Abstract
In this article, we study a boundary value problem of a class of linear singular systems of fractional nabla difference equations whose coefficients are constant matrices. By taking into consideration the cases that the matrices are square with the leading coefficient matrix singular, square with an identically zero matrix pencil and nonsquare, we provide necessary and sufficient conditions for the existence and uniqueness of solutions. More analytically, we study the conditions under which the boundary value problem has a unique solution, infinite solutions and no solutions. Furthermore, we provide a formula for the case of the unique solution. Finally, numerical examples are given to justify our theory.
Keywords:
boundary conditions; singular systems; fractional calculus; nabla operator; difference equations; linear; discrete time system1 Introduction
Difference equations of fractional order have recently proven to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetism, and so forth [17]. There has been a significant development in the study of fractional differential/difference equations and inclusions in recent years; see the monographs of Baleanu et al.[1], Kaczorek [4], Klamka et al.[8], Malinowska et al.[5], Podlubny [7], and the survey by Agarwal et al.[9]. For some recent contributions on fractional differential/difference equations, see [1,4,5,827] and the references therein. In this article we provide an introductory study for a boundary value problem of a class of singular fractional nabla discrete time systems. If we define by , α integer, and n such that or , then the nabla fractional operator in the case of RiemannLiouville fractional difference of nth order for any is defined by, see [5,1012,23],
where the raising power function is defined by
The following problem will then be considered. The singular fractional discrete time systems of the form
with known boundary conditions
where (i.e., the algebra of matrices with elements in the field ℱ) with , , , and . For the sake of simplicity, we set and . The matrices F and G can be nonsquare (when ) or square () and F singular (). The main purpose will be to provide necessary and sufficient conditions for the existence and uniqueness of solutions of the above boundary value problem, i.e., to study the conditions under which the system has unique, infinite and no solutions and to provide a formula for the case of the unique solution (if it exists). Many authors use matrix pencil theory to study linear discrete time systems with constant matrices; see, for instance, [2843]. A matrix pencil is a family of matrices , parametrized by a complex number s, see [39,41,44,45]. When G is square and , where is the identity matrix, the zeros of the function are the eigenvalues of G. Consequently, the problem of finding the nontrivial solutions of the equation
is called the generalized eigenvalue problem. Although the generalized eigenvalue problem looks like a simple generalization of the usual eigenvalue problem, it exhibits some important differences. In the first place, it is possible for F, G to be nonsquare matrices. Moreover, even with F, G square it is possible (in the case F is singular) for to be identically zero, independent of s. Finally, even if we assume F, G square matrices with a nonzero pencil, it is possible (when F is singular) for the problem to have infinite eigenvalues. To see this, write the generalized eigenvalue problem in the reciprocal form
If F is singular with a null vector X, then , so that X is an eigenvector of the reciprocal problem corresponding to eigenