# Generalization of Fuzzy Laplace Transforms of Fuzzy Fractional Derivatives about the General Fractional Ordern

Authors: Amal Khalaf Haydar; Ruaa Hameed Hassan
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

The main aim in this paper is to use all the possible arrangements of objects such thatr
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.


---

## Body

## 1. 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 Section2. 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 f 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 f are found. In Section 5, conclusions are drawn.


## 2. Basic Concepts

In this section, we give the basic concepts which are needed in the next sections. We denoteC
F
a
,
b as the space of all continuous fuzzy-valued functions on a
,
b. Also, we denote the space of all Lebesgue integrable fuzzy-valued functions on the bounded interval a
,
b
⊂
R by L
F
a
,
b.Theorem 1 (see [6]).
Letn be a positive integer. Let D
n
-
1
f be continuous on J
=
0
,
∞; C is the class of piecewise continuous functions on J
′
=
0
,
∞ and integrable on any finite subinterval of J
=
0
,
∞ and let υ
>
0
. Then, one finds the following:
(
a
) If D
n
f is of class C, then(1)D-υfx=D-υ-nDnfx+Qnx,υand
(
b
) if D
n
f is continuous on J, then for x
>
0
(2)DnD-υfx=D-υDnfx+Qnx,υ-n,where(3)Qnx,υ=∑k=0n-1xυ+kΓυ+k+1Dkf0.Definition 2 (see [7]).
A fuzzy numberu in parametric form is a pair u
_
,
u
- of functions u
_
r
,
u
-
r,  0
≤
r
≤
1, which satisfy the following requirements.(1)
u
_
r is a bounded nondecreasing left continuous function in 0,1 and right continuous at 0.
(2)
u
-
r is a bounded nonincreasing left continuous function in 0,1 and right continuous at 0.
(3)
Consideru
_
r
≤
u
-
r,  0
≤
r
≤
1.We denote the set of all real numbers by R and the set of all fuzzy numbers on R is indicated by E.Definition 3 (see [8]).
Letx
,
y
∈
E. If there exists z
∈
E such that x
=
y
+
z, then z is called the H-difference of x and y, and it is denoted by x
⊖
y. The sign “⊖” always stands for H-difference and also note that x
⊖
y
≠
x
+
-
1
y.Definition 4 (see [9]).
Letf
x be continuous fuzzy-valued function; suppose that f
x
e
-
s
x is improper fuzzy Rimann-integrable on 0
,
∞ then ∫
0
∞
f
x
e
-
s
x
d
x is called fuzzy Laplace transforms and is denoted as L
f
x
=
∫
0
∞
f
x
e
-
s
x
d
x, s
>
0.
We have(4)∫0∞fxe-sxdx=∫0∞f_x;re-sxdx,∫0∞f-x;re-sxdx;also by using the definition of classical Laplace transform:(5)lf_x;r=∫0∞f_x;re-sxdx,lf-x;r=∫0∞f-x;re-sxdxthen, we follow:(6)Lfx=lf_x;r,lf-x;r.Definition 5 (see [2]).
Letf
∈
C
F
a
,
b
∩
L
F
a
,
b. The fuzzy Riemann-Liouville integral of fuzzy-valued function f is defined as follows:(7)Ia+βfx=1Γβ∫axftdtx-t1-β,x>a,0<β≤1.


## 3. Generalization of Fuzzy Laplace Transforms of the Fuzzy Riemann-Liouville Fractional Derivatives of Ordern
-
1
<
β
<
n

In this section, we define Riemann-Liouville fractional derivatives of the general fractional order0
<
β
<
n and we find fuzzy Laplace transforms for Riemann-Liouville fractional derivatives of the general fractional order n
-
1
<
β
<
n for fuzzy-valued function f under H-differentiability.Definition 6.
Letf
x
∈
C
F
0
,
b
∩
L
F
0
,
b, and β and β are values of β rounded up and down to the nearest integer number, respectively. One can see that ϕ
x
=
1
/
Γ
β
-
β
∫
0
x
f
t
d
t
/
x
-
t
1
-
β
+
β, and the functions ϕ
i
1
,
i
2
,
…
,
i
j
,
1 and ϕ
i
1
,
i
2
,
…
,
i
j
,
2 are defined as:(8)ϕi1,i2,…,ij,1x0=limh→0+⁡ϕi1,i2,…,ijx0+h⊖ϕi1,i2,…,ijx0h=limh→0+⁡ϕi1,i2,…,ijx0⊖ϕi1,i2,…,ijx0-hh,(9)ϕi1,i2,…,ij,2x0=limh→0+⁡ϕi1,i2,…,ijx0⊖ϕi1,i2,…,ijx0+h-h=limh→0+⁡ϕi1,i2,…,ijx0-h⊖ϕi1,i2,…,ijx0-h,for j
=
0,1
,
…
,
n
-
2, such that i
1
,
i
2
,
…
,
i
j are all the possible arrangements of j objects which have the number given in the rule:(10)jPr1,r2=j!r1!r2!,r1+r2=j,where r
1 of them equal 1 (meaning Riemann-Liouville type derivative in the first form) and r
2 of them equal 2 (meaning Riemann-Liouville type derivative in the second form) and ϕ
i
1
,
…
,
i
0
=
ϕ. f
x is the Riemann-Liouville type fuzzy fractional differentiable function of order 0
<
β
<
n, β
≠
1,2
,
…
,
n
-
1, at x
0
∈
0
,
b, if there exists an element D
β
RL
f
x
0
∈
C
F such that for all 0
≤
r
≤
1 and for h
>
0 sufficiently near zero. Then:(
1
) If i
β
=
1, then(11)DβRLfx0=limh→0+⁡ϕi1,i2,…,iβx0+h⊖ϕi1,i2,…,iβx0h=limh→0+⁡ϕi1,i2,…,iβx0⊖ϕi1,i2,…,iβx0-hh.
(
2
) If i
β
=
2, then(12)DβRLfx0=limh→0+⁡ϕi1,i2,…,iβx0⊖ϕi1,i2,…,iβx0+h-h=limh→0+⁡ϕi1,i2,…,iβx0-h⊖ϕi1,i2,…,iβx0-h,for k
-
1
<
β
<
k,  k
=
1,2
,
…
,
n, such that i
1
,
i
2
,
…
,
i
β are all the possible arrangements of β objects which have the number given by the rule:(13)βPr1,r2=β!r1!r2!,r1+r2=β.If the fuzzy-valued function f
x is differentiable as in Definition 6 cases defined in (11), it is the Riemann-Liouville type differentiable in the first form and denoted by D
i
1
,
i
2
,
…
,
i
β
,
1
β
RL
f
x
0. If f
x is differentiable as in Definition 6 cases defined in (12), it is the Riemann-Liouville type differentiable in the second form and denoted by D
i
1
,
i
2
,
…
,
i
β
,
2
β
RL
f
x
0.
We note that if we taken
=
1
(
0
<
β
<
1
) in Definition 6 we get Definition  3.2 [2] which is introduced by Salahshour et al.Theorem 7.
Letf
x
∈
C
F
0
,
b
∩
L
F
0
,
b be a fuzzy-valued function such that f
x
=
f
_
x
;
r
,
f
-
x
;
r for r
∈
0,1, x
0
∈
0
,
b, and ϕ
x
=
1
/
Γ
β
-
β
∫
0
x
f
t
d
t
/
x
-
t
1
-
β
+
β. Suppose that 0
<
β
<
n and m is the number of repetitions of number 2 among i
1
,
i
2
,
…
,
i
β for k
-
1
<
β
<
k,  k
=
1,2
,
…
,
n, say, i
k
1
,
i
k
2
,
…
,
i
k
m, such that k
1
<
k
2
<
⋯
<
k
m; that is, i
k
1
=
i
k
2
=
⋯
=
i
k
m
=
2 and 0
≤
m
≤
β. Then, one has the following:Ifm is an even number, then(14)Di1,i2,…,iββRLfx0=DβRLf_x0;r,DβRLf-x0;r.
Ifm is an odd number, then(15)Di1,i2,…,iββRLfx0=DβRLf-x0;r,DβRLf_x0;r,where(16)DβRLf_x0;r=1Γβ-βddxβ∫0xf_t;rx-t1-β+βdtx=x0,DβRLf-x0;r=1Γβ-βddxβ∫0xf-t;rx-t1-β+βdtx=x0.Proof.
Suppose thatm is an even number, and then m
=
2
s, s
∈
N. Now, we have two probabilities as follows.
The first probability isD
i
1
,
…
,
i
k
1
,
…
,
i
k
2
,
…
,
i
k
2
s
,
…
,
i
β
β
RL
f
x
0 is the Riemann-Liouville type fuzzy fractional differentiable function in the first form (i
β
=
1), and then from (11) of Definition 6, we have:(17)
ϕ
i
1
,
…
,
i
β
x
0
+
h
⊖
ϕ
i
1
,
…
,
i
β
x
0
=
ϕ
_
i
1
,
…
,
i
β
x
0
+
h
;
r
-
ϕ
_
i
1
,
…
,
i
β
x
0
;
r
,
ϕ
-
i
1
,
…
,
i
β
x
0
+
h
;
r
-
ϕ
-
i
1
,
…
,
i
β
x
0
;
r
,
ϕ
i
1
,
…
,
i
β
x
0
⊖
ϕ
i
1
,
…
,
i
β
x
0
-
h
=
ϕ
_
i
1
,
…
,
i
β
x
0
;
r
-
ϕ
_
i
1
,
…
,
i
β
x
0
-
h
;
r
,
ϕ
-
i
1
,
…
,
i
β
x
0
;
r
-
ϕ
-
i
1
,
…
,
i
β
x
0
-
h
;
r
.Multiplying both sides by 1
/
h, h
>
0, we obtain:(18)
ϕ
i
1
,
…
,
i
β
x
0
+
h
⊖
ϕ
i
1
,
…
,
i
β
x
0
h
=
ϕ
_
i
1
,
…
,
i
β
x
0
+
h
;
r
-
ϕ
_
i
1
,
…
,
i
β
x
0
;
r
h
,
ϕ
-
i
1
,
…
,
i
β
x
0
+
h
;
r
-
ϕ
-
i
1
,
…
,
i
β
x
0
;
r
h
,
ϕ
i
1
,
…
,
i
β
x
0
⊖
ϕ
i
1
,
…
,
i
β
x
0
-
h
h
=
ϕ
_
i
1
,
…
,
i
β
x
0
;
r
-
ϕ
_
i
1
,
…
,
i
β
x
0
-
h
;
r
h
,
ϕ
-
i
1
,
…
,
i
β
x
0
;
r
-
ϕ
-
i
1
,
…
,
i
β
x
0
-
h
;
r
h
.By taking h
→
0
+ on both sides of the above equations, we get:(19)DβRLfx0=ddxϕ_i1,…,iβx0;r,ddxϕ-i1,…,iβx0;r.Now, since ϕ
i
1
,
…
,
i
k
1
-
1
x
0 is equal to the limits defined in (8) of Definition 6, then by applying (8) for k
1
-
1 times, we get(20)ϕi1,…,ik1-1x0=ϕ_k1-1x0;r,ϕ-k1-1x0;r.Since ϕ
i
1
,
…
,
i
k
1
x
0 is equal to the limits defined in (9) of Definition 6, then by applying (9) once, we get:(21)ϕi1,…,ik1x0=ϕ-k1x0;r,ϕ_k1x0;r.Since ϕ
i
1
,
…
,
i
k
2
-
1
x
0 is equal to the limits defined in (8) of Definition 6, then by applying (8) for k
2
-
1
-
k
1 times, we get:(22)ϕi1,…,ik2-1x0=ϕ-k2-1x0;r,ϕ_k2-1x0;r.Since ϕ
i
1
,
…
,
i
k
2
x
0 is equal to the limits defined in (9) of Definition 6, then by applying (9) once, we get:(23)ϕi1,…,ik2x0=ϕ_k2x0;r,ϕ-k2x0;r.
In other words, from (23) we note that, after applying (8) and (9) for any even number from i
k
1
,
i
k
2
,
…
,
i
k
m, we will get an equation similar to (23). Therefore, for ϕ
i
1
,
…
,
i
k
2
s
x
0, we have:(24)ϕi1,…,ik2sx0=ϕ_k2sx0;r,ϕ-k2sx0;r,since 2
s is an even number.
Finally, sinceϕ
i
1
,
…
,
i
β
x
0 is equal to the limits defined in (8) of Definition 6, then by applying (8) for β
-
k
2
s times, we get:(25)ϕi1,…,iβx0=ϕ_βx0;r,ϕ-βx0;r.Then,(26)ϕ_i1,…,iβx0;r=ϕ_βx0;r,ϕ-i1,…,iβx0;r=ϕ-βx0;r.Substituting (26) in (19) yields the following:(27)DβRLfx0=ddxβϕ_x0;r,ddxβϕ-x0;r=DβRLf_x0;r,DβRLf-x0;r.The second probability is D
i
1
,
…
,
i
k
1
,
…
,
i
k
2
,
…
,
i
k
2
s
-
1
,
…
i
β
β
RL
f
x
0 is the Riemann-Liouville type fuzzy fractional differentiable function in the second form (i
β
=
2), and then, by applying (12) of Definition 6, we can get(28)DβRLfx0=ddxϕ-i1,…,iβx0;r,ddxϕ_i1,…,iβx0;r.Since 2
s
-
2 is an even number, then by replacing 2
s by 2
s
-
2 in (24), we get:(29)ϕi1,…,ik2s-2x0=ϕ_k2s-2x0;r,ϕ-k2s-2x0;r.Similarly, by applying (8) k
2
s
-
1
-
1
-
k
2
s
-
2 times for ϕ
i
1
,
…
,
i
k
2
s
-
1
-
1
x
0 and (9) once for ϕ
i
1
,
…
,
i
k
2
s
-
1
x
0, we get:(30)ϕi1,…,ik2s-1x0=ϕ-k2s-1x0;r,ϕ_k2s-1x0;r.Finally, since ϕ
i
1
,
…
,
i
β
x
0 is equal to the limits defined in (8) of Definition 6, then, by applying (8) for β
-
k
2
s
-
1 times, we get:(31)ϕi1,…,iβx0=ϕ-βx0;r,ϕ_βx0;r.Then,(32)ϕ_i1,…,iβx0;r=ϕ-βx0;r,ϕ-i1,…,iβx0;r=ϕ_βx0;r.Substituting (32) in (28) yields:(33)DβRLfx0=ddxβϕ_x0;r,ddxβϕ-x0;r=DβRLf_x0;r,DβRLf-x0;r.If m is an odd number, the proof is similar.We note that if we taken
=
1
(
0
<
β
<
1
) in Theorem 7, we get Theorem  3.2 [2] which is found by Salahshour et al.Corollary 8.
Letf
x
∈
C
F
0
,
b
∩
L
F
0
,
b be a fuzzy-valued function and f
x
=
f
_
x
;
r
,
f
-
x
;
r for r
∈
0,1 and x
0
∈
0
,
b. Suppose that 0
<
β
<
3 and m is the number of repetitions of number 2 among i
1
,
i
2
,
…
,
i
β for k
-
1
<
β
<
k,  k
=
1,2
,
3. Then, one has the following.
Iff
x is i
-
β
R
L-differentiable fuzzy-valued function, then for 0
<
β
<
1
(34)D1βRLfx0=DβRLf_x0;r,DβRLf-x0;r.
Iff
x is i
i
-
β
R
L-differentiable fuzzy-valued function, then for 0
<
β
<
1
(35)D2βRLfx0=DβRLf-x0;r,DβRLf_x0;r.
IfD
1
β
R
L
f
x is i
-
β
R
L-differentiable fuzzy-valued function, then for 1
<
β
<
2
(36)D1,1βRLfx0=DβRLf_x0;r,DβRLf-x0;r.
IfD
1
β
R
L
f
x is i
i
-
β
R
L-differentiable fuzzy-valued function, then for 1
<
β
<
2
(37)D1,2βRLfx0=DβRLf-x0;r,DβRLf_x0;r.
IfD
2
β
R
L
f
x is i
-
β
R
L-differentiable fuzzy-valued function, then for 1
<
β
<
2
(38)D2,1βRLfx0=DβRLf-x0;r,DβRLf_x0;r.
