# An Eigenvalue-Eigenvector Method for Solving a System of Fractional Differential Equations with Uncertainty

Authors: M. R. Balooch Shahriyar; F. Ismail; S. Aghabeigi; A. Ahmadian; S. Salahshour
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

A new method is proposed for solving systems of fuzzy fractional differential equations (SFFDEs) with fuzzy initial conditions involving fuzzy Caputo differentiability. For this purpose, three cases are introduced based on the eigenvalue-eigenvector approach; then it is shown that the solution of system of fuzzy fractional differential equations is vector of fuzzy-valued functions. Then the method is validated by solving several examples.


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## Body

## 1. Introduction

Recently, a lot of research has been focused on the application of fractional calculus, and such application is in the modelling of many physical and chemical processes as well as in engineering [1–5].It has been found that the behavior of many physical systems can be properly described by using the fractional order system theory. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. The advantages or the real objects of the fractional order systems are that we have more degrees of freedom in the model and that a “memory” is included in the model. For example, the nonlinear oscillation of earthquake can be modeled with fractional derivatives [6]. In mechanics, fractional calculus plays an important role; for example, it has been successfully employed to model damping forces with memory effects to describe state feedback controllers [7, 8] and dynamics of interfaces between nanoparticles and substrates [9]. Due to its tremendous scope and applications in several disciplines, a considerable attention has been given to exact and numerical solutions of fractional differential equations [10–18]. The analytic results on the existence and uniqueness of solutions to the fractional differential equations have been studied by many authors [19, 20]. From the numerical point of view, several methods have been presented to achieve the goal of highly accurate and reliable solutions for the fractional differential equations. The most commonly used methods are fractional differential transform method [21], operational matrix method [22, 23], finite difference method [24], and Haar wavelets method [25].On the other hand, fuzzy differential equations have received considerable attention in dealing with various problems. So the development in this field has risen from the theoretical and practical perspectives [26–33].Recently, Agarwal et al. [34] proposed the concept of solutions for fractional differential equations with uncertainty which was followed by the authors in [35, 36]. They have considered Riemann-Liouville’s differentiability to solve FFDEs which is a combination of the Hukuhara difference and the Riemann-Liouville derivative. In [37, 38], the authors considered the generalization of H-differentiability for the fractional case. A lot of research has been devoted to find the accurate and efficient methods for solving fuzzy fractional differential equations (FFDEs). It is well known that the exact solutions of most of the FFDEs cannot be found easily; therefore, in the recent years, attempts have been made to address this problem [39–41]. It is with this motivation that we introduce in this paper an eigenvalue-eigenvector method for solving fuzzy fractional differential equations (FFDEs).Also regarding some defects of H-differentiability [42], Bede et al. [43–46] discussed the limitations of the H-differentiability due to the shortcoming of the Hukuhara differentiability. So we motivated our interest to adopt our proposed method based on the generalized differentiability in the sense of fractional order which was introduced by [37, 38, 41].In this paper, we intend to investigate the solutions of systems of fractional differential equations with uncertainty which is calledsystem of fuzzy fractional differential equations (SFFDEs). Here, we use the Riemann-Liouville derivative in the fuzzy concept, which applied the concept of the system of fractional derivatives under Caputo’s differentiability by applying the Hukuhara difference, which is denoted as fuzzy Caputo’s derivative. Similar to the deterministic cases, the construction of Caputo’s derivatives is based on the definitions of the Riemann-Liouville derivatives in fuzzy cases.This paper is organized as follows. In Section2, we review the well-known definitions of fuzzy numbers, and some basic concepts are given. In Section 3, fuzzy Caputo’s derivative is introduced, and the relation between Riemann-Liouville and Caputo’s derivatives and some of its properties is considered. Consequently, the eigenvalue-eigenvector method for solving fuzzy fractional order linear systems with initial values under fuzzy Caputo’s derivative is given, and three cases of eigenvalues are considered in Section 4. The proposed method is illustrated by solving several examples in Section 5 to depict the applicability and validity of the proposed method. Finally, conclusion is drawn in Section 6.


## 2. Preliminaries

The basic definition of fuzzy numbers is given in [47, 48].We denote the set of all real numbers byℝ, and the set of all fuzzy number on ℝ is indicated by 𝔼. A fuzzy number is a mapping u:ℝ→[0,1] with the following properties:(a)
uis upper semicontinuous,(b)
u is fuzzy convex; that is, u(λx+(1-λ)y)≥min{u(x),u(y)} for all x,y∈ℝ,λ∈[0,1],(c)
u is normal; that is, ∃x0∈ℝ for which u(x0)=1,(d)
suppu={x∈ℝ∣u(x)>0} is the support of the u, and its closure cl(supp u) is compact.An equivalent parametric definition is also given in [49–51] as follows.Definition 1.
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)
u_(r)≤u¯(r),  0≤r≤1.Moreover, we can also present ther-cut representation of fuzzy number as [u]r=[u_(r),u¯(r)] for all 0≤r≤1.According to Zadeh’s extension principle, operation of addition on𝔼 is defined by
(1)(u⊕v)(x)=supy∈ℝmin{u(y),v(x-y)},x∈ℝ,
and scalar multiplication of a fuzzy number is given by
(2)(k⊙u)(x)={u(xk),k>0,0~,k=0,
where 0~∈𝔼.The Hausdorff distance between fuzzy numbers is given byd:𝔼×𝔼→ℝ+⋃‍{0},
(3)d(u,v)=supr∈[0,1]max{|u_(r)-v_(r)|,|u¯(r)-v¯(α)|},
where u=(u_(r),u¯(r)) and v=(v_(r),v¯(r))⊂ℝ is utilized in [43]. Then, it is easy to see that d is a metric in 𝔼 and has the following properties (see [33]):(1)
d(u+w,v+w)=d(u,v), for all u,v,w∈𝔼,(2)
d(ku,kv)=|k|d(u,v), for all k∈ℝ,u,v∈𝔼,(3)
d(u+v,w+e)≤d(u,w)+d(v,e), for all u,v,w,e∈𝔼,(4)
(d,𝔼) is a complete metric space.Definition 2.
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.In this paper, the sign “⊖” always stands for H-difference, and also note that x⊖y≠x+(-1)y.


