Generalization of Fuzzy Laplace Transforms of Fuzzy Fractional Derivatives about the General Fractional Order n − 1 < β < n

The main aim in this paper is to use all the possible arrangements of objects such that r 1 of them are equal to 1 and r 2 (the others) of them are equal to 2, in order to generalize the definitions of Riemann-Liouville and Caputo fractional derivatives (about order 0 < β < n) for a fuzzy-valued function. Also, we find fuzzy Laplace transforms for Riemann-Liouville and Caputo fractional derivatives about the general fractional order n − 1 < β < n under H-differentiability. Some fuzzy fractional initial value problems (FFIVPs) are solved using the above two generalizations.


Introduction
Fuzzy Fractional Differential Equations (FFDEs) can offer a more comprehensive account of the process or phenomenon.This has recently captured much attention in FFDEs.As the derivative of a function is defined in the sense of Riemann-Liouville, Grünwald-Letnikov, or Caputo in fractional calculus, the used derivative is to be specified and defined in FFDEs as well [1].
Many researchers have worked on the field of Fuzzy Fractional Differential Equations (FFDEs); for example, Salahshour et al. [2] dealt with the solutions of FFDEs under Riemann-Liouville H-differentiability by fuzzy Laplace transforms; Mazandarani and Kamyad [1] presented the solution to FFIVP under Caputo-type fuzzy fractional derivatives by a modified fractional Euler method; Wu and Baleanu [3] proposed a novel modification of the variational iteration method (VIM) by means of the Laplace transform; they extended the method successfully to fractional differential equations; Ahmadian et al. [4] reveal a computational method based on using a Tau method with Jacobi polynomials for the solution of fuzzy linear fractional differential equations of order 0 < V < 1, and Allahviranloo et al. [5] introduced the fuzzy Caputo fractional differential equations under the generalized Hukuhara differentiability.This paper is arranged as follows.Basic concepts are given in Section 2. In Section 3, the general formula of the fuzzy Riemann-Liouville fractional derivatives and the general formula of Laplace transforms of the fuzzy Riemann-Liouville fractional derivatives for a fuzzy-valued function  are found.In Section 4, the general formula of the fuzzy Caputo fractional derivatives and the general formula of Laplace transforms of the fuzzy Caputo fractional derivatives for a fuzzy-valued function  are found.In Section 5, conclusions are drawn.

Basic Concepts
In this section, we give the basic concepts which are needed in the next sections.We denote   [, ] as the space of all continuous fuzzy-valued functions on [, ].Also, we denote the space of all Lebesgue integrable fuzzy-valued functions on the bounded interval [, ] ⊂ R by   [, ].
We denote the set of all real numbers by R and the set of all fuzzy numbers on R is indicated by .Definition 3 (see [8]).Let ,  ∈ .If there exists  ∈  such that  =  + , then  is called the H-difference of  and , and it is denoted by  ⊖ .The sign "⊖" always stands for H-difference and also note that  ⊖  ̸ =  + (−1).

Generalization of Fuzzy Laplace Transforms of the Fuzzy Riemann-Liouville
Fractional Derivatives of Order  − 1 <  < for  = 0, 1, . . .,  − 2, such that  1 ,  2 , . . .,   are all the possible arrangements of  objects which have the number given in the rule: where  1 of them equal 1 (meaning Riemann-Liouville type derivative in the first form) and  2 of them equal 2 (meaning Riemann-Liouville type derivative in the second form) and   1 ,..., 0 = .() is the Riemann-Liouville type fuzzy fractional differentiable function of order 0 <  < ,  ̸ = 1, 2, . . .,  − 1, at  0 ∈ (0,), if there exists an element ( RL   )( 0 ) ∈   such that for all 0 ≤  ≤ 1 and for ℎ > 0 sufficiently near zero.Then: (2) If  ⌈⌉ = 2, then for  − 1 <  < ,  = 1, 2, . . ., , such that  1 ,  2 , . . .,  ⌊⌋ are all the possible arrangements of ⌊⌋ objects which have the number given by the rule: If the fuzzy-valued function () is differentiable as in Definition 6 cases defined in (11), it is the Riemann-Liouville type differentiable in the first form and denoted by is differentiable as in Definition 6 cases defined in (12), it is the Riemann-Liouville type differentiable in the second form and denoted by We note that if we take  = 1 (0 <  < 1) in Definition 6 we get Definition 3.2 [2] which is introduced by Salahshour et al.
If () is  [() − ]-differentiable fuzzy-valued function, then for 0 <  < 1 where Then, one has the following: If  is an even number, we have: If  is an odd number, we have: such that ..,  1 ,...,  2 ,...,   ,...,  )().Suppose that  is an odd number; then, from Theorem 7, when  − 1 <  < , we get: Therefore, we get: Then, from (54), we get: We know from Laplace transform of the Riemann-Liouville fractional derivative of order  > 0 that The above equation can be written as: In a similar manner, we can get: Example 11.Consider the following FFIVP: We note that By taking fuzzy Laplace transform for both sides of (69), we get Now, by using Theorem 9 when  = 2 we have 2 2 = 4 cases as follows.
If  is an even number, the proof is similar.

Conclusions
The general formulas for fuzzy Riemann-Liouville and Caputo fractional derivatives about the general order 0 <  <  for fuzzy-valued function  are found by using all the possible arrangements of objects such that  1 of them equal 1 and  2 (the others) of them equal 2. Also, the general formulas for fuzzy Laplace transforms of Riemann-Liouville and Caputo fractional derivatives about the general order −1 <  <  are found under Hukuhara difference (H-difference).