then applying case (4) of the above theorem, we obtain

Theorem 4.6. ([29]) Let Φ is a fuzzy-valued function with γ-cuts representation
. When we take a strongly generalized conformable derivative of a fuzzyvalued function of order Ψ, we have two possible cases. Φ Ψ-1 1 (v), and Φ Ψ-1 2 (v). For the derivative of order 2Ψ, we have again two possible cases on each Ψ-derivative, which are
Proof. We prove the case (1) . Other cases can be proved in a similar pattern.

Fuzzy Conformable Derivative of Order PΨ
Throughout this subsection, we will use the terms given in the form ( [28]) Remark 5.2. For p = 1, the above definition becomes equal to the definition of the fuzzy conformable strongly generalized derivative of order Ψ. If Φ iΨ (v 0 ) is differentiable of type (Ψ-1) for even values of i for i = 1, 2, · · · p, and differentiable of type (Ψ-2) for other values in the above definition, then Continuing in this way,finally we reach the step If Φ iΨ (v 0 ) is differentiable of type (Ψ-1) for odd values of i for i = 1, 2, · · · p − 1, and differentiable of type (Ψ-2) for other values in above definition, then The above definition has two cases associated with odd, and even values of p. (i) If Φ iΨ (v 0 ) is differentiable of type (Ψ-1) for even values of i for i = 1, 2, · · · p, and differentiable of type (Ψ-2) for other values in the above definition, then continuing in this way,finally, we reach the step for odd values of i for i = 1, 2, · · · p − 1, and differentiable of type (Ψ-2) for other values in above definition, then continuing in this way, finally, we reach the step

Case 2:
For odd values of p, we have two possibilities.
for odd values of i for i = 1, 2, · · · p, and differentiable of type (Ψ-2) for other values in the above definition, then continuing in this way, finally, we reach the step for even values of i for i = 1, 2, · · · p − 1, and differentiable of type (Ψ-2) for other values in the above definition, then continuing in this way, finally, we reach the step Remark 5.5. If we put p = 1 in the above equation, then we get the definition of the strongly generalized conformable derivative of order Ψ.
Theorem 5.6. Define a fuzzy-valued function by where η is a fuzzy number. Now if ψ is a differentiable real-valued function of order pΨ, then Φ is the strongly generalized differentiable of order pΨ, and we have Theorem 5.7. If p is odd, then we have four cases.
1. If a kth element of Φ iΨ (v) is differentiable of type (Ψ-1) for odd values of i, and differentiable of type (Ψ-2) for other values, and an ith element of ψ jΨ (v) is differentiable of type (Ψ-1) for odd values of j. Then Φ (v) + ψ (v) is differentiable of order pΨ, and we have 2. If a kth element of Φ iΨ (v) is differentiable of type (Ψ-1) for even values of i, and an ith element of ψ jΨ (v) is differentiable of type (Ψ-1) for odd values of j. Then Φ (v) + ψ (v) is differentiable of order pΨ, and we have 1. If fuzzy-valued functions Φ and it's first fuzzy conformable derivative are differentiable of type (Ψ-1), and ψ, and it's derivatives of order Ψ, and 2Ψ are differentiable of type (Ψ-2) or if one of Φ, and it's derivatives of order Ψ, and 2Ψ are differentiable of type (Ψ-1), and other two are differentiable of type (Ψ-2), and one of ψ, and it's derivatives of order Ψ, and 2Ψ are differentiable of type (Ψ-2), and other two are differentiable of type (Ψ-1). Now by applying the above theorem, and we have

2.
If fuzzy-valued functions Φ it's derivatives of order Ψ, and 2Ψ are differentiable of type (Ψ-2), and ψ, and it's derivatives of order Ψ, and 2Ψare differentiable of type (Ψ-1) or if one of Φ, and derivatives of order Ψ, and 2Ψare differentiable of type (Ψ-2), and other two are differentiable of type (Ψ-1), and one of ψ, and it's derivatives of order Ψ, and 2Ψ are differentiable of type (Ψ-1), and the other two are differentiable of type (Ψ-2). Now by applying the above theorem, exist, and we have 3. If one of Φ, and it's derivatives of order Ψ, and 2Ψ are differentiable of type (Ψ-2), and other two are differentiable of type (Ψ-1), and one of ψ, and it's derivatives of order Ψ, and 2Ψ are differentiable of type (Ψ-2). Now by applying the above theorem,

4.
If one of Φ, and it's derivatives of order Ψ, and 2Ψ are differentiable of type (Ψ-1), and the other two are differentiable of type (Ψ-2), and all of ψ, and it's derivatives of order Ψ, and 2Ψare differentiable of type (Ψ-1). Now by applying the above theorem, Theorem 5.9. If p is odd, then we have four cases.

2.
If one of Φ and it's first three Ψ-derivatives are differentiable of type (Ψ-2) and other two are differentiable of type (Ψ-1), and one of ψ, and it's first three Ψ-derivatives are differentiable of type (Ψ-1), and other two are differentiable of type (Ψ-2) or if one of Φ, and it's first three Ψ-derivatives are differentiable of type (Ψ-1), and other three are differentiable of type (Ψ-2), and one of ψ, and it's first three Ψ-derivatives are differentiable of type (Ψ-1), and other three are differentiable of type (Ψ-2) or vice versa. Then by applying above theorem, and we have 3. If one of Φ it's first three Ψ-derivatives are differentiable of type (Ψ-2), and other three are differentiable of type (Ψ-2), and all of ψ, and it's derivatives are differentiable of type (Ψ-2) or all are differentiable of type (Ψ-1). Then by applying above the-

4.
If all of Φ and it's Ψ-derivatives are differentiable of type (Ψ-2), and one of ψ and it's first three Ψderivatives are differentiable of type (Ψ-1), and other two are fuzzy Ψdifferentiable of type (Ψ-2) or if Φ and it's Ψderivatives are differentiable of type (Ψ-1) and one of ψ and it's three Ψderivatives are differentiable of type (Ψ-2), and other three are differentiable of type (Ψ-1). Then by applying above theorem, 1 v .

Conclusion
Our choice has certain advantages concerning other techniques for solving the fuzzy conformable differential equations. First, solutions have a decreasing length of support. Second, it can also solve the fuzzy conformable partial differential equations. Third, a numerical solution of the fuzzy conformable differential equations can be obtained using this technique, which is not possible with the fuzzy differential inclusion. The solution obtained by using conformable strongly generalized differentiability is not unique. But it can be seen as an advantage because we can choose among those solutions, the best one that better reflects the real-life situation. The conformable strongly generalized derivative has certain advantages over other fuzzy fractional derivatives. For example, a solution of the fuzzy differential equations can easily be obtained using a conformable strongly generalized derivative. It possesses the fundamental properties of the fuzzy derivatives. For example, the derivative of a product of fuzzy-valued functions, fuzzy chain rule, fuzzy mean value theorem, fuzzy Roll's theorem, etc. In the case of other fractional derivatives, some functions are not infinitely differentiable at some point, so the fuzzy Taylor series does not exist for those fuzzy-valued functions, where using fuzzy conformable calculus, these functions are infinitely differentiable.
In the future, we aim to introduce the fuzzy conformable integral calculus, and, using these results, we will solve the fuzzy conformable differential equations of order Ψ, 2Ψ, and pΨ. This combination of two theories of fuzzy calculus and conformable calculus will help us to obtain better results in the modeling of real-life phenomena.