## 3. Fuzzy Caputo’s Derivative

In this section, the concept of fuzzy Caputo’s derivatives is considered using the Hukuhara difference. We denoteC𝔽[a,b] as a space of all fuzzy-valued functions which are continuous on [a,b]. Also, we denote the space of all Lebesgue integrable fuzzy-value functions on the bounded interval [a,b]⊂ℝ by L𝔽[a,b]. We denote the space of fuzzy-value functions f(x) which have continuous H-derivative up to order n-1 on [a,b] such that f(n-1)(x)∈AC𝔽([a,b]) by AC(n)𝔽([a,b]).Now, we define the fuzzy Riemann-Liouville integral of fuzzy-valued function as follows.Definition 3.
Letf(x)∈C𝔽[a,b]∩L𝔽[a,b], the fuzzy Riemann-Liouville integral of fuzzy-valued function f is defined as follows:
(4)(Ia+βf)(x)=1Γ(β)∫ax‍f(t)dt(x-t)1-β,x>a,0<β<1.Sincef(x;r)=[f_(x;r),f¯(x;r)], for all 0≤r≤1, then we can indicate the fuzzy Riemann-Liouville integral of fuzzy-valued function f based on the lower and upper functions as follows.Theorem 4.
Letf(x)∈C𝔽[a,b]∩L𝔽[a,b], the fuzzy Riemann-Liouville integral of fuzzy-valued function f is defined as follows:
(5)(Ia+βf)(x;r)=[(Ia+βf_)(x;r),(Ia+βf¯)(x;r)],0≤r≤1,
where
(6)(Ia+βf_)(x;r)=1Γ(β)∫ax‍f_(t;r)dt(x-t)1-β,0≤r≤1,(Ia+βf¯)(x;r)=1Γ(β)∫ax‍f¯(t;r)dt(x-t)1-β,0≤r≤1.Letf∈C𝔽((0,a])∩L𝔽(0,a) be a given function such that f(t,r)=[f_(t;r),f¯(t;r)] for all t∈(0,a] and 0≤r≤1. We define the fuzzy fractional Riemann-Liouville derivative of order 0<β<1 of f,
(7)RLDβf(t)=1Γ(1-β)ddt∫0t‍(t-s)-βf(s)ds,
and in the parametric form,
(8)RLDβf(t,r)=:1Γ(1-β)[ddt∫0t‍(t-s)-βf_(s;r)ds,ddt∫0t‍(t-s)-βf¯(s;r)ds],
provided that the equation defines a fuzzy number RLDβf(t)∈E. In fact,
(9)RLDβf(t,r):=[RLDβf_(t;r),RLDβf¯(t;r)].
Obviously,  RLDβf(t)=(d/dt)I1-βf(t) for t∈(0,a].Definition 5 (the relation between Riemann-Liouville and Caputo’s operators).
Letf(x)∈C(n)𝔽[a,b]∩L𝔽[a,b] be a fuzzy-valued function. One defines the fuzzy fractional Caputo’s derivative of order 0<β<1 of f,
(10)(CDa+βf)(x)=(RLDa+β[f(t)⊖∑k=0n-1‍xkk!f(k)(a)])(x),
provided that the equation defines a fuzzy number CDa+βf(x)∈E:
(11)(CDa+βf)(x;r)=[(CDa+βf_)(x;r),(CDa+βf¯)(x;r)],
where
(12)(CDa+βf_)(x;r)=RLDa+β[f_(t;r)-∑k=0n-1‍xkk!f_(k)(a;r)](x;r)=1Γ(n-β)dndxn∫ax‍f_(t;r)-∑k=0n-1(tk/k!)f_(k)(a;r)(x-t)β-n+1dt,(CDa+βf¯)(x;r)=RLDa+β[f¯(t;r)-∑k=0n-1‍xkk!f¯(k)(a;r)](x;r)=1Γ(n-β)dndxn∫ax‍f¯(t,r)-∑k=0n-1(tk/k!)f¯(k)(a;r)(x-t)β-n+1dt,
for all 0≤r≤1 and 0<β<1. Obviously, CDa+βf(t)=In-βDnf(t) for n-1<β<n,n∈ℕ.Definition 6 (see [52]).
Then×n linear system is
(13)a11x1+a12x2+⋯+a1nxn=y1,a11x1+a12x2+⋯+a1nxn=y1,⋮an1x1+an2x2+⋯+annxn=yn,
where the coefficient matrix A=(aij),1≤i,j≤n, is a crisp n×n matrix, and yi∈𝔼,1≤i≤n, is called a fuzzy system of linear equations (FSLEs).Definition 7.
A fuzzy number vector(x1,x2,…,xn)t given by xi=(x_i(r),x¯i(r)),1≤i≤n,0≤r≤1, is called a solution of the FSLE if
(14)∑j=1n‍aijxj_=∑j=1n‍aijxj_=y_i,∑j=1n‍aijxj¯=∑j=1n‍aijxj¯=y¯i.Considering theith equation of the system (13)
(15)ai1(x_1,x¯1)+⋯+aii(x_i,x¯i)+⋯+ain(x_n,x¯n)=(y_i,y¯i),
we have
(16)ai1x1_+⋯+aiixi_+⋯+ainxn_=yi_(r),ai1x1¯+⋯+aiixi¯+⋯+ainxn¯=yi¯(r),1≤i≤n,0≤r≤1.
From (16), we have two crisp n×n linear systems for all i that can be extended to a 2n×2n crisp linear system as follows:
(17)SX=Y⟹[s1≥0s2≤0s2≤0s1≥0][X_X¯]=[Y_Y¯],
where sij are determined as follows:
(18)ifaij≥0⟹sij=aij,si+n,j+n=aij,ifaij<0⟹si,j+n=aij,si+n,j=aij,
and any sij which is not determined is zero.In this paper, the following system will be solved:(19)dβ1x~1dtβ1=f1(x~1,x~2,…,x~n)=a11x~1+a12x~2+⋯+a1nx~n,dβ2x~2dtβ2=f2(x~1,x~2,…,x~n)=a21x~1+a22x~2+⋯+a2nx~n,⋮dβnx~ndtβn=fn(x~1,x~2,…,x~n)=an1x~1+an2x~2+⋯+annx~n.
Thus,
(20)dβdtβX~(t)=AX~(t),X~(0)=X0∈𝔼n,
where X~∈𝔼n, the matrix A=(aij),1≤i,j≤n,  β=[β1,β2,…,βn] indicates the fractional orders, dβ/dtβ=[dβ1/dtβ1,dβ2/dtβ2,…,dβn/dtβn], and dβi/dtβi is the fuzzy Caputo’s fractional derivative of order βi, where 0<βi<1, for i=1,2,…,n. To obtain the solution of (dβ/dtβ)X~(t)=AX~(t), the eigenvalue-eigenvector method is used.


## 4. Fuzzy Fractional Order Linear Systems

In this section, we drive the general solution for fuzzy fractional order linear system as follows:(21)dβdtβX~(t)=AX~(t),0<t≤a,
where X~∈𝔼n,a>0, the coefficient matrix A=(aij),1≤i,j≤n, is a crisp n×n matrix and x~i∈𝔼,1≤i≤n, and dβ/dtβ is the fuzzy Caputo’s fractional derivative, where 0<β<1. Simply to construct the general solution of the system (21), we proceed by analogy with treatment of homogeneous integer order fuzzy linear systems with the constant coefficient where the exponential function Exp(t) is replaced by the Mittag-Leffler function Eβ(tβ). Thus, we seek solutions of the form
(22)X~(t)=ξEβ(λtβ),
where the constant λ and the vector ξ are to be determined. Substituting form (22) for Xin the system (21) gives
(23)ξλEβ(λtβ)=AξEβ(λtβ).
Upon canceling the nonzero factor Eβ(λtβ), we obtain Aξ=λξ or
(24)(A-λI)ξ=0,
where I is the n×n identity matrix. Therefore, the vector X~ given by (22) is a solution of the system (21) provided that λ and the vector ξ are associated eigenvectors of the matrix A. In the following Section, three cases for the eigenvalue of matrix A are discussed.


### 4.1. Real and Distinct Eigenvalues

In this case, suppose thatλi, for i=1,…,n, are the real eigenvalues of matrix A. Therefore, the solution of (20) is as follows:
(25)X~(t)=∑i=1n‍c~ivi(t),
where c~i are fuzzy numbers, vi(t)=Eβ(λitβ)ξi for i=1,…,n and λi and ξi are the real eigenvalues and eigenvector of matrix A, respectively.By setting initial valuest=a, in (25),
(26)X~(a)=∑i=1n‍c~iEβ(λitβ)ξi=α~
is obtained, where ξi=[ξi1,ξi2,…,ξin]t and α~=[α~1,α~2,…,α~n]t. Thus, c~i, i=1,…,n. From the following, fuzzy systems are obtained:
(27)c~1Eβλ1aβξ11+c~2Eβλ2aβξ21+⋯+c~nEβλnaβξn1=α~1,c~1Eβλ1aβξ12+c~2Eβλ2aβξ22+⋯+c~nEβλnaβξn2=α~2,⋮c~1Eβλ1aβξ1n+c~2Eβλ2aβξ2n+⋯+c~nEβλnaβξnn=α~2.
The parametric form of (27) is as follows:
(28)(c_1(r),c¯1(r))Eβλ1aβξ11+(c_2(r),c¯2(r))Eβλ2aβξ21+⋯+(c_n(r),c¯n(r))Eβλnaβξn1=(α_1(r),α¯1(r)),(c_1(r),c¯1(r))Eβλ1aβξ12+(c_2(r),c¯2(r))Eβλ2aβξ22+⋯+(c_n(r),c¯n(r))Eβλnaβξn2=(α_2(r),α¯2(r)),⋮(c_1(r),c¯1(r))Eβλ1aβξ1n+(c_2(r),c¯2(r))Eβλ2aβξ2n+⋯+(c_n(r),c¯n(r))Eβλnaβξnn=(α_n(r),α¯n(r)).
Now similar to (17), there is a 2n×2n crisp system. Therefore, c~i=(c_i(r),c¯i(r)),i=1,2,…,n are obtained from (28) and are set in (25). Finally the solution of (20) will be obtained from X~(t)=[x~1,x~2,…,x~n]t.Theorem 8.
The solution of fuzzy system (20) with real eigenvalues is a fuzzy number (25).Proof.
It is shown thatX~(t)=∑i=1n‍c~iEβ(λitβ)ξi for i=1,…,n and 0<β<1 is the solution of (dβ/dtβ)X~=AX~. Let C~=[c~1,c~2,…,c~n] with c~i which are the fuzzy numbers and [C~]r=Πi=1n[c~i]r, then
(29)X_(t;r)=min{∑i=1n‍ciEβ(λitβ)ξi∣ci∈[C~]r}=∑i=1n‍ciEβ(λitβ)ξi_=∑i=1n‍ciEβ(λitβ)ξi_,X¯(t;r)=max{∑i=1n‍ciEβ(λitβ)ξi∣ci∈[C~]r}=∑i=1n‍ciEβ(λitβ)ξi¯=∑i=1n‍ciEβ(λitβ)ξi¯.
With differentiation of (29), we are obtained:
(30)dβdtβX_(t;r)=∑i=1n‍λiciEβ(λitβ)ξi_,0≤r≤1,dβdtβX¯(t;r)=∑i=1n‍λiciEβ(λitβ)ξi¯,0≤r≤1.
Since λi is an eigenvalue and ξi is its corresponding eigenvalue of matrix A, then Aξi=λξi,i=1,….,n. Therefore,
(31)dβdtβX_(t;r)=∑i=1n‍λiciEβ(λitβ)ξi_=∑i=1n‍λiξiciEβ(λitβ)_=A∑i=1n‍ciEβ(λitβ)ξi_=AX_(t;r),dβdtβX¯(t;r)=∑i=1n‍λiciEβ(λitβ)ξi¯=∑i=1n‍λiξiciEβ(λitβ)¯=A∑i=1n‍ciEβ(λitβ)ξi¯=AX¯(t;r).
Such that
(32)(dβdtβX_(t;r);dβdtβX¯(t;r))=A(X_(t;r);X¯(t;r)),--------------0<β<1,0≤r≤1.
This means that (dβ/dtβ)X~(t)=AX~(t). From (7), (10), and (dβ/dtβ)X~(t)=[(dβ1/dtβ1)x~1,  (dβ2/dtβ2)x~2,…,  (dβn/dtβn)x~n]t, it is clear that X~(t) is a fuzzy number vector.