IfD
2
β
R
L
f
x is i
i
-
β
R
L-differentiable fuzzy-valued function, then for 1
<
β
<
2
(39)D2,2βRLfx0=DβRLf_x0;r,DβRLf-x0;r.
IfD
1,1
β
R
L
f
x is i
-
β
R
L-differentiable fuzzy-valued function, then for 2
<
β
<
3
(40)D1,1,1βRLfx0=DβRLf_x0;r,DβRLf-x0;r.
IfD
1,1
β
R
L
f
x is i
i
-
β
R
L-differentiable fuzzy-valued function, then for 2
<
β
<
3
(41)D1,1,2βRLfx0=DβRLf-x0;r,DβRLf_x0;r.
IfD
1,2
β
R
L
f
x is i
-
β
R
L-differentiable fuzzy-valued function, then for 2
<
β
<
3
(42)D1,2,1βRLfx0=DβRLf-x0;r,DβRLf_x0;r.
IfD
1,2
β
R
L
f
x is i
i
-
β
R
L-differentiable fuzzy-valued function, then for 2
<
β
<
3
(43)D1,2,2βRLfx0=DβRLf_x0;r,DβRLf-x0;r.
IfD
2,1
β
R
L
f
x is i
-
β
R
L-differentiable fuzzy-valued function, then for 2
<
β
<
3
(44)D2,1,1βRLfx0=DβRLf-x0;r,DβRLf_x0;r.
IfD
2,1
β
R
L
f
x is i
i
-
β
R
L-differentiable fuzzy-valued function, then for 2
<
β
<
3
(45)D2,1,2βRLfx0=DβRLf_x0;r,DβRLf-x0;r.
IfD
2,2
β
R
L
f
x is i
-
β
R
L-differentiable fuzzy-valued function, then for 2
<
β
<
3
(46)D2,2,1βRLfx0=DβRLf_x0;r,DβRLf-x0;r.
IfD
2,2
β
R
L
f
x is i
i
-
β
R
L-differentiable fuzzy-valued function, then for 2
<
β
<
3
(47)D2,2,2βRLfx0=DβRLf-x0;r,DβRLf_x0;r,where(48)DβRLf_x0;r=1Γβ-βddxβ∫0xf_t;rdtx-t1-β+βx=x0,DβRLf-x0;r=1Γβ-βddxβ∫0xf-t;rdtx-t1-β+βx=x0.Theorem 9.
Suppose thatf
x
∈
C
F
[
0
,
∞
)
∩
L
F
[
0
,
∞
) is fuzzy-valued function; f
x
=
f
_
x
;
r
,
f
-
x
;
r for r
∈
0,1. One supposes that n
-
1
<
β
<
n and m is the number of repetitions of 2 among i
1
,
i
2
,
…
,
i
n, say, i
k
1
,
i
k
2
,
…
,
i
k
m, such that k
1
<
k
2
<
⋯
<
k
m; that is, i
k
1
=
i
k
2
=
⋯
=
i
k
m
=
2 and 0
≤
m
≤
n. Then, one has the following:Ifm is an even number, we have:(49)LDi1,i2,…,inβRLfx=sβLfx⊖sn-1Dβ-nRLf0⊗∑k=0n-2skDβ-k-1RLf0,such that(50)⊗=⊖,ifthenumberofrepetitionsof2amongi1,i2,…,in-k+1isanevennumber,-,ifthenumberofrepetitionsof2amongi1,i2,…,in-k+1isanoddnumber.
Ifm is an odd number, we have:(51)LDi1,i2,…,inβRLfx=-sn-1Dβ-nRLf0⊖-sβLfx⊗∑k=0n-2skDβ-k-1RLf0,such that(52)⊗=⊖,ifthenumberofrepetitionsof2amongi1,i2,…,in-k+1isanoddnumber,-,ifthenumberofrepetitionsof2amongi1,i2,…,in-k+1isanevennumber.Proof.
ByD
i
1
,
i
2
,
…
,
i
n
β
RL
f
x we mean D
i
1
,
…
,
i
k
1
,
…
,
i
k
2
,
…
,
i
k
m
,
…
,
i
n
β
RL
f
x. Suppose that m is an odd number; then, from Theorem 7, when n
-
1
<
β
<
n, we get:(53)Di1,i2,…,inβRLfx=DβRLf-x;r,DβRLf_x;r.Therefore, we get:(54)DβRLf_x;r=DβRLf-x;r,DβRLf¯x;r=DβRLf_x;r.Then, from (54), we get:(55)LDi1,i2,…,inβRLfx=LDβRLf_x;r,DβRLf¯x;r=lDβRLf-x;r,lDβRLf_x;r.
