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

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.

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 called system 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 Section 2, 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 by ℝ, 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)supp 𝑢={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 number 𝑢 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 the 𝑟-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 on 𝔼 is defined by (𝑢⊕𝑣)(𝑥)=sup_𝑦∈ℝmin𝑢𝑦,𝑣𝑥−𝑦,𝑥∈ℝ, (1) and scalar multiplication of a fuzzy number is given by (𝑥)=𝑢𝑥𝑘⊙𝑢)(𝑘,𝑘>0,0,𝑘=0, (2) where 0∈𝔼.

The Hausdorff distance between fuzzy numbers is given by 𝑑∶𝔼×𝔼→ℝ₊⋃{0}, 𝑑(𝑢,𝑣)=sup_𝑟∈[0,1]max𝑢(𝑟)−𝑣(𝑟),|𝑢(𝑟)−𝑣(𝛼)|, (3) 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. Let 𝑥,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 denote 𝐶^𝔽[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 𝐿^𝔽[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)∈𝐴𝐶^𝔽([𝑎,𝑏]) by 𝐴𝐶^(𝑛)𝔽([𝑎,𝑏]).

Now, we define the fuzzy Riemann-Liouville integral of fuzzy-valued function as follows.

Definition 3. Let 𝑓(𝑥)∈𝐶^𝔽[𝑎,𝑏]∩𝐿^𝔽[𝑎,𝑏], the fuzzy Riemann-Liouville integral of fuzzy-valued function 𝑓 is defined as follows: 𝐼^𝛽_𝑎+𝑓(𝑥)=1Γ𝛽^𝑥_𝑎𝑓(𝑡)𝑑𝑡(𝑥−𝑡)^1−𝛽,𝑥>𝑎,0<𝛽<1. (4)

Since 𝑓(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. Let 𝑓(𝑥)∈𝐶^𝔽[𝑎,𝑏]∩𝐿^𝔽[𝑎,𝑏], the fuzzy Riemann-Liouville integral of fuzzy-valued function 𝑓 is defined as follows: 𝐼^𝛽_𝑎+𝑓(𝑥;𝑟)=𝐼^𝛽_𝑎+𝑓(𝑥;𝑟),𝐼^𝛽_𝑎+𝑓(𝑥;𝑟),0≤𝑟≤1, (5) where 𝐼^𝛽_𝑎+𝑓()=1𝑥;𝑟Γ𝛽^𝑥_𝑎𝑓(𝑡;𝑟)𝑑𝑡(𝑥−𝑡)^1−𝛽𝐼,0≤𝑟≤1,_𝛽𝑎+𝑓()=1𝑥;𝑟Γ𝛽^𝑥_𝑎𝑓(𝑡;𝑟)𝑑𝑡(𝑥−𝑡)^1−𝛽,0≤𝑟≤1. (6)

Let 𝑓∈𝐶^𝔽((0,𝑎])∩𝐿^𝔽(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, ^RL𝐷^𝛽𝑓(𝑡)=1Γ𝑑1−𝛽𝑑𝑡^𝑡₀(𝑡−𝑠)^−𝛽𝑓(𝑠)𝑑𝑠, (7) and in the parametric form, ^RL𝐷^𝛽𝑓()1𝑡,𝑟=∶Γ𝑑1−𝛽𝑑𝑡^𝑡₀(𝑡−𝑠)^−𝛽𝑓()𝑑𝑠;𝑟𝑑𝑠,𝑑𝑡^𝑡₀(𝑡−𝑠)^−𝛽𝑓(𝑠;𝑟)𝑑𝑠, (8) provided that the equation defines a fuzzy number ^RL𝐷^𝛽𝑓(𝑡)∈𝐸. In fact, ^RL𝐷^𝛽𝑓(𝑡,𝑟)∶=^RL𝐷^𝛽𝑓(𝑡;𝑟),^RL𝐷^𝛽𝑓(𝑡;𝑟). (9) Obviously, ^RL𝐷^𝛽𝑓(𝑡)=(𝑑/𝑑𝑡)𝐼^1−𝛽𝑓(𝑡) for 𝑡∈(0,𝑎].

Definition 5 (the relation between Riemann-Liouville and Caputo’s operators). Let 𝑓(𝑥)∈𝐶^(𝑛)𝔽[𝑎,𝑏]∩𝐿^𝔽[𝑎,𝑏] be a fuzzy-valued function. One defines the fuzzy fractional Caputo’s derivative of order 0<𝛽<1 of 𝑓, ^𝐶𝐷^𝛽_𝑎+𝑓(𝑥)=^RL𝐷^𝛽_𝑎+𝑓(𝑡)⊖^𝑛−1_𝑘=0𝑥^𝑘𝑓𝑘!^(𝑘)(𝑎)(𝑥), (10) provided that the equation defines a fuzzy number ^𝐶𝐷^𝛽_𝑎+𝑓(𝑥)∈𝐸: ^𝐶𝐷^𝛽_𝑎+𝑓(𝑥;𝑟)=^𝐶𝐷^𝛽_𝑎+𝑓(𝑥;𝑟),^𝐶𝐷^𝛽_𝑎+𝑓(𝑥;𝑟), (11) where ^𝐶𝐷^𝛽_𝑎+𝑓()=𝑥;𝑟_RL𝐷^𝛽_𝑎+𝑓(𝑡;𝑟)−^𝑛−1_𝑘=0𝑥^𝑘𝑓𝑘!^(𝑘)()=1𝑎;𝑟)(𝑥;𝑟Γ𝑑𝑛−𝛽𝑑𝑥^𝑛𝑛𝑥_𝑎𝑓(𝑡;𝑟)−∑^𝑛−1_𝑘=0𝑡^𝑘/𝑘!𝑓^(𝑘)(𝑎;𝑟)(𝑥−𝑡)^{𝛽−𝑛+1}𝑑𝑡,_𝐶𝐷^𝛽_𝑎+𝑓()=𝑥;𝑟_RL𝐷^𝛽_𝑎+𝑓(𝑡;𝑟)−^𝑛−1_𝑘=0𝑥^𝑘𝑘!𝑓^(𝑘)()=1𝑎;𝑟)(𝑥;𝑟Γ𝑑𝑛−𝛽𝑑𝑥^𝑛𝑛𝑥_𝑎𝑓(𝑡,𝑟)−∑^𝑛−1_𝑘=0𝑡^𝑘/𝑘!𝑓^(𝑘)(𝑎;𝑟)(𝑥−𝑡)^{𝛽−𝑛+1}𝑑𝑡, (12) for all 0≤𝑟≤1 and 0<𝛽<1. Obviously, ^𝐶𝐷^𝛽_𝑎+𝑓(𝑡)=𝐼^𝑛−𝛽𝐷^𝑛𝑓(𝑡) for 𝑛−1<𝛽<𝑛,𝑛∈ℕ.

Definition 6 (see [52]). The 𝑛×𝑛 linear system is 𝑎₁₁𝑥₁+𝑎₁₂𝑥₂+⋯+𝑎_1𝑛𝑥_𝑛=𝑦₁,𝑎₁₁𝑥₁+𝑎₁₂𝑥₂+⋯+𝑎_1𝑛𝑥_𝑛=𝑦₁,⋮𝑎_𝑛1𝑥₁+𝑎_𝑛2𝑥₂+⋯+𝑎_𝑛𝑛𝑥_𝑛=𝑦_𝑛, (13) where the coefficient matrix 𝐴=(𝑎_𝑖𝑗),1≤𝑖,𝑗≤𝑛, is a crisp 𝑛×𝑛 matrix, and 𝑦_𝑖∈𝔼,1≤𝑖≤𝑛, is called a fuzzy system of linear equations (FSLEs).

Definition 7. A fuzzy number vector (𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑡 given by 𝑥_𝑖=(𝑥_𝑖(𝑟),𝑥_𝑖(𝑟)),1≤𝑖≤𝑛,0≤𝑟≤1, is called a solution of the FSLE if ^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗=^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗=𝑦_𝑖,_𝑛𝑗=1𝑎_𝑖𝑗𝑥_𝑗=^𝑛_𝑗=1𝑎_𝑖𝑗𝑥_𝑗=𝑦_𝑖. (14)

Considering the 𝑖th equation of the system (13) 𝑎_𝑖1𝑥₁,𝑥₁+⋯+𝑎_𝑖𝑖𝑥_𝑖,𝑥_𝑖+⋯+𝑎_𝑖𝑛𝑥_𝑛,𝑥_𝑛=𝑦_𝑖,𝑦_𝑖, (15) we have 𝑎_𝑖1𝑥₁+⋯+𝑎_𝑖𝑖𝑥_𝑖+⋯+𝑎_𝑖𝑛𝑥_𝑛=𝑦_𝑖(𝑟),𝑎_𝑖1𝑥₁+⋯+𝑎_𝑖𝑖𝑥_𝑖+⋯+𝑎_𝑖𝑛𝑥_𝑛=𝑦_𝑖(𝑟),1≤𝑖≤𝑛,0≤𝑟≤1. (16) 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: 𝑠𝐒𝐗=𝐘⟹¹≥0𝑠₂𝑠≤0₂≤0𝑠₁𝑋≥0𝑋=𝑌𝑌, (17) where 𝑠_𝑖𝑗 are determined as follows: if𝑎_𝑖𝑗≥0⟹𝑠_𝑖𝑗=𝑎_𝑖𝑗,𝑠_{𝑖+𝑛,𝑗+𝑛}=𝑎_𝑖𝑗,if𝑎_𝑖𝑗<0⟹𝑠_{𝑖,𝑗+𝑛}=𝑎_𝑖𝑗,𝑠_{𝑖+𝑛,𝑗}=𝑎_𝑖𝑗, (18) and any 𝑠_𝑖𝑗 which is not determined is zero.