### 4.2. Complex Eigenvalues

In this case, suppose that some eigenvalues ofλi, for i=1,2,…,k, are complex numbers. Since the entries of matrix A are real, therefore characteristic polynomial has real coefficients; therefore, complex roots are in conjugate pairs.Lemma 9.
Let the entries of matrixA be real and λ an eigenvalue of matrix A, where λ=μ+iν,ν≠0, and ξ=a+ib are the corresponding eigenvectors of λ, then u1(t)=Re[ξEβ(μ+iν)tβ)], u2(t)=Im[ξEβ(μ+iν)tβ)] are solutions.
Therefore, from the above-mentioned lemma, the solution of each pair of conjugate complex eigenvaluesλ=μ+iν is as follows:
(33)w~(t)=c~1Re[ξEβ(μ+iν)tβ)]+c~2Im[ξEβ(μ+iν)tβ)],
where ξ is the corresponding eigenvector of eigenvalue λ. Hence, the solution of (33) is as follows:
(34)X~(t)=∑i=1i=k/2‍w~i(t)+i∑i=k/2+1i=n‍v~i(t),
where w~i(t)=c~i1Re(ξiEβ(λitβ))+c~i2Im(ξiEβ(λitβ)) from each pair of conjugate complex eigenvalues and vi(t)=c~iξiEβ(λitβ) from real eigenvalues are obtained. Then by setting initial values t=a in (34) and by solving a fuzzy system similar to (28), fuzzy coefficients are obtained. By setting fuzzy coefficient in (34), X~(t) is obtained; finally the solution of (20) will be obtained from X~(t)=[x1,x2,…,xn]t.Theorem 10.
The solution of fuzzy system (20) with complex eigenvalues is a fuzzy number (34).Proof.
It is shown thatX~(t)=∑i=1i=k/2‍w~i(t)+i∑i=k/2i=n‍v~i(t) is the solution of (dβ/dtβ)X~=AX~.
LetC~1=(c~11,c~21,…,c~k/2,1),C~2=(c~12,c~22,…,c~k/2,2), and C~=(c~k/2+1,c~k/2+2,…,c~n) with c~i1,c~i2,c~i which are the fuzzy numbers and [C~1]r=Πi=1k/2[c~i1]r,[C~2]r=Πi=1k/2[c~i2]r, and [C~]r=Πi=k/2+1k/2[c~i]r,(35)X_(t;r)=min{∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))+∑i=k/2+1n‍ciξiEβ(λitβ)∣ci1∈[C~1]r,ci2∈[C~2]r,ci∈[C~]r}=∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))+∑i=k/2+1n‍ciξiEβ(λitβ)_=∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))_+∑i=k/2+1n‍ciξiEβ(λitβ)_,X¯(t;r)=max{∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))+∑i=k/2+1n‍ciξiEβ(λitβ)∣ci1∈[C~1]r,ci2∈[C~2]r,ci∈[C~]r}=∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))+∑i=k/2+1n‍ciξiEβ(λitβ)¯=∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))¯+∑i=k/2+1n‍ciξiEβ(λitβ)¯.With differentiation of the above equations we obtain the following
(36)dβdtβX_(t;r)=∑i=1k/2‍ci1Re(ξiλiEβ(λitβ))+ci2Im(ξiλiEβ(λitβ))_+∑i=k/2+1n‍ciλiξiEβ(λitβ)_,0≤r≤1,dβdtβX¯(t;r)=∑i=1k/2‍ci1Re(ξiλiEβ(λitβ))+ci2Im(ξiλiEβ(λitβ))¯+∑i=k/2+1n‍ciλiξiEβ(λitβ)¯,0≤r≤1.
Since λi is an eigenvalue and ξi is its corresponding eigenvalue of matrix A, then Aξi=λξi,i=1,….,n. Therefore,
(37)dβdtβX_(t;r)=∑i=1k/2‍ci1Re(AξiEβ(λitβ))+ci2Im(AξiEβ(λitβ))_+∑i=k/2+1n‍ciAξiEβ(λitβ)_=A(∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))_+∑i=k/2+1n‍ciξiEβ(λitβ)_)=AX_(t;r),dβdtβX¯(t;r)=∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))¯+∑i=k/2+1n‍ciξiEβ(λitβ)¯=A(∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))¯------+∑i=k/2+1n‍ciξiEβ(λitβ)¯)=AX¯(t;r).
Then,
(38)(dβdtβX_(t;r);dβdtβX¯(t;r))=A(X_(t;r);X¯(t;r)),---------------0<β<1,0≤r≤1.
This means that (dβ/dtβ)X~(t)=AX~(t). From (7), (10), and (dβ/dtβ)X~(t)=[(dβ1/dtβ1)x~1,  (dβ2/dtβ2)x~2,…,(dβn/dtβn)x~n]t, it is clear that X~(t) is a fuzzy number vector.