We know from Laplace transform of the Riemann-Liouville fractional derivative of orderβ
>
0 that(56)lDβRLf_x;r=sβlf_x;r-∑k=0n-1skDβ-k-1RLf_0;r=sβlf_x;r-sn-1Dβ-nRLf_0;r-∑k=0n-2skDβ-k-1RLf_0;r=sβlf_x;r-sn-1Dβ-nRLf_0;r-∑k=0n-2sn-2-kDβ-n+k+1RLf_0;r.The above equation can be written as:(57)lDβRLf_x;r=sβlf_x;r-sn-1Dβ-nRLf_0;r-∑k=0k1-2sn-2-kDβ-n+k+1RLf_0;r-∑k=k1-1k2-2sn-2-kDβ-n+k+1RLf_0;r-⋯-∑k=km-1-1km-2sn-2-kDβ-n+k+1RLf_0;r-∑k=km-1n-2sn-2-kDβ-n+k+1RLf_0;r.In a similar manner, we can get:(58)lDβRLf-x;r=sβlf-x;r-sn-1Dβ-nRLf-0;r-∑k=0k1-2sn-2-kDβ-n+k+1RLf-0;r-∑k=k1-1k2-2sn-2-kDβ-n+k+1RLf-0;r-⋯-∑k=km-1-1km-2sn-2-kDβ-n+k+1RLf-0;r-∑k=km-1n-2sn-2-kDβ-n+k+1RLf-0;r.Since i
k
1
=
i
k
2
=
⋯
=
i
k
m
=
2 and m is an odd number, then we have the following equations:(59)Dβ-n+k+1RLf_0;r=Dβ-n+k+1RLf_0;r,Dβ-n+k+1RLf-0;r=Dβ-n+k+1RLf¯0;r,for0≤k≤k1-2,Dβ-n+k+1RLf_0;r=Dβ-n+k+1RLf¯0;r,Dβ-n+k+1RLf-0;r=Dβ-n+k+1RLf_0;r,fork1-1≤k≤k2-2,⋮Dβ-n+k+1RLf_0;r=Dβ-n+k+1RLf_0;r,Dβ-n+k+1RLf-0;r=Dβ-n+k+1RLf¯0;r,forkm-1-1≤k≤km-2,Dβ-n+k+1RLf_0;r=Dβ-n+k+1RLf¯0;r,Dβ-n+k+1RLf-0;r=Dβ-n+k+1RLf_0;r,forkm-1≤k≤n-2.The last one of the above equations yields from Theorem 7 because m is an odd number. Using (57), (58), and the above equations, (55) becomes:(60)LDi1,i2,…,inβRLfx=-sn-1Dβ-nRLf0⊖-sβLfx⊗∑k=0n-2sn-2-kDβ-n+k+1RLf0=-sn-1Dβ-nRLf0⊖-sβLfx⊗∑k=0n-2skDβ-k-1RLf0,where ⊗ is defined as in (52).
Ifm is an even number, the proof is similar.We note that if we taken
=
1
(
0
<
β
<
1
) in Theorem 9, we get Theorem  4.4 [2] which is found by Salahshour et al.Corollary 10.
Suppose thatf
x
∈
C
F
0
,
∞
∩
L
F
0
,
∞. One supposes that 2
<
β
<
3. Then, one finds the following.
IfD
1,1
β
R
L
f
x is i
-
β
R
L-differentiable fuzzy-valued function, then(61)LD1,1,1βRLfx=sβLfx⊖s2Dβ-3RLf0⊖Dβ-1RLf0⊖sDβ-2RLf0.
IfD
1,1
β
R
L
f
x is i
i
-
β
R
L-differentiable fuzzy-valued function, then(62)LD1,1,2βRLx=-s2Dβ-3RLf0⊖-sβLfx-Dβ-1RLf0-sDβ-2RLf0.
IfD
1,2
β
R
L
f
x is i
-
β
R
L-differentiable fuzzy-valued function, then(63)LD1,2,1βRLfx=-s2Dβ-3RLf0⊖-sβLfx⊖Dβ-1RLf0-sDβ-2RLf0.
IfD
1,2
β
R
L
f
x is i
i
-
β
R
L-differentiable fuzzy-valued function, then(64)LD1,2,2βRLfx=sβLfx⊖s2Dβ-3RLf0-Dβ-1RLf0⊖sDβ-2RLf0.
IfD
2,1
β
R
L
f
x is i
-
β
R
L-differentiable fuzzy-valued function, then(65)LD2,1,1βRLfx=-s2Dβ-3RLf0⊖-sβLfx⊖Dβ-1RLf0⊖sDβ-2RLf0.
IfD
2,1
β
R
L
f
x is i
i
-
β
R
L-differentiable fuzzy-valued function, then(66)LD2,1,2βRLfx=sβLfx⊖s2Dβ-3RLf0-Dβ-1RLf0-sDβ-2RLf0.
IfD
2,2
β
R
L
f
x is i
-
β
R
L-differentiable fuzzy-valued function, then(67)LD2,2,1βRLfx=sβLfx⊖s2Dβ-3RLf0⊖Dβ-1RLf0-sDβ-2RLf0.
IfD
2,2
β
R
L
f
x is i
i
-
β
R
L-differentiable fuzzy-valued function, then(68)LD2,2,2βRLfx=-s2Dβ-3RLf0⊖-sβLfx-Dβ-1RLf0⊖sDβ-2RLf0.Example 11.
Consider the following FFIVP:(69)DβRLyx=σ;σ=2r-2,2-2r,1<β<2,(70)Dβ-1RLy0=Dβ-2RLy0=y0β-1RL∈E.We note that(71)Dβ-1RLy_0;r=Dβ-2RLy_0;r=y_0β-1RLr,Dβ-1RLy¯0;r=Dβ-2RLy-0;r=y-0β-1RLr.By taking fuzzy Laplace transform for both sides of (69), we get(72)LDβRLyx=Lσ.Now, by using Theorem 9 when n
=
2 we have 2
2
=
4 cases as follows.Case 1.