In this paper, the following system will be solved: 𝑑^𝛽1𝑥₁𝑑𝑡^𝛽1=𝑓₁𝑥₁,𝑥₂,…,𝑥_𝑛=𝑎₁₁𝑥₁+𝑎₁₂𝑥₂+⋯+𝑎_1𝑛𝑥_𝑛,𝑑^𝛽2𝑥₂𝑑𝑡^𝛽2=𝑓₂𝑥₁,𝑥₂,…,𝑥_𝑛=𝑎₂₁𝑥₁+𝑎₂₂𝑥₂+⋯+𝑎_2𝑛𝑥_𝑛,⋮𝑑^𝛽𝑛𝑥_𝑛𝑑𝑡^𝛽𝑛=𝑓_𝑛𝑥₁,𝑥₂,…,𝑥_𝑛=𝑎_𝑛1𝑥₁+𝑎_𝑛2𝑥₂+⋯+𝑎_𝑛𝑛𝑥_𝑛. (19) Thus, 𝑑^𝛽𝑑𝑡^𝛽𝐗(𝑡)=𝐴𝐗(𝑡),𝐗(0)=𝐗₀∈𝔼^𝑛, (20) where 𝐗∈𝔼^𝑛, the matrix 𝐴=(𝑎_𝑖𝑗),1≤𝑖,𝑗≤𝑛,??𝛽=[𝛽₁,𝛽₂,…,𝛽_𝑛] indicates the fractional orders, 𝑑^𝛽/𝑑𝑡^𝛽=[𝑑^𝛽1/𝑑𝑡^𝛽1,𝑑^𝛽2/𝑑𝑡^𝛽2,…,𝑑^𝛽𝑛/𝑑𝑡^𝛽𝑛], and 𝑑^𝛽𝑖/𝑑𝑡^𝛽𝑖 is the fuzzy Caputo’s fractional derivative of order 𝛽_𝑖, where 0<𝛽_𝑖<1, for 𝑖=1,2,…,𝑛. To obtain the solution of (𝑑^𝛽/𝑑𝑡^𝛽)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: 𝑑^𝛽𝑑𝑡^𝛽𝐗(𝑡)=𝐴𝐗(𝑡),0<𝑡≤𝑎, (21) where 𝐗∈𝔼^𝑛,𝑎>0, the coefficient matrix 𝐴=(𝑎_𝑖𝑗),1≤𝑖,𝑗≤𝑛, is a crisp 𝑛×𝑛 matrix and 𝑥_𝑖∈𝔼,1≤𝑖≤𝑛, and 𝑑^𝛽/𝑑𝑡^𝛽 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 𝐸_𝛽(𝑡^𝛽). Thus, we seek solutions of the form 𝐗(𝑡)=𝜉𝐸_𝛽𝜆𝑡^𝛽, (22) where the constant 𝜆 and the vector 𝜉 are to be determined. Substituting form (22) for 𝐗in the system (21) gives 𝜉𝜆𝐸_𝛽𝜆𝑡^𝛽=𝐴𝜉𝐸_𝛽𝜆𝑡^𝛽. (23) Upon canceling the nonzero factor 𝐸_𝛽(𝜆𝑡^𝛽), we obtain 𝐴𝜉=𝜆𝜉 or (𝐴−𝜆𝐼)𝜉=0, (24) 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 that 𝜆_𝑖, for 𝑖=1,…,𝑛, are the real eigenvalues of matrix 𝐴. Therefore, the solution of (20) is as follows: 𝐗(𝑡)=^𝑛_𝑖=1𝑐_𝑖̃𝑣_𝑖(𝑡), (25) where 𝑐_𝑖̃ are fuzzy numbers, 𝑣_𝑖(𝑡)=𝐸_𝛽(𝜆_𝑖𝑡^𝛽)𝜉_𝑖 for 𝑖=1,…,𝑛 and 𝜆_𝑖 and 𝜉_𝑖 are the real eigenvalues and eigenvector of matrix 𝐴, respectively.

By setting initial values 𝑡=𝑎, in (25), 𝐗(𝑎)=^𝑛_𝑖=1𝑐_𝑖̃𝐸_𝛽𝜆_𝑖𝑡^𝛽𝜉_𝑖=𝛼 (26) is obtained, where 𝜉_𝑖=[𝜉_𝑖1,𝜉_𝑖2,…,𝜉_𝑖𝑛]^𝑡 and 𝛼=[𝛼₁,𝛼₂,…,𝛼_𝑛]^𝑡. Thus, 𝑐_𝑖̃, 𝑖=1,…,𝑛. From the following, fuzzy systems are obtained: 𝑐₁̃𝐸_𝛽𝜆₁𝑎^𝛽𝜉₁₁+𝑐₂̃𝐸_𝛽𝜆₂𝑎^𝛽𝜉₂₁+⋯+𝑐_𝑛̃𝐸_𝛽𝜆_𝑛𝑎^𝛽𝜉_𝑛1=𝛼₁,𝑐₁̃𝐸_𝛽𝜆₁𝑎^𝛽𝜉₁₂+𝑐₂̃𝐸_𝛽𝜆₂𝑎^𝛽𝜉₂₂+⋯+𝑐_𝑛̃𝐸_𝛽𝜆_𝑛𝑎^𝛽𝜉_𝑛2=𝛼₂,⋮𝑐₁̃𝐸_𝛽𝜆₁𝑎^𝛽𝜉_1𝑛+𝑐₂̃𝐸_𝛽𝜆₂𝑎^𝛽𝜉_2𝑛+⋯+𝑐_𝑛̃𝐸_𝛽𝜆_𝑛𝑎^𝛽𝜉_𝑛𝑛=𝛼₂. (27) The parametric form of (27) is as follows: 𝑐₁(𝑟),𝑐₁(𝑟)𝐸_𝛽𝜆₁𝑎^𝛽𝜉₁₁+𝑐₂(𝑟),𝑐₂(𝑟)𝐸_𝛽𝜆₂𝑎^𝛽𝜉₂₁+⋯+𝑐_𝑛(𝑟),𝑐_𝑛(𝑟)𝐸_𝛽𝜆_𝑛𝑎^𝛽𝜉_𝑛1=𝛼₁(𝑟),𝛼₁(𝑟),𝑐₁(𝑟),𝑐₁(𝑟)𝐸_𝛽𝜆₁𝑎^𝛽𝜉₁₂+𝑐₂(𝑟),𝑐₂(𝑟)𝐸_𝛽𝜆₂𝑎^𝛽𝜉₂₂+⋯+𝑐_𝑛(𝑟),𝑐_𝑛(𝑟)𝐸_𝛽𝜆_𝑛𝑎^𝛽𝜉_𝑛2=𝛼₂(𝑟),𝛼₂(𝑟),⋮𝑐₁(𝑟),𝑐₁(𝑟)𝐸_𝛽𝜆₁𝑎^𝛽𝜉_1𝑛+𝑐₂(𝑟),𝑐₂(𝑟)𝐸_𝛽𝜆₂𝑎^𝛽𝜉_2𝑛+⋯+𝑐_𝑛(𝑟),𝑐_𝑛(𝑟)𝐸_𝛽𝜆_𝑛𝑎^𝛽𝜉_𝑛𝑛=𝛼_𝑛(𝑟),𝛼_𝑛(𝑟). (28) Now similar to (17), there is a 2𝑛×2𝑛 crisp system. Therefore, 𝑐_𝑖̃=(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)=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑡.

