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

1 Department of Mathematics, Kerman Branch, Islamic Azad University, P.O. Box 9189945, Kerman, Iran 2 Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia 3 Department of Mathematics, Mobarakeh Branch, Islamic Azad University, P.O. Box 9176754, Mobarakeh, Iran 4Department of Mathematics, Science Faculty, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia 5 Young Researchers and Elite Club, Mobarakeh Branch, Islamic Azad University, P.O. Box 9176754, Mobarakeh, Iran


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][2][3][4][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][11][12][13][14][15][16][17][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][27][28][29][30][31][32][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 dif ferential equations (FFDEs).It is well known that the exact

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  : R → [0, 1] with the following properties: is the support of the , and its closure cl(supp ) is compact.
According to Zadeh's extension principle, operation of addition on E is defined by and scalar multiplication of a fuzzy number is given by where 0 ∈ E.
The Hausdorff distance between fuzzy numbers is given by  : where  = ((), ()) and V = (V(), V()) ⊂ R is utilized in [43].Then, it is easy to see that  is a metric in E and has the following properties (see [33]): (1) ( + , In this paper, the sign "⊖" always stands for H-difference, and also note that  ⊖  ̸ =  + (−1).

Fuzzy Fractional Order Linear Systems
In this section, we drive the general solution for fuzzy fractional order linear system as follows: where X ∈ E  ,  > 0, the coefficient matrix  = (  ), 1 ≤ ,  ≤ , is a crisp  ×  matrix and x ∈ E, 1 ≤  ≤ , and   /  is the fuzzy Caputo's fractional derivative, where 0 <  < 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() is replaced by the Mittag-Leffler function   (  ).Thus, we seek solutions of the form where the constant  and the vector  are to be determined.Substituting form (22) for X in the system (21) gives Upon canceling the nonzero factor   (  ), we obtain  =  or where  is the  ×  identity matrix.Therefore, the vector X given by ( 22) is a solution of the system (21) provided that  and the vector  are associated eigenvectors of the matrix .In the following Section, three cases for the eigenvalue of matrix  are discussed.
Therefore, from the above-mentioned lemma, the solution of each pair of conjugate complex eigenvalues  =  + ] is as follows: where  is the corresponding eigenvector of eigenvalue .Hence, the solution of (33) is as follows: where w () = c1 Re(    (    )) + c2 Im(    (    )) from each pair of conjugate complex eigenvalues and V  () = c     (    ) from real eigenvalues are obtained.Then by setting initial values  =  in (34) and by solving a fuzzy system similar to (28), fuzzy coefficients are obtained.By setting fuzzy coefficient in (34), X() is obtained; finally the solution of (20) will be obtained from X() = [ 1 ,  2 , . . .,   ]  .
Theorem 10.The solution of fuzzy system (20) with complex eigenvalues is a fuzzy number (34).

Multiple Eigenvalues.
In this case, suppose that some eigenvalues of matrix  are multiple.Suppose that  0 is an eigenvalue of matrix  with multiplicity , and the corresponding eigenvectors of eigenvalue  0 are  1 ,  2 , . . .,   , if all   are linearly independent, then If   and  are linearly independent vectors, that is,  < , then the following lemma is brought.
Lemma 11.Let  0 be an eigenvalue of matrix  with multiple  > 1, and let the numbers of   which are linearly independent be less than  0 , therefore at least one non-zero vector exists such that If  is satisfied in (40), the solution is as follows: based on the properties of the Mittag-Leffler type functions, where In general, if matrix  has a repeated eigenvalue  0 of multiplicity  with  linearly independent eigenvectors, where  < , then the following (1)   ( 0   ) +  2   ( 0   ) , . . .
are  −  linearly independent solutions of the system (20).
Hence, with the above-mentioned lemma, the solution of ( 20) is as follows: where V   () =  ()  (    )  ( −   )  +     (    ) for   which are satisfied in Lemma 11 and V  () =     (    ) for real eigenvalues are obtained.Then by setting initial values  = , in (44) and by solving a fuzzy system similar to (28), fuzzy coefficient is obtained.By setting fuzzy coefficient in (44), X() is obtained, and finally the solution of (20) will be obtained from X() = [ x1 , x2 , . . ., x ]  .Theorem 12.The solution of fuzzy system (20) with multiple eigenvalues is a fuzzy number (44).

Examples
Example 1.Consider that the system is with initial value where 0 <  < 1.

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.