LetD
1
β
RL
y
x be i
-
β
R
L-differentiable. By using Theorem 9, when m
=
0 (even), (72) becomes(73)sβLyx⊖sDβ-2RLy0⊖Dβ-1RLy0=Lσ.Then, we get(74)sβly_x;r-sDβ-2RLy_0;r-Dβ-1RLy_0;r=2r-2s,sβly-x;r-sDβ-2RLy-0;r-Dβ-1RLy¯0;r=2-2rs.The solution of FFIVP (69) is as follows:(75)y_x;r=2r-2xβΓβ+1+y_0β-1RLrxβ-1Γβ+xβ-2Γβ-1,y-x;r=2-2rxβΓβ+1+y-0β-1RLrxβ-1Γβ+xβ-2Γβ-1.Case 2.
LetD
1
β
RL
y
x be i
i
-
β
R
L-differentiable. By using Theorem 9, when m
=
1 (odd), (72) becomes(76)-sDβ-2RLy0⊖-sβLyx-Dβ-1RLy0=Lσ.Then, we get:(77)-sDβ-2RLy_0;r+sβly_x;r-Dβ-1RLy_0;r=2-2rs,-sDβ-2RLy-0;r+sβly-x;r-Dβ-1RLy¯0;r=2r-2s.The solution of FFIVP (69) is as follows:(78)y_x;r=2-2rxβΓβ+1+y_0β-1RLrxβ-1Γβ+xβ-2Γβ-1,y-x;r=2r-2xβΓβ+1+y-0β-1RLrxβ-1Γβ+xβ-2Γβ-1.Case 3.
LetD
2
β
RL
y
x be i
-
β
R
L-differentiable. By using Theorem 9, when m
=
1 (odd), (72) becomes:(79)-sDβ-2RLy0⊖-sβLyx⊖Dβ-1RLy0=Lσ.Then, we get:(80)-sDβ-2RLy_0;r+sβly_x;r-Dβ-1RLy¯0;r=2-2rs,-sDβ-2RLy-0;r+sβly-x;r-Dβ-1RLy_0;r=2r-2s.The solution of FFIVP (69) is as follows:(81)y_x;r=2-2rxβΓβ+1+y-0β-1RLrxβ-1Γβ+y_0β-1RLrxβ-2Γβ-1,y-x;r=2r-2xβΓβ+1+y_0β-1RLrxβ-1Γβ+y-0β-1RLrxβ-2Γβ-1.Case 4.
LetD
2
β
RL
y
x be i
i
-
β
R
L-differentiable. By using Theorem 9, when m
=
2 (even), (72) becomes:(82)sβLyx⊖sDβ-2RLy0-Dβ-1RLy0=Lσ.Then, we get:(83)sβly_x;r-sDβ-2RLy_0;r-Dβ-1RLy¯0;r=2r-2s,sβly-x;r-sDβ-2RLy-0;r-Dβ-1RLy_0;r=2-2rs.The solution of FFIVP (69) is as follows:(84)y_x;r=2r-2xβΓβ+1+y-0β-1RLrxβ-1Γβ+y_0β-1RLrxβ-2Γβ-1,y-x;r=2-2rxβΓβ+1+y_0β-1RLrxβ-1Γβ+y-0β-1RLrxβ-2Γβ-1.


## 4. Generalization of Fuzzy Laplace Transforms of the Fuzzy Caputo Fractional Derivatives of Ordern
-
1
<
β
<
n

In this section, we define Caputo fractional derivatives of the general fractional order0
<
β
<
n and we find fuzzy Laplace transforms for Caputo fractional derivatives of the general fractional order n
-
1
<
β
<
n for fuzzy-valued function f under H-differentiability.Remark 12.
To get Caputo type fuzzy fractional derivatives of order0
<
β
<
n for f
(
x
)
∈
C
F
0
,
b
∩
L
F
0
,
b, we take(85)Gx=1Γβ-β∫0xftdtx-t1-β+β⊖∑k=0βDkf0xβ-β+kΓ1+β-β+kinstead of ϕ
x
=
1
/
Γ
β
-
β
∫
0
x
f
t
d
t
/
x
-
t
1
-
β
+
β in Definition 6.We note that if we taken
=
2
(
0
<
β
<
2
) in Remark 12, we get Definition  3.1 [1] which is introduced by Mazandarani and Kamyad.Theorem 13.
Letf
x
∈
C
F
0
,
b
∩
L
F
0
,
b be a fuzzy-valued function such that f
x
=
f
_
x
;
r
,
f
-
x
;
r for r
∈
0,1, x
0
∈
0
,
b, and G
x is defined as in (85).
Suppose that0
<
β
<
n and m is the number of repetitions of number 2 among i
1
,
i
2
,
…
,
i
β for k
-
1
<
β
<
k, k
=
1,2
,
…
,
n, say, i
k
1
,
i
k
2
,
…
,
i
k
m, such that k
1
<
k
2
<
⋯
<
k
m; that is, i
k
1
=
i
k
2
=
⋯
=
i
k
m
=
2 and 0
≤
m
≤
β. Then, one can find the following:Ifm is an even number, then(86)Di1,i2,…,iββCfx0=DβCf_x0;r,DβCf-x0;r.
Ifm is an odd number, then(87)Di1,i2,…,iββCfx0=DβCf-x0;r,DβCf_x0;r,where(88)DβCf_x0;r=1Γβ-β∫0xDβf_t;rx-t1-β+βdtx=x0,DβCf-x0;r=1Γβ-β∫0xDβf-t;rx-t1-β+βdtx=x0,Dkft=dkftdtk.Proof.
Letm be an even number. If we make the same steps in the proof of Theorem 7 when m is an even number, we get(89)DβCfx0=DβG_x0;r,DβG-x0;r,where D
=
d
/
d
x.