Theorem 8. The solution of fuzzy system (20) with real eigenvalues is a fuzzy number (25).

Proof. It is shown that 𝐗(𝑡)=∑^𝑛_𝑖=1𝑐_𝑖̃𝐸_𝛽(𝜆_𝑖𝑡^𝛽)𝜉_𝑖 for 𝑖=1,…,𝑛 and 0<𝛽<1 is the solution of (𝑑^𝛽/𝑑𝑡^𝛽)𝐗=𝐴𝐗. Let 𝐶=[𝑐₁̃,𝑐₂̃,…,𝑐_𝑛̃] with 𝑐_𝑖̃ which are the fuzzy numbers and [𝐶]_𝑟=Π^𝑛_𝑖=1[𝑐_𝑖̃]_𝑟, then 𝑋(𝑡;𝑟)=min^𝑛_𝑖=1𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽𝜉_𝑖∣𝑐_𝑖∈𝐶_𝑟=^𝑛_𝑖=1𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽𝜉_𝑖=^𝑛_𝑖=1𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽𝜉_𝑖,𝑋(𝑡;𝑟)=max^𝑛_𝑖=1𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽𝜉_𝑖∣𝑐_𝑖∈𝐶_𝑟=^𝑛_𝑖=1𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽𝜉_𝑖=^𝑛_𝑖=1𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽𝜉_𝑖. (29) With differentiation of (29), we are obtained: 𝑑^𝛽𝑑𝑡^𝛽𝑋(𝑡;𝑟)=^𝑛_𝑖=1𝜆_𝑖𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽𝜉_𝑖𝑑,0≤𝑟≤1,_𝛽𝑑𝑡^𝛽𝑋(𝑡;𝑟)=^𝑛_𝑖=1𝜆_𝑖𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽𝜉_𝑖,0≤𝑟≤1. (30) Since 𝜆_𝑖 is an eigenvalue and 𝜉_𝑖 is its corresponding eigenvalue of matrix 𝐴, then 𝐴𝜉_𝑖=𝜆𝜉_𝑖,𝑖=1,….,𝑛. Therefore, 𝑑^𝛽𝑑𝑡^𝛽𝑋(𝑡;𝑟)=^𝑛_𝑖=1𝜆_𝑖𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽𝜉_𝑖=^𝑛_𝑖=1𝜆_𝑖𝜉_𝑖𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽=𝐴^𝑛_𝑖=1𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽𝜉_𝑖=𝐴𝑋(),𝑑𝑡;𝑟_𝛽𝑑𝑡^𝛽𝑋(𝑡;𝑟)=^𝑛_𝑖=1𝜆_𝑖𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽𝜉_𝑖=^𝑛_𝑖=1𝜆_𝑖𝜉_𝑖𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽=𝐴^𝑛_𝑖=1𝑐_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽𝜉_𝑖=𝐴𝑋(𝑡;𝑟). (31) Such that 𝑑^𝛽𝑑𝑡^𝛽𝑋();𝑑𝑡;𝑟^𝛽𝑑𝑡^𝛽𝑋(𝑡;𝑟)=𝐴𝑋(𝑡;𝑟);𝑋(𝑡;𝑟),0<𝛽<1,0≤𝑟≤1. (32) This means that (𝑑^𝛽/𝑑𝑡^𝛽)𝐗(𝑡)=𝐴𝐗(𝑡). From (7), (10), and (𝑑^𝛽/𝑑𝑡^𝛽)𝐗(𝑡)=[(𝑑^𝛽1/𝑑𝑡^𝛽1)𝑥₁,??(𝑑^𝛽2/𝑑𝑡^𝛽2)𝑥₂,…,??(𝑑^𝛽𝑛/𝑑𝑡^𝛽𝑛)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 of 𝜆_𝑖, 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 matrix 𝐴 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[𝜉𝐸_𝛽(𝜇+𝑖𝜈)𝑡^𝛽)], 𝑢₂(𝑡)=Im[𝜉𝐸_𝛽(𝜇+𝑖𝜈)𝑡^𝛽)] are solutions.
Therefore, from the above-mentioned lemma, the solution of each pair of conjugate complex eigenvalues 𝜆=𝜇+𝑖𝜈 is as follows: 𝑤(𝑡)=𝑐₁̃Re𝜉𝐸_𝛽𝜇+𝑖𝜈𝑡^𝛽+𝑐₂̃Im𝜉𝐸_𝛽𝜇+𝑖𝜈𝑡^𝛽, (33) where 𝜉 is the corresponding eigenvector of eigenvalue 𝜆. Hence, the solution of (33) is as follows: 𝐗(𝑡)=^𝑖=𝑘/2_𝑖=1𝑤_𝑖(𝑡)+𝑖^𝑖=𝑛_{𝑖=𝑘/2+1}𝑣_𝑖̃(𝑡), (34) where 𝑤_𝑖(𝑡)=𝑐̃_𝑖1Re(𝜉_𝑖𝐸_𝛽(𝜆_𝑖𝑡^𝛽))+𝑐̃_𝑖2Im(𝜉_𝑖𝐸_𝛽(𝜆_𝑖𝑡^𝛽)) from each pair of conjugate complex eigenvalues and 𝑣_𝑖(𝑡)=𝑐_𝑖̃𝜉_𝑖𝐸_𝛽(𝜆_𝑖𝑡^𝛽) 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,…,𝑥_𝑛]^𝑡.

Theorem 10. The solution of fuzzy system (20) with complex eigenvalues is a fuzzy number (34).

Proof. It is shown that 𝐗(𝑡)=∑^𝑖=𝑘/2_𝑖=1𝑤_𝑖(𝑡)+𝑖∑^𝑖=𝑛_𝑖=𝑘/2𝑣_𝑖̃(𝑡) is the solution of (𝑑^𝛽/𝑑𝑡^𝛽)𝐗=𝐴𝐗.
Let 𝐶₁=(𝑐̃₁₁,𝑐̃₂₁,…,𝑐̃_𝑘/2,1),𝐶₂=(𝑐̃₁₂,𝑐̃₂₂,…,𝑐̃_𝑘/2,2), and 𝐶=(𝑐̃_𝑘/2+1,𝑐̃_𝑘/2+2,…,𝑐_𝑛̃) with 𝑐̃_𝑖1,𝑐̃_𝑖2,𝑐_𝑖̃ which are the fuzzy numbers and [𝐶₁]_𝑟=Π^𝑘/2_𝑖=1[𝑐̃_𝑖1]_𝑟,[𝐶₂]_𝑟=Π^𝑘/2_𝑖=1[𝑐̃_𝑖2]_𝑟, and [𝐶]_𝑟=Π^𝑘/2_{𝑖=𝑘/2+1}[𝑐_𝑖̃]_𝑟,𝑋(𝑡;𝑟)=min∑^𝑘/2_𝑖=1𝑐_𝑖1Re𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+𝑐_𝑖2Im𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+∑^𝑛_{𝑖=𝑘/2+1}𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽∣𝑐_𝑖1∈𝐶_1𝑟,𝑐_𝑖2∈𝐶_2𝑟,𝑐_𝑖∈𝐶_𝑟=∑^𝑘/2_𝑖=1𝑐_𝑖1Re𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+𝑐_𝑖2Im𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+∑^𝑛_{𝑖=𝑘/2+1}𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽=∑^𝑘/2_𝑖=1𝑐_𝑖1Re𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+𝑐_𝑖2Im𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+∑^𝑛_{𝑖=𝑘/2+1}𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽,𝑋(𝑡;𝑟)=max∑^𝑘/2_𝑖=1𝑐_𝑖1Re𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+𝑐_𝑖2Im𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+∑^𝑛_{𝑖=𝑘/2+1}𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽∣𝑐_𝑖1∈𝐶_1𝑟,𝑐_𝑖2∈𝐶_2𝑟,𝑐_𝑖∈𝐶_𝑟=∑^𝑘/2_𝑖=1𝑐_𝑖1Re𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+𝑐_𝑖2Im𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+∑^𝑛_{𝑖=𝑘/2+1}𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽=∑^𝑘/2_𝑖=1𝑐_𝑖1Re𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+𝑐_𝑖2Im𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+∑^𝑛_{𝑖=𝑘/2+1}𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽. (35)With differentiation of the above equations we obtain the following 𝑑^𝛽𝑑𝑡^𝛽𝑋()=𝑡;𝑟_𝑘/2𝑖=1𝑐_𝑖1Re𝜉_𝑖𝜆_i𝐸_𝛽𝜆_𝑖𝑡^𝛽+𝑐_𝑖2Im𝜉_𝑖𝜆_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+^𝑛_{𝑖=𝑘/2+1}𝑐_𝑖𝜆_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽𝑑,0≤𝑟≤1,_𝛽𝑑𝑡^𝛽𝑋()=𝑡;𝑟_𝑘/2𝑖=1𝑐_𝑖1Re𝜉_𝑖𝜆_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+𝑐_𝑖2Im𝜉_𝑖𝜆_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+^𝑛_{𝑖=𝑘/2+1}𝑐_𝑖𝜆_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽,0≤𝑟≤1. (36) Since 𝜆_𝑖 is an eigenvalue and 𝜉_𝑖 is its corresponding eigenvalue of matrix 𝐴, then 𝐴𝜉_𝑖=𝜆𝜉_𝑖,𝑖=1,….,𝑛. Therefore, 𝑑^𝛽𝑑𝑡^𝛽𝑋()=𝑡;𝑟_𝑘/2𝑖=1𝑐_𝑖1Re𝐴𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+𝑐_𝑖2Im𝐴𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+^𝑛_{𝑖=𝑘/2+1}𝑐_𝑖𝐴𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽=𝐴^𝑘/2_𝑖=1𝑐_𝑖1Re𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+𝑐_𝑖2Im𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+^𝑛_{𝑖=𝑘/2+1}𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽=𝐴𝑋(),𝑑𝑡;𝑟_𝛽𝑑𝑡^𝛽𝑋()=𝑡;𝑟_𝑘/2𝑖=1𝑐_𝑖1Re𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+𝑐_𝑖2Im𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+^𝑛_{𝑖=𝑘/2+1}𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽=𝐴^𝑘/2_𝑖=1𝑐_𝑖1Re𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+𝑐_𝑖2Im𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+^𝑛_{𝑖=𝑘/2+1}𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽=𝐴𝑋(𝑡;𝑟). (37) Then, 𝑑^𝛽𝑑𝑡^𝛽𝑋();𝑑𝑡;𝑟^𝛽𝑑𝑡^𝛽𝑋(𝑡;𝑟)=𝐴𝑋(𝑡;𝑟);𝑋(𝑡;𝑟),0<𝛽<1,0≤𝑟≤1. (38) This means that (𝑑^𝛽/𝑑𝑡^𝛽)𝐗(𝑡)=𝐴𝐗(𝑡). From (7), (10), and (𝑑^𝛽/𝑑𝑡^𝛽)𝐗(𝑡)=[(𝑑^𝛽1/𝑑𝑡^𝛽1)𝑥₁,??(𝑑^𝛽2/𝑑𝑡^𝛽2)𝑥₂,…,(𝑑^𝛽𝑛/𝑑𝑡^𝛽𝑛)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 matrix 𝐴 are multiple. Suppose that 𝜆₀ is an eigenvalue of matrix 𝐴 with multiplicity 𝑘, and the corresponding eigenvectors of eigenvalue 𝜆₀ are 𝜉₁,𝜉₂,…,𝜉_𝑘, if all 𝜉_𝑖 are linearly independent, then 𝐗(𝑡)=𝑐₁̃𝜉₁𝐸_𝛽𝜆₀𝑡^𝛽+𝑐₂̃𝜉₂𝐸_𝛽𝜆₀𝑡^𝛽+⋯+𝑐_𝑘̃𝜉_𝑘𝐸_𝛽𝜆₀𝑡^𝛽. (39) If 𝜉_𝑖 and 𝑚 are linearly independent vectors, that is, 𝑚<𝑘, then the following lemma is brought.

Lemma 11. Let 𝜆₀ be an eigenvalue of matrix 𝐴 with multiple 𝑘>1, and let the numbers of 𝜉_𝑖 which are linearly independent be less than 𝑚₀, therefore at least one non-zero vector exists such that (𝐴−𝜆𝐼)𝜉≠0,(𝐴−𝜆𝐼)²𝜉=0. (40) If 𝜉 is satisfied in (40), the solution is as follows: 𝑣^′(𝑡)=𝐸_𝛽⁽¹⁾𝜆₀𝑡^𝛽𝑡^𝛽𝐴−𝜆₀𝐼𝜉+𝜉𝐸_𝛽𝜆₀𝑡^𝛽, (41) based on the properties of the Mittag-Leffler type functions, where 𝐸_𝛽⁽¹⁾(𝑧)=𝑑𝐸𝑑𝑡^𝛽(𝑧). (42)

In general, if matrix 𝐴 has a repeated eigenvalue 𝜆₀ of multiplicity 𝑘 with 𝑚 linearly independent eigenvectors, where 𝑚<𝑘, then the following 𝑥₁(𝑡)=𝜉₁𝐸_𝛽𝜆₀𝑡^𝛽,𝑥₂(𝑡)=𝜉₁𝑡^𝛽𝐸_𝛽⁽¹⁾𝜆₀𝑡^𝛽+𝜉₂𝐸_𝛽𝜆₀𝑡^𝛽,⋮𝑥_𝑘−𝑚(𝑡)=𝜉₁𝑡^{𝛽(𝑘−𝑚−1)}𝐸_𝛽^{(𝑘−𝑚−1)}𝜆₀𝑡^𝛽+𝜉₂𝑡^{𝛽(𝑘−𝑚−2)}𝐸_𝛽^{(𝑘−𝑚−2)}𝜆₀𝑡^𝛽+⋯+𝜉_𝑘−𝑚𝐸_𝛽𝜆₀𝑡^𝛽, (43) are 𝑘−𝑚 linearly independent solutions of the system (20).

Hence, with the above-mentioned lemma, the solution of (20) is as follows: 𝐗(𝑡)=^𝑖=𝑘_𝑖=1𝑐_𝑖̃𝑣^′_𝑖(𝑡)+^𝑖=𝑛_𝑖=𝑘+1𝑐_𝑖̃𝑣_𝑖(𝑡), (44) where 𝑣^′_𝑖(𝑡)=𝐸_𝛽^(𝑘)(𝜆_𝑖𝑡^𝛽)𝑡^𝛽(𝐴−𝜆_𝑖𝐼)𝜉_𝑖+𝜉_𝑖𝐸_𝛽(𝜆_𝑖𝑡^𝛽) for 𝜆_𝑖 which are satisfied in Lemma 11 and 𝑣_𝑖(𝑡)=𝜉_𝑖𝐸_𝛽(𝜆_𝑖𝑡^𝛽) 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)=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑡.

Theorem 12. The solution of fuzzy system (20) with multiple eigenvalues is a fuzzy number (44).

Proof. It is shown that 𝐗(𝑡)=∑^𝑖=𝑘_𝑖=1𝑐_𝑖̃𝑣^′_𝑖(𝑡)+∑^𝑖=𝑛_𝑖=𝑘+1𝑐_𝑖̃𝑣_𝑖(𝑡) is the solution of (𝑑^𝛽/𝑑𝑡^𝛽)𝐗=𝐴𝐗. Let 𝐶=[𝑐₁̃,𝑐₂̃,…,𝑐_𝑛̃] with 𝑐_𝑖̃ which are the fuzzy numbers and [𝐶]_𝑟=Π^𝑛_𝑖=1[𝑐_𝑖̃]_𝑟, then 𝑋(𝑡;𝑟)=min^𝑘_𝑖=1𝑐_𝑖𝐸_𝛽^(𝑘)𝜆_𝑖𝑡^𝛽𝑡^𝛽𝐴−𝜆_𝑖𝐼𝜉_𝑖+𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+^𝑖=𝑛_𝑖=𝑘+1𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽∣𝑐_𝑖∈𝐶_𝑟=^𝑘_𝑖=1𝑐_𝑖𝐸_𝛽^(𝑘)𝜆_𝑖𝑡^𝛽𝑡^𝛽𝐴−𝜆_𝑖𝐼𝜉_𝑖+𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+^𝑖=𝑛_𝑖=𝑘+1𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽=^𝑘_𝑖=1𝑐_𝑖𝐸_𝛽^(𝑘)𝜆_𝑖𝑡^𝛽𝑡^𝛽𝐴−𝜆_𝑖𝐼𝜉_𝑖+𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+^𝑖=𝑛_𝑖=𝑘+1𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽,𝑋(𝑡;𝑟)=max^𝑘_𝑖=1𝑐_𝑖𝐸_𝛽^(𝑘)𝜆_𝑖𝑡^𝛽𝑡^𝛽𝐴−𝜆_𝑖𝐼𝜉_𝑖+𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+^𝑖=𝑛_𝑖=𝑘+1𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽∣𝑐_𝑖∈𝐶_𝑟=^𝑘_𝑖=1𝑐_𝑖𝐸_𝛽^(𝑘)𝜆_𝑖𝑡^𝛽𝑡^𝛽𝐴−𝜆_𝑖𝐼𝜉_𝑖+𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+^𝑖=𝑛_𝑖=𝑘+1𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽=^𝑘_𝑖=1𝑐_𝑖𝐸_𝛽^(𝑘)𝜆_𝑖𝑡^𝛽𝑡^𝛽𝐴−𝜆_𝑖𝐼𝜉_𝑖+𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+^𝑖=𝑛_𝑖=𝑘+1𝑐_𝑖𝜉_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽. (45)With differentiation of the above equations, we obtain𝑑^𝛽𝑑𝑡^𝛽𝑋(𝑡;𝑟)=^𝑘_𝑖=1𝑐_𝑖𝜆_𝑖𝐸_𝛽^(𝑘)𝜆_𝑖𝑡^𝛽𝑡^𝛽𝐴−𝜆_𝑖𝐼𝜉_𝑖+𝐸_𝛽^(𝑘)𝜆_𝑖𝑡^𝛽𝐴−𝜆_𝑖𝐼𝜉_𝑖+𝜉_𝑖𝜆_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽+^𝑖=𝑛_𝑖=𝑘+1𝑐_𝑖𝜉_𝑖𝜆_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽,𝑑^𝛽𝑑𝑡^𝛽𝑋(𝑡;𝑟)=^𝑘_𝑖=1𝑐_𝑖[𝜆_𝑖𝐸_𝛽^(𝑘)𝜆_𝑖𝑡^𝛽𝑡^𝛽𝐴−𝜆_𝑖𝐼𝜉_𝑖+𝐸_𝛽^(𝑘)(𝜆_𝑖𝑡^𝛽)(𝐴−𝜆_𝑖𝐼)𝜉_𝑖+𝜉_𝑖𝜆_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽]+^𝑖=𝑛_𝑖=𝑘+1𝑐_𝑖𝜉_𝑖𝜆_𝑖𝐸_𝛽𝜆_𝑖𝑡^𝛽. (46)Since 𝜆_𝑖 is an eigenvalue and 𝜉_𝑖 is its corresponding eigenvalue of matrix 𝐴, then 𝐴𝜉_𝑖=𝜆𝜉_𝑖. Therefore, 𝑑^𝛽𝑑𝑡^𝛽𝑋();𝑑𝑡;𝑟^𝛽𝑑𝑡^𝛽𝑋(𝑡;𝑟)=𝐴𝑋(𝑡;𝑟);𝑋(𝑡;𝑟),0<𝛽<1,0≤𝑟≤1. (47) This means that (𝑑^𝛽/𝑑𝑡^𝛽)𝐗(𝑡)=𝐴𝐗(𝑡). From (7), (10), (𝑑^𝛽/𝑑𝑡^𝛽)𝐗(𝑡)=[(𝑑^𝛽1/𝑑𝑡^𝛽1)𝑥₁,??(𝑑^𝛽2/𝑑𝑡^𝛽2)𝑥₂,…,??(𝑑^𝛽𝑛/𝑑𝑡^𝛽𝑛)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 𝑑^𝛽𝑥𝑑𝑡^𝛽𝑑^𝛽𝑦𝑑𝑡^𝛽𝑥𝑦,𝑥(0)𝑦(0)=,=𝐴,𝐴=2−14−3𝑟+13−𝑟 (48) where 0<𝛽<1.
The eigenvalues of the matrix 𝐴 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 𝐗(𝑡)=𝑐₁̃11𝐸_𝛽𝑡^𝛽+𝑐₂̃14𝐸_𝛽−2𝑡^𝛽, (49) where 𝑐₁̃ and 𝑐₂̃ are fuzzy coefficients. This is obtained by setting the initial values in parametric form, then the following system is obtained: .1100𝑐140000110014¹𝑐₂𝑐₁𝑐₂=𝑟+12+𝑟3−𝑟5−𝑟, (50) where 𝑐₁̃=(𝑟+2/3;7/3−𝑟) and 𝑐₂̃=(1/3;2/3) are obtained from the above system and are set in (25). Therefore, the solution of SFFDEs is as follows: 𝐗(𝑡)=2𝑟+3𝐸_𝛽𝑡^𝛽+13𝐸_𝛽−2𝑡^𝛽,73−𝑟𝐸_𝛽𝑡^𝛽2+43𝐸_𝛽−2𝑡^𝛽. (51)