### 4.3. Multiple Eigenvalues

In this case, suppose that some eigenvalues of matrixA are multiple. Suppose that λ0 is an eigenvalue of matrix A with multiplicity k, and the corresponding eigenvectors of eigenvalue λ0 are ξ1,ξ2,…,ξk, if all ξi are linearly independent, then
(39)X~(t)=c~1ξ1Eβλ0tβ+c~2ξ2Eβλ0tβ+⋯+c~kξkEβλ0tβ.
If ξi and m are linearly independent vectors, that is, m<k, then the following lemma is brought.Lemma 11.
Letλ0 be an eigenvalue of matrix A with multiple k>1, and let the numbers of ξi which are linearly independent be less than m0, therefore at least one non-zero vector exists such that
(40)(A-λI)ξ≠0,(A-λI)2ξ=0.
If ξ is satisfied in (40), the solution is as follows:
(41)v′(t)=Eβ(1)(λ0tβ)tβ(A-λ0I)ξ+ξEβ(λ0tβ),
based on the properties of the Mittag-Leffler type functions, where
(42)Eβ(1)(z)=ddtEβ(z).In general, if matrixA has a repeated eigenvalue λ0 of multiplicity k with m linearly independent eigenvectors, where m<k, then the following
(43)x1(t)=ξ1Eβ(λ0tβ),x2(t)=ξ1tβEβ(1)(λ0tβ)+ξ2Eβ(λ0tβ),⋮xk-m(t)=ξ1tβ(k-m-1)Eβ(k-m-1)(λ0tβ)+ξ2tβ(k-m-2)Eβ(k-m-2)(λ0tβ)+⋯+ξk-mEβ(λ0tβ),
are k-m linearly independent solutions of the system (20).Hence, with the above-mentioned lemma, the solution of (20) is as follows:
(44)X~(t)=∑i=1i=k‍c~ivi′(t)+∑i=k+1i=n‍c~ivi(t),
where vi′(t)=Eβ(k)(λitβ)tβ(A-λiI)ξi+ξiEβ(λitβ) for λi which are satisfied in Lemma 11 and vi(t)=ξiEβ(λitβ) for real eigenvalues are obtained. Then by setting initial values t=a, in (44) and by solving a fuzzy system similar to (28), fuzzy coefficient is obtained. By setting fuzzy coefficient in (44), X~(t) is obtained, and finally the solution of (20) will be obtained from X~(t)=[x~1,x~2,…,x~n]t.Theorem 12.
The solution of fuzzy system (20) with multiple eigenvalues is a fuzzy number (44).Proof.
It is shown thatX~(t)=∑i=1i=k‍c~ivi′(t)+∑i=k+1i=n‍c~ivi(t) is the solution of (dβ/dtβ)X~=AX~. Let C~=[c~1,c~2,…,c~n] with c~i which are the fuzzy numbers and [C~]r=Πi=1n[c~i]r, then (45)X_(t;r)=min{∑i=1k‍ciEβ(k)(λitβ)(tβ(A-λiI)ξi)+ξiEβ(λitβ)+∑i=k+1i=n‍ciξiEβ(λitβ)∣ci∈[C~]r}=∑i=1k‍ciEβ(k)(λitβ)(tβ(A-λiI)ξi)+ξiEβ(λitβ)+∑i=k+1i=n‍ciξiEβ(λitβ)_=∑i=1k‍ciEβ(k)(λitβ)(tβ(A-λiI)ξi)+ξiEβ(λitβ)_+∑i=k+1i=n‍ciξiEβ(λitβ)_,X¯(t;r)=max{∑i=1k‍ciEβ(k)(λitβ)(tβ(A-λiI)ξi)+ξiEβ(λitβ)+∑i=k+1i=n‍ciξiEβ(λitβ)∣ci∈[C~]r}=∑i=1k‍ciEβ(k)(λitβ)(tβ(A-λiI)ξi)+ξiEβ(λitβ)+∑i=k+1i=n‍ciξiEβ(λitβ)¯=∑i=1k‍ciEβ(k)(λitβ)(tβ(A-λiI)ξi)+ξiEβ(λitβ)¯+∑i=k+1i=n‍ciξiEβ(λitβ)¯.With differentiation of the above equations, we obtain(46)dβdtβX_(t;r)=∑i=1k‍ci[λiEβ(k)(λitβ)(tβ(A-λiI)ξi+Eβ(k)(λitβ)(A-λiI)ξi+ξiλiEβ(λitβ)]_+∑i=k+1i=n‍ciξiλiEβ(λitβ)_,dβdtβX¯(t;r)=∑i=1k‍ci[λiEβ(k)(λitβ)(tβ(A-λiI)ξi+Eβ(k)(λitβ)(A-λiI)ξi+ξiλiEβ(λitβ)]¯+∑i=k+1i=n‍ciξiλiEβ(λitβ)¯.Since λi is an eigenvalue and ξi is its corresponding eigenvalue of matrix A, then Aξi=λξi. Therefore,
(47)(dβdtβX_(t;r);dβdtβX¯(t;r))=A(X_(t;r);X¯(t;r)),--------------0<β<1,0≤r≤1.
This means that (dβ/dtβ)X~(t)=AX~(t). From (7), (10), (dβ/dtβ)X~(t)=[(dβ1/dtβ1)x~1,  (dβ2/dtβ2)x~2,…,  (dβn/dtβn)x~n]t, it is clear that X~(t) is a fuzzy number vector.


## 4.1. Real and Distinct Eigenvalues

In this case, suppose thatλi, for i=1,…,n, are the real eigenvalues of matrix A. Therefore, the solution of (20) is as follows:
(25)X~(t)=∑i=1n‍c~ivi(t),
where c~i are fuzzy numbers, vi(t)=Eβ(λitβ)ξi for i=1,…,n and λi and ξi are the real eigenvalues and eigenvector of matrix A, respectively.By setting initial valuest=a, in (25),
(26)X~(a)=∑i=1n‍c~iEβ(λitβ)ξi=α~
is obtained, where ξi=[ξi1,ξi2,…,ξin]t and α~=[α~1,α~2,…,α~n]t. Thus, c~i, i=1,…,n. From the following, fuzzy systems are obtained:
(27)c~1Eβλ1aβξ11+c~2Eβλ2aβξ21+⋯+c~nEβλnaβξn1=α~1,c~1Eβλ1aβξ12+c~2Eβλ2aβξ22+⋯+c~nEβλnaβξn2=α~2,⋮c~1Eβλ1aβξ1n+c~2Eβλ2aβξ2n+⋯+c~nEβλnaβξnn=α~2.
The parametric form of (27) is as follows:
(28)(c_1(r),c¯1(r))Eβλ1aβξ11+(c_2(r),c¯2(r))Eβλ2aβξ21+⋯+(c_n(r),c¯n(r))Eβλnaβξn1=(α_1(r),α¯1(r)),(c_1(r),c¯1(r))Eβλ1aβξ12+(c_2(r),c¯2(r))Eβλ2aβξ22+⋯+(c_n(r),c¯n(r))Eβλnaβξn2=(α_2(r),α¯2(r)),⋮(c_1(r),c¯1(r))Eβλ1aβξ1n+(c_2(r),c¯2(r))Eβλ2aβξ2n+⋯+(c_n(r),c¯n(r))Eβλnaβξnn=(α_n(r),α¯n(r)).
Now similar to (17), there is a 2n×2n crisp system. Therefore, c~i=(c_i(r),c¯i(r)),i=1,2,…,n are obtained from (28) and are set in (25). Finally the solution of (20) will be obtained from X~(t)=[x~1,x~2,…,x~n]t.Theorem 8.
The solution of fuzzy system (20) with real eigenvalues is a fuzzy number (25).Proof.
It is shown thatX~(t)=∑i=1n‍c~iEβ(λitβ)ξi for i=1,…,n and 0<β<1 is the solution of (dβ/dtβ)X~=AX~. Let C~=[c~1,c~2,…,c~n] with c~i which are the fuzzy numbers and [C~]r=Πi=1n[c~i]r, then
(29)X_(t;r)=min{∑i=1n‍ciEβ(λitβ)ξi∣ci∈[C~]r}=∑i=1n‍ciEβ(λitβ)ξi_=∑i=1n‍ciEβ(λitβ)ξi_,X¯(t;r)=max{∑i=1n‍ciEβ(λitβ)ξi∣ci∈[C~]r}=∑i=1n‍ciEβ(λitβ)ξi¯=∑i=1n‍ciEβ(λitβ)ξi¯.
With differentiation of (29), we are obtained:
(30)dβdtβX_(t;r)=∑i=1n‍λiciEβ(λitβ)ξi_,0≤r≤1,dβdtβX¯(t;r)=∑i=1n‍λiciEβ(λitβ)ξi¯,0≤r≤1.
Since λi is an eigenvalue and ξi is its corresponding eigenvalue of matrix A, then Aξi=λξi,i=1,….,n. Therefore,
(31)dβdtβX_(t;r)=∑i=1n‍λiciEβ(λitβ)ξi_=∑i=1n‍λiξiciEβ(λitβ)_=A∑i=1n‍ciEβ(λitβ)ξi_=AX_(t;r),dβdtβX¯(t;r)=∑i=1n‍λiciEβ(λitβ)ξi¯=∑i=1n‍λiξiciEβ(λitβ)¯=A∑i=1n‍ciEβ(λitβ)ξi¯=AX¯(t;r).
Such that
(32)(dβdtβX_(t;r);dβdtβX¯(t;r))=A(X_(t;r);X¯(t;r)),--------------0<β<1,0≤r≤1.
This means that (dβ/dtβ)X~(t)=AX~(t). From (7), (10), and (dβ/dtβ)X~(t)=[(dβ1/dtβ1)x~1,  (dβ2/dtβ2)x~2,…,  (dβn/dtβn)x~n]t, it is clear that X~(t) is a fuzzy number vector.


## 4.2. Complex Eigenvalues

In this case, suppose that some eigenvalues ofλi, for i=1,2,…,k, are complex numbers. Since the entries of matrix A are real, therefore characteristic polynomial has real coefficients; therefore, complex roots are in conjugate pairs.Lemma 9.
Let the entries of matrixA be real and λ an eigenvalue of matrix A, where λ=μ+iν,ν≠0, and ξ=a+ib are the corresponding eigenvectors of λ, then u1(t)=Re[ξEβ(μ+iν)tβ)], u2(t)=Im[ξEβ(μ+iν)tβ)] are solutions.
Therefore, from the above-mentioned lemma, the solution of each pair of conjugate complex eigenvaluesλ=μ+iν is as follows:
(33)w~(t)=c~1Re[ξEβ(μ+iν)tβ)]+c~2Im[ξEβ(μ+iν)tβ)],
where ξ is the corresponding eigenvector of eigenvalue λ. Hence, the solution of (33) is as follows:
(34)X~(t)=∑i=1i=k/2‍w~i(t)+i∑i=k/2+1i=n‍v~i(t),
where w~i(t)=c~i1Re(ξiEβ(λitβ))+c~i2Im(ξiEβ(λitβ)) from each pair of conjugate complex eigenvalues and vi(t)=c~iξiEβ(λitβ) from real eigenvalues are obtained. Then by setting initial values t=a in (34) and by solving a fuzzy system similar to (28), fuzzy coefficients are obtained. By setting fuzzy coefficient in (34), X~(t) is obtained; finally the solution of (20) will be obtained from X~(t)=[x1,x2,…,xn]t.Theorem 10.
The solution of fuzzy system (20) with complex eigenvalues is a fuzzy number (34).Proof.
It is shown thatX~(t)=∑i=1i=k/2‍w~i(t)+i∑i=k/2i=n‍v~i(t) is the solution of (dβ/dtβ)X~=AX~.
LetC~1=(c~11,c~21,…,c~k/2,1),C~2=(c~12,c~22,…,c~k/2,2), and C~=(c~k/2+1,c~k/2+2,…,c~n) with c~i1,c~i2,c~i which are the fuzzy numbers and [C~1]r=Πi=1k/2[c~i1]r,[C~2]r=Πi=1k/2[c~i2]r, and [C~]r=Πi=k/2+1k/2[c~i]r,(35)X_(t;r)=min{∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))+∑i=k/2+1n‍ciξiEβ(λitβ)∣ci1∈[C~1]r,ci2∈[C~2]r,ci∈[C~]r}=∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))+∑i=k/2+1n‍ciξiEβ(λitβ)_=∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))_+∑i=k/2+1n‍ciξiEβ(λitβ)_,X¯(t;r)=max{∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))+∑i=k/2+1n‍ciξiEβ(λitβ)∣ci1∈[C~1]r,ci2∈[C~2]r,ci∈[C~]r}=∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))+∑i=k/2+1n‍ciξiEβ(λitβ)¯=∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))¯+∑i=k/2+1n‍ciξiEβ(λitβ)¯.With differentiation of the above equations we obtain the following
(36)dβdtβX_(t;r)=∑i=1k/2‍ci1Re(ξiλiEβ(λitβ))+ci2Im(ξiλiEβ(λitβ))_+∑i=k/2+1n‍ciλiξiEβ(λitβ)_,0≤r≤1,dβdtβX¯(t;r)=∑i=1k/2‍ci1Re(ξiλiEβ(λitβ))+ci2Im(ξiλiEβ(λitβ))¯+∑i=k/2+1n‍ciλiξiEβ(λitβ)¯,0≤r≤1.
Since λi is an eigenvalue and ξi is its corresponding eigenvalue of matrix A, then Aξi=λξi,i=1,….,n. Therefore,
(37)dβdtβX_(t;r)=∑i=1k/2‍ci1Re(AξiEβ(λitβ))+ci2Im(AξiEβ(λitβ))_+∑i=k/2+1n‍ciAξiEβ(λitβ)_=A(∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))_+∑i=k/2+1n‍ciξiEβ(λitβ)_)=AX_(t;r),dβdtβX¯(t;r)=∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))¯+∑i=k/2+1n‍ciξiEβ(λitβ)¯=A(∑i=1k/2‍ci1Re(ξiEβ(λitβ))+ci2Im(ξiEβ(λitβ))¯------+∑i=k/2+1n‍ciξiEβ(λitβ)¯)=AX¯(t;r).
Then,
(38)(dβdtβX_(t;r);dβdtβX¯(t;r))=A(X_(t;r);X¯(t;r)),---------------0<β<1,0≤r≤1.
This means that (dβ/dtβ)X~(t)=AX~(t). From (7), (10), and (dβ/dtβ)X~(t)=[(dβ1/dtβ1)x~1,  (dβ2/dtβ2)x~2,…,(dβn/dtβn)x~n]t, it is clear that X~(t) is a fuzzy number vector.