Thus,(90)
D
β
C
f
x
0
=
D
β
1
Γ
β
-
β
∫
0
x
f
_
t
;
r
d
t
x
-
t
1
-
β
+
β
-
∑
k
=
0
β
D
k
f
_
0
;
r
x
β
-
β
+
k
Γ
1
+
β
-
β
+
k
x
=
x
0
,
D
β
1
Γ
β
-
β
∫
0
x
f
-
t
;
r
d
t
x
-
t
1
-
β
+
β
-
∑
k
=
0
β
D
k
f
-
0
;
r
x
β
-
β
+
k
Γ
1
+
β
-
β
+
k
x
=
x
0
.By using Definition 5, we have:(91)
D
β
C
f
x
0
=
D
β
D
-
β
-
β
f
_
x
0
;
r
-
∑
k
=
0
β
D
k
f
_
0
;
r
D
β
x
β
-
β
+
k
Γ
1
+
β
-
β
+
k
x
=
x
0
,
D
β
D
-
β
-
β
f
-
x
0
;
r
-
∑
k
=
0
β
D
k
f
-
0
;
r
D
β
x
β
-
β
+
k
Γ
1
+
β
-
β
+
k
x
=
x
0
,where D
-
β
-
β
f
_
x
0
;
r and D
-
β
-
β
f
-
x
0
;
r are the Riemann-Liouville fractional integrals of the functions f
_
x
;
r and f
-
x
;
r at x
=
x
0, respectively. By using (b) of Theorem 1 with n
=
β, ν
=
β
-
β, and the equation D
n
x
m
=
Γ
m
+
1
/
Γ
m
+
1
-
n
x
m
-
n, we get:(92)
D
β
C
f
x
0
=
D
-
β
-
β
D
β
f
_
x
0
;
r
+
Q
_
β
x
0
,
-
β
-
∑
k
=
0
β
D
k
f
_
0
;
r
x
k
-
β
Γ
1
-
β
+
k
x
=
x
0
,
D
-
β
-
β
D
β
f
-
x
0
;
r
+
Q
¯
β
x
0
,
-
β
-
∑
k
=
0
β
D
k
f
-
0
;
r
x
k
-
β
Γ
1
-
β
+
k
x
=
x
0
.Thus,(93)
D
β
C
f
x
0
=
D
-
β
-
β
D
β
f
_
x
0
;
r
,
D
-
β
-
β
D
β
f
-
x
0
;
r
=
D
β
C
f
_
x
0
;
r
,
D
β
C
f
-
x
0
;
r
.If m is an odd number, the proof is similar.We note that if we taken
=
2 (0
<
β
<
2) in Theorem 13 we get Theorem  3.1 [1] which is found by Mazandarani and Kamyad.Theorem 14.
Suppose thatf
x
∈
C
F
0
,
∞
∩
L
F
0
,
∞ is fuzzy-valued function f
x
=
f
_
x
;
r
,
f
-
x
;
r for r
∈
0,1. One supposes that n
-
1
<
β
<
n and m is the number of repetitions of 2 among i
1
,
i
2
,
…
,
i
n, say, i
k
1
,
i
k
2
,
…
,
i
k
m, such that k
1
<
k
2
<
⋯
<
k
m; that is, i
k
1
=
i
k
2
=
⋯
=
i
k
m
=
2 and 0
≤
m
≤
n. Then, one has the following.
Ifm is an even number, we have:(94)LDi1,i2,…,inβCfx=sβLfx⊖sβ-1f0⊗∑k=1n-1sβ-k+1fk0,such that(95)⊗=⊖,ifthenumberofrepetitionsofnumber2amongi1,i2,…,ikisanevennumber,-,ifthenumberofrepetitionsofnumber2amongi1,i2,…,ikisanoddnumber.
Ifm is an odd number, we have:(96)LDi1,i2,…,inβCfx=-sβ-1f0⊖-sβLfx⊗∑k=1n-1sβ-k+1fk0,such that(97)⊗=⊖,ifthenumberofrepetitionsofnumber2amongi1,i2,…,ikisanoddnumber,-,ifthenumberofrepetitionsofnumber2amongi1,i2,…,ikisanevennumber.Proof.
ByD
i
1
,
i
2
,
…
,
i
n
β
C
f
x we mean D
i
1
,
…
,
i
k
1
,
…
,
i
k
2
,
…
,
i
k
m
,
…
,
i
n
β
C
f
x. Suppose that m is an odd number, and then from Theorem 13, when n
-
1
<
β
<
n, we get:(98)Di1,i2,…,inβCfx=DβCf-x;r,DβCf_x;r.Therefore, we get:(99)DβCf_x;r=DβCf-x;r,DβCf¯x;r=DβCf_x;r.Then, from (99), we get:(100)LDi1,i2,…,inβCfx=LDβCf_x;r,DβCf¯x;r=lDβCf-x;r,lDβCf_x;r.We know from Laplace transform of the Caputo fractional derivative of order β
>
0 that(101)lDβCf_x;r=sβlf_x;r-∑k=0n-1sβ-k-1f_k0;r=sβlf_x;r-sβ-1f_0;r-∑k=1n-1sβ-k-1f_k0;r.The above equation can be written as:(102)lDβCf_x;r=sβlf_x;r-sβ-1f_0;r-∑k=1k1-1sβ-k-1f_k0;r-∑k=k1k2-1sβ-k-1f_k0;r-⋯-∑k=km-1km-1sβ-k-1f_k0;r-∑k=kmn-1sβ-k-1f_k0;r.In a similar manner, we can get:(103)lDβCf-x;r=sβlf-x;r-sβ-1f-0;r-∑k=1k1-1sβ-k-1f-k0;r-∑k=k1k2-1sβ-k-1f-k0;r-⋯-∑k=km-1km-1sβ-k-1f_k0;r-∑k=kmn-1sβ-k-1f_k0;r.Since i
k
1
=
i
k
2
=
⋯
=
i
k
m
=
2 and m is an odd number, we have the following equations:(104)f_k0;r=fk_0;r,f-k0;r=fk¯0;r;1≤k≤k1-1,f_k0;r=fk¯0;r,f-k0;r=fk_0;r;k1≤k≤k2-1,⋮f_k0;r=fk_0;r,f-k0;r=fk¯0;r;km-1≤k≤km-1,f_k0;r=fk¯0;r,f-k0;r=fk_0;r;km≤k≤n-1.The last one of the equations in (104) yields because m is an odd number. Using (102), (103), and (104); then (100) becomes:(105)LDi1,i2,…,inβCfx=-sβ-1f0⊖-sβLfx⊗∑k=1n-1sβ-k-1fk0,where ⊗ is defined as (97).