Example 2. Consider that the system is with initial value 𝑑^𝛽𝑥𝑑𝑡^𝛽𝑑^𝛽𝑦𝑑𝑡^𝛽𝑑^𝛽𝑧𝑑𝑡^𝛽𝑥𝑦𝑧=𝐴,𝐴=0100015−3−1, (52) where 0<𝛽<1.
The eigenvalues of the matrix 𝐴 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: 410025−30251000−15−250110000−3425010025−15−25⋅𝑐100000110¹𝑐₂𝑐₃𝑐₁𝑐₂𝑐₃=,𝑐̃0.75+0.25𝑟1.5+0.5𝑟3.75+0.25𝑟1.25−0.25𝑟2.5−0.5𝑟4.25−0.25𝑟₁=(),𝑐̃1.553571428+0.071428𝑟,1.696428572−0.0714285714𝑟₂=(),𝑐̃2.196428572+0.1785714286𝑟,2.553571428−0.1785714286𝑟₃=(),−3.107142858+0.9821428567𝑟,−1.142857142−0.9821428527𝑟 (53) which are obtained from the above system and are set in (25). Therefore, the solution of SFFDEs is as follows: 𝑋(𝑡;𝑟)=(1.553571428+0.071428𝑟)𝐸_𝛽𝑡^𝛽+(2.196428572+0.1785714286𝑟)2−325−151×𝐸_𝛽(−1+2𝑖)𝑡^𝛽+𝐸_𝛽(−1−2𝑖)𝑡^𝛽−(2.553571428−0.1785714286𝑟)42𝑖25−250×𝐸_𝛽(−1+2𝑖)𝑡^𝛽−𝐸_𝛽(−1−2𝑖)𝑡^𝛽+(−3.107142858+0.9821428567𝑟)2425−250×𝐸_𝛽(−1+2𝑖)𝑡^𝛽+𝐸_𝛽(−1−2𝑖)𝑡^𝛽+(−3.107142858+0.9821428567𝑟)2𝑖−325−151×𝐸_𝛽(−1+2𝑖)𝑡^𝛽−𝐸_𝛽(−1−2𝑖)𝑡^𝛽,𝑋(𝑡;𝑟)=(1.696428572−0.0714285714𝑟)𝐸_𝛽𝑡^𝛽+(2.553571428−0.1785714286𝑟)2−325−151×𝐸_𝛽(−1+2𝑖)𝑡^𝛽+𝐸_𝛽(−1−2𝑖)𝑡^𝛽−(2.196428572+0.1785714286𝑟)42𝑖25−250×𝐸_𝛽(−1+2𝑖)𝑡^𝛽−𝐸_𝛽(−1−2𝑖)𝑡^𝛽+(−1.142857142−0.9821428527𝑟)2425−250×𝐸_𝛽(−1+2𝑖)𝑡^𝛽+𝐸_𝛽(−1−2𝑖)𝑡^𝛽+(−1.142857142−0.9821428527𝑟)2𝑖−325−151×𝐸_𝛽(−1+2𝑖)𝑡^𝛽−𝐸_𝛽(−1−2𝑖)𝑡^𝛽. (54)

Example 3. Consider the system 𝑑^𝛽𝑥𝑑𝑡^𝛽𝑑^𝛽𝑦𝑑𝑡^𝛽𝑑^𝛽𝑧𝑑𝑡^𝛽𝑥𝑦𝑧=𝐵,𝐵=11121−10−11, (55) where 0<𝛽<1.
The eigenvalues of the matrix 𝐴 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 𝐗(𝑡)=𝑐₁̃42𝐸−3_𝛽−𝑡^𝛽+𝑐₂̃01𝐸−1^𝛽2𝑡^𝛽+𝑐₃̃01𝑡−1^𝛽𝐸_𝛽⁽¹⁾2𝑡^𝛽+101𝐸_𝛽2𝑡^𝛽, (56) where 𝑐₁̃,𝑐₂̃, and 𝑐₃̃ are fuzzy coefficients. In particular, if we take 𝛽=1, then the general solution (56) can be written as 𝐗(𝑡)=𝑐₁̃42()+𝑐̃−3exp−𝑡₂01()+𝑐̃−1exp2𝑡₃01()+101().−1𝑡exp2𝑡exp2𝑡 (57)

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 Section 5, one can conclude that the solution of the system of fuzzy fractional differential is a fuzzy number.

Acknowledgments

The authors would like to express their thanks to Professor Tofigh Allahviranloo, the lead Guest Editor, for his consideration about this paper and the anonymous reviewers for their helpful suggestions and comments.