## 4.3. Multiple Eigenvalues

In this case, suppose that some eigenvalues of matrixA are multiple. Suppose that λ0 is an eigenvalue of matrix A with multiplicity k, and the corresponding eigenvectors of eigenvalue λ0 are ξ1,ξ2,…,ξk, if all ξi are linearly independent, then
(39)X~(t)=c~1ξ1Eβλ0tβ+c~2ξ2Eβλ0tβ+⋯+c~kξkEβλ0tβ.
If ξi and m are linearly independent vectors, that is, m<k, then the following lemma is brought.Lemma 11.
Letλ0 be an eigenvalue of matrix A with multiple k>1, and let the numbers of ξi which are linearly independent be less than m0, therefore at least one non-zero vector exists such that
(40)(A-λI)ξ≠0,(A-λI)2ξ=0.
If ξ is satisfied in (40), the solution is as follows:
(41)v′(t)=Eβ(1)(λ0tβ)tβ(A-λ0I)ξ+ξEβ(λ0tβ),
based on the properties of the Mittag-Leffler type functions, where
(42)Eβ(1)(z)=ddtEβ(z).In general, if matrixA has a repeated eigenvalue λ0 of multiplicity k with m linearly independent eigenvectors, where m<k, then the following
(43)x1(t)=ξ1Eβ(λ0tβ),x2(t)=ξ1tβEβ(1)(λ0tβ)+ξ2Eβ(λ0tβ),⋮xk-m(t)=ξ1tβ(k-m-1)Eβ(k-m-1)(λ0tβ)+ξ2tβ(k-m-2)Eβ(k-m-2)(λ0tβ)+⋯+ξk-mEβ(λ0tβ),
are k-m linearly independent solutions of the system (20).Hence, with the above-mentioned lemma, the solution of (20) is as follows:
(44)X~(t)=∑i=1i=k‍c~ivi′(t)+∑i=k+1i=n‍c~ivi(t),
where vi′(t)=Eβ(k)(λitβ)tβ(A-λiI)ξi+ξiEβ(λitβ) for λi which are satisfied in Lemma 11 and vi(t)=ξiEβ(λitβ) for real eigenvalues are obtained. Then by setting initial values t=a, in (44) and by solving a fuzzy system similar to (28), fuzzy coefficient is obtained. By setting fuzzy coefficient in (44), X~(t) is obtained, and finally the solution of (20) will be obtained from X~(t)=[x~1,x~2,…,x~n]t.Theorem 12.
The solution of fuzzy system (20) with multiple eigenvalues is a fuzzy number (44).Proof.
It is shown thatX~(t)=∑i=1i=k‍c~ivi′(t)+∑i=k+1i=n‍c~ivi(t) is the solution of (dβ/dtβ)X~=AX~. Let C~=[c~1,c~2,…,c~n] with c~i which are the fuzzy numbers and [C~]r=Πi=1n[c~i]r, then (45)X_(t;r)=min{∑i=1k‍ciEβ(k)(λitβ)(tβ(A-λiI)ξi)+ξiEβ(λitβ)+∑i=k+1i=n‍ciξiEβ(λitβ)∣ci∈[C~]r}=∑i=1k‍ciEβ(k)(λitβ)(tβ(A-λiI)ξi)+ξiEβ(λitβ)+∑i=k+1i=n‍ciξiEβ(λitβ)_=∑i=1k‍ciEβ(k)(λitβ)(tβ(A-λiI)ξi)+ξiEβ(λitβ)_+∑i=k+1i=n‍ciξiEβ(λitβ)_,X¯(t;r)=max{∑i=1k‍ciEβ(k)(λitβ)(tβ(A-λiI)ξi)+ξiEβ(λitβ)+∑i=k+1i=n‍ciξiEβ(λitβ)∣ci∈[C~]r}=∑i=1k‍ciEβ(k)(λitβ)(tβ(A-λiI)ξi)+ξiEβ(λitβ)+∑i=k+1i=n‍ciξiEβ(λitβ)¯=∑i=1k‍ciEβ(k)(λitβ)(tβ(A-λiI)ξi)+ξiEβ(λitβ)¯+∑i=k+1i=n‍ciξiEβ(λitβ)¯.With differentiation of the above equations, we obtain(46)dβdtβX_(t;r)=∑i=1k‍ci[λiEβ(k)(λitβ)(tβ(A-λiI)ξi+Eβ(k)(λitβ)(A-λiI)ξi+ξiλiEβ(λitβ)]_+∑i=k+1i=n‍ciξiλiEβ(λitβ)_,dβdtβX¯(t;r)=∑i=1k‍ci[λiEβ(k)(λitβ)(tβ(A-λiI)ξi+Eβ(k)(λitβ)(A-λiI)ξi+ξiλiEβ(λitβ)]¯+∑i=k+1i=n‍ciξiλiEβ(λitβ)¯.Since λi is an eigenvalue and ξi is its corresponding eigenvalue of matrix A, then Aξi=λξi. Therefore,
(47)(dβdtβX_(t;r);dβdtβX¯(t;r))=A(X_(t;r);X¯(t;r)),--------------0<β<1,0≤r≤1.
This means that (dβ/dtβ)X~(t)=AX~(t). From (7), (10), (dβ/dtβ)X~(t)=[(dβ1/dtβ1)x~1,  (dβ2/dtβ2)x~2,…,  (dβn/dtβn)x~n]t, it is clear that X~(t) is a fuzzy number vector.