Ifm is an even number, the proof is similar.Corollary 15.
Suppose thatf
x
∈
C
F
0
,
∞
∩
L
F
0
,
∞. One supposes that 2
<
β
<
3. Then:IfD
1,1
β
C
f
x is i
-
β
C-differentiable fuzzy-valued function, then(106)LD1,1,1βCfx=sβLfx⊖sβ-1f0⊖sβ-2f′0⊖sβ-3f′′0.
IfD
1,1
β
C
f
x is i
i
-
β
C-differentiable fuzzy-valued function, then(107)LD1,1,2βCfx=-sβ-1f0⊖-sβLfx-sβ-2f′0-sβ-3f′′0.
IfD
1,2
β
C
f
x is i
-
β
C-differentiable fuzzy-valued function, then(108)LD1,2,1βCfx=-sβ-1f0⊖-sβLfx-sβ-2f′0⊖sβ-3f′′0.
IfD
1,2
β
C
f
x is i
i
-
β
C-differentiable fuzzy-valued function, then(109)LD1,2,2βCfx=sβLfx⊖sβ-1f0⊖sβ-2f′0-sβ-3f′′0.
IfD
2,1
β
C
f
x is i
-
β
C-differentiable fuzzy-valued function, then(110)LD2,1,1βCfx=-sβ-1f0⊖-sβLfx⊖sβ-2f′0⊖sβ-3f′′0.
IfD
2,1
β
C
f
x is i
i
-
β
C-differentiable fuzzy-valued function, then(111)LD2,1,2βCfx=sβLfx⊖sβ-1f0-sβ-2f′0-sβ-3f′′0.
IfD
2,2
β
C
f
x is i
-
β
C-differentiable fuzzy-valued function, then(112)LD2,2,1βCfx=sβLfx⊖sβ-1f0-sβ-2f′0⊖sβ-3f′′0.
IfD
2,2
β
C
f
x is i
i
-
β
C-differentiable fuzzy-valued function, then(113)LD2,2,2βCfx=-sβ-1f0⊖-sβLfx⊖sβ-2f′0-sβ-3f′′0.Example 16.
Consider the following FFIVP:(114)D3/4Cyx=-yx,0<β<1,(115)y0=r,2-r.We note that:(116)y_0;r=r,y-0;r=2-r.By taking fuzzy Laplace transform for both sides of (114), we get:(117)LD3/4Cyx=-Lyx.Now, by using Theorem 14 when n
=
1 we have 2
1
=
2 cases as follows.Case 1.
Lety
x be i
-
β
C-differentiable fuzzy-valued function. By using Theorem 14, when m
=
0 (even), (117) becomes:(118)s3/4Lyx⊖s-1/4y0=-Lyx.Then, we get:(119)s3/4ly_x;r-s-1/4y_0;r=-ly-x;r,s3/4ly-x;r-s-1/4y-0;r=-ly_x;r.The solution of the above system is(120)ly_x;r=rs1/2-2-rs-1/4s3/2-1,ly-x;r=2-rs1/2-rs-1/4s3/2-1.The solution of FFIVP (114) is as follows:(121)y_x;r=rE3/2,1x3/2-2-rx3/4E3/2,7/4x3/2,y-x;r=2-rE3/2,1x3/2-rx3/4E3/2,7/4x3/2,where E
α
,
β
z denotes the Mittag-Leffler function.Case 2.
Lety
x be i
i
-
β
C-differentiable fuzzy-valued function. By using Theorem 14, when m
=
1 (odd), (117) becomes:(122)-s-1/4y0⊖-s3/4Lyx=-Lyx.Then, we get:(123)-s-1/4y-0;r+s3/4ly-x;r=-ly-x;r,-s-1/4y_0;r+s3/4ly_x;r=-ly_x;r.The solution of FFIVP (114) is as follows:(124)y_x;r=rE3/4,1-x3/4,y-x;r=2-rE3/4,1-x3/4.


## 5. Conclusions

The general formulas for fuzzy Riemann-Liouville and Caputo fractional derivatives about the general order0
<
β
<
n for fuzzy-valued function f are found by using all the possible arrangements of objects such that r
1 of them equal 1 and r
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 n
-
1
<
β
<
n are found under Hukuhara difference (H-difference).



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*Source: 6380978-2016-01-06.xml*