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Published: 20.05.2021  Balooch Shahriyar, F.

## Eigenvalues and eigenvectors

Balooch Shahriyar, F. Ismail, S. Aghabeigi, A. Ahmadian, S. Box , Kerman, Iran. Box , Mobarakeh, Iran. 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. 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. 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. 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. 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. This paper is organized as follows. In Section 2 , we review the well-known definitions of fuzzy numbers, and some basic concepts are given.

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. The basic definition of fuzzy numbers is given in [ 47 , 48 ]. We denote the set of all real numbers by , and the set of all fuzzy number on is indicated by. A fuzzy number is a mapping with the following properties: a is upper semicontinuous, b is fuzzy convex; that is, for all , c is normal; that is, for which , d supp is the support of the , and its closure cl supp is compact.

An equivalent parametric definition is also given in [ 49 — 51 ] as follows. Definition 1. A fuzzy number in parametric form is a pair of functions , , which satisfy the following requirements: 1 is a bounded nondecreasing left continuous function in and right continuous at 0, 2 is a bounded nonincreasing left continuous function in and right continuous at 0, 3 ,?? Moreover, we can also present the -cut representation of fuzzy number as for all. The Hausdorff distance between fuzzy numbers is given by , where and is utilized in [ 43 ].

Then, it is easy to see that is a metric in and has the following properties see [ 33 ] : 1 , for all , 2 , for all , 3 , for all , 4 is a complete metric space. Definition 2. If there exists such that , then is called the H-difference of and , and it is denoted by. We denote as a space of all fuzzy-valued functions which are continuous on. Also, we denote the space of all Lebesgue integrable fuzzy-value functions on the bounded interval by.

We denote the space of fuzzy-value functions which have continuous H-derivative up to order on such that by. Definition 3. Let , the fuzzy Riemann-Liouville integral of fuzzy-valued function is defined as follows:. Since , for all , then we can indicate the fuzzy Riemann-Liouville integral of fuzzy-valued function based on the lower and upper functions as follows. Theorem 4. Let , the fuzzy Riemann-Liouville integral of fuzzy-valued function is defined as follows: where. Let be a given function such that for all and.

We define the fuzzy fractional Riemann-Liouville derivative of order of , and in the parametric form, provided that the equation defines a fuzzy number. In fact, Obviously, for. Let be a fuzzy-valued function. Obviously, for. Definition 6 see [ 52 ]. The linear system is where the coefficient matrix , is a crisp matrix, and , is called a fuzzy system of linear equations FSLEs.

Definition 7. A fuzzy number vector given by , is called a solution of the FSLE if. Considering the th equation of the system 13 we have From 16 , we have two crisp linear systems for all that can be extended to a crisp linear system as follows: where are determined as follows: and any which is not determined is zero.

In this paper, the following system will be solved: Thus, where , the matrix ,?? To obtain the solution of , the eigenvalue-eigenvector method is used. 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 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 in the system 21 gives Upon canceling the nonzero factor , we obtain or where is the identity matrix.

Therefore, the vector 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. In this case, suppose that , for , are the real eigenvalues of matrix.

Therefore, the solution of 20 is as follows: where are fuzzy numbers, for and and are the real eigenvalues and eigenvector of matrix , respectively. By setting initial values , in 25 , is obtained, where and. Thus, ,. From the following, fuzzy systems are obtained: The parametric form of 27 is as follows: Now similar to 17 , there is a crisp system.

Therefore, are obtained from 28 and are set in Finally the solution of 20 will be obtained from. Theorem 8. The solution of fuzzy system 20 with real eigenvalues is a fuzzy number It is shown that for and is the solution of. Let with which are the fuzzy numbers and , then With differentiation of 29 , we are obtained: Since is an eigenvalue and is its corresponding eigenvalue of matrix , then. Therefore, Such that This means that. In this case, suppose that some eigenvalues of , for , are complex numbers.

Since the entries of matrix are real, therefore characteristic polynomial has real coefficients; therefore, complex roots are in conjugate pairs. Lemma 9. Let the entries of matrix be real and an eigenvalue of matrix , where , and are the corresponding eigenvectors of , then , are solutions.

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 from each pair of conjugate complex eigenvalues and 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 , is obtained; finally the solution of 20 will be obtained from.

Theorem The solution of fuzzy system 20 with complex eigenvalues is a fuzzy number It is shown that is the solution of. Let , and with which are the fuzzy numbers and , and , With differentiation of the above equations we obtain the following Since is an eigenvalue and is its corresponding eigenvalue of matrix , then. Therefore, Then, This means that. In this case, suppose that some eigenvalues of matrix are multiple. Suppose that is an eigenvalue of matrix with multiplicity , and the corresponding eigenvectors of eigenvalue are , if all are linearly independent, then If and are linearly independent vectors, that is, , then the following lemma is brought.

Lemma Let be an eigenvalue of matrix with multiple , and let the numbers of which are linearly independent be less than , 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 of multiplicity with linearly independent eigenvectors, where , then the following are linearly independent solutions of the system Hence, with the above-mentioned lemma, the solution of 20 is as follows: where for which are satisfied in Lemma 11 and for real eigenvalues are obtained. ## 7. Eigenvalues and Eigenvectors

Geometrically , an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. If T is a linear transformation from a vector space V over a field F into itself and v is a nonzero vector in V , then v is an eigenvector of T if T v is a scalar multiple of v. This can be written as. There is a direct correspondence between n -by- n square matrices and linear transformations from an n -dimensional vector space into itself, given any basis of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices , or the language of linear transformations. As we will see in the examples below, for a given matrix A there are only a few Example: Find the eigenvalues and associated eigenvectors of the matrix. A.

The properties of the eigenvalues and their corresponding eigenvectors are also discussed and used in solving questions. Free Mathematics Tutorials. About the author Download E-mail. In this example the eigenvalues are: a , e and g. Left multiply both sides of the above equation by matrix A.

On the previous page, Eigenvalues and eigenvectors - physical meaning and geometric interpretation applet we saw the example of an elastic membrane being stretched, and how this was represented by a matrix multiplication, and in special cases equivalently by a scalar multiplication. That example demonstrates a very important concept in engineering and science - eigenvalues and eigenvectors - which is used widely in many applications, including calculus, search engines, population studies, aeronautics and so on. Let A be any square matrix.

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## OtГіn R.

Methods for the calculation of eigenvalues and eigenvectors Part II.

Chapter 6. Eigenvalues and Eigenvectors. Summary To solve the eigenvalue problem for an n by n matrix, follow these steps: 1. Compute the determinant of A.