## 5. Examples

Example 1.
Consider that the system is with initial value(48)(dβx~dtβdβy~dtβ)=A(x~y~),A=(2-14-3),(x~(0)y~(0))=(r+13-r),
where 0<β<1.
The eigenvalues of the matrixA are λ1=1, and λ2=-2 and their corresponding eigenvectors are x1=[1,1]t, and x2=[1,4]t, respectively. Therefore, the general solution of the system (48) is
(49)X~(t)=c~1(11)Eβ(tβ)+c~2(14)Eβ(-2tβ),
where c~1 and c~2 are fuzzy coefficients. This is obtained by setting the initial values in parametric form, then the following system is obtained:
(50)[1100140000110014].[c_1c_2c¯1c¯2]=[r+12+r3-r5-r],
where c~1=(r+2/3;7/3-r) and c~2=(1/3;2/3) are obtained from the above system and are set in (25). Therefore, the solution of SFFDEs is as follows:
(51)X~(t)=((r+23)Eβ(tβ)+(13)Eβ(-2tβ),(73-r)Eβ(tβ)+4(23)Eβ(-2tβ)).Example 2.
Consider that the system is with initial value(52)(dβx~dtβdβy~dtβdβz~dtβ)=A(x~y~z~),A=(0100015-3-1),
where 0<β<1.
The eigenvalues of the matrixA are λ1=1,λ2=-1+2i, and λ3=-1-2i, and their corresponding eigenvectors are ξ1=[1,1,1]t,  ξ2=[(-3/25)+(4/25)i,  (-1/5)-(2/5)i,1]t,  and ξ3=[(-3/25)-(4/25)i,  (-1/5)+(2/5)i,1]t, which are eigenvalues and eigenvectors of matrix A, respectively, and by setting the initial values in parametric form, the following system is obtained:
(53)[104250-32501000-15-251100000-3250104250-15-25100000110]·[c_1c_2c_3c¯1c¯2c¯3]=[0.75+0.25r1.5+0.5r3.75+0.25r1.25-0.25r2.5-0.5r4.25-0.25r],c~1=(1.553571428+0.071428r,1.696428572-0.0714285714r),c~2=(2.196428572+0.1785714286r,2.553571428-0.1785714286r),c~3=(-3.107142858+0.9821428567r,-1.142857142-0.9821428527r),
which are obtained from the above system and are set in (25). Therefore, the solution of SFFDEs is as follows:
(54)X_(t;r)=(1.553571428+0.071428r)Eβ(tβ)+(2.196428572+0.1785714286r)2(-325-151)×(Eβ((-1+2i)tβ)+Eβ((-1-2i)tβ))-(2.553571428-0.1785714286r)2i(425-250)×(Eβ((-1+2i)tβ)-Eβ((-1-2i)tβ))+(-3.107142858+0.9821428567r)2(425-250)×(Eβ((-1+2i)tβ)+Eβ((-1-2i)tβ))+(-3.107142858+0.9821428567r)2i(-325-151)×(Eβ((-1+2i)tβ)-Eβ((-1-2i)tβ)),X¯(t;r)=(1.696428572-0.0714285714r)Eβ(tβ)+(2.553571428-0.1785714286r)2(-325-151)×(Eβ((-1+2i)tβ)+Eβ((-1-2i)tβ))-(2.196428572+0.1785714286r)2i(425-250)×(Eβ((-1+2i)tβ)-Eβ((-1-2i)tβ))+(-1.142857142-0.9821428527r)2(425-250)×(Eβ((-1+2i)tβ)+Eβ((-1-2i)tβ))+(-1.142857142-0.9821428527r)2i(-325-151)×(Eβ((-1+2i)tβ)-Eβ((-1-2i)tβ)).Example 3.
Consider the system(55)(dβx~dtβdβy~dtβdβz~dtβ)=B(x~y~z~),B=(11121-10-11),
where 0<β<1.
The eigenvalues of the matrixA are λ1=1,λ2=λ3=2, and their corresponding eigenvectors are ξ1=[-3,4,2]t,ξ2=ξ3=[0,1,-1]t, respectively. Therefore, the general solution of the system (55) is
(56)X~(t)=c~1(-342)Eβ(-tβ)+c~2(01-1)Eβ(2tβ)+c~3[(01-1)tβEβ(1)(2tβ)+(101)Eβ(2tβ)],
where c~1,c~2, and c~3 are fuzzy coefficients. In particular, if we take β=1, then the general solution (56) can be written as
(57)X~(t)=c~1(-342)exp(-t)+c~2(01-1)exp(2t)+c~3[(01-1)texp(2t)+(101)exp(2t)].


## 6. Conclusion

In this paper, we investigated an analytical method (eigenvalue-eigenvector) for solving a system of fuzzy fractional differential equation under fuzzy Caputo’s derivative. To this end, we exploited generalized H-differentiability and derived the solutions based on this concept. To illustrate the effectiveness of the proposed method, several examples were solved. From Section5, one can conclude that the solution of the system of fuzzy fractional differential is a fuzzy number.



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*Source: 579761-2013-07-09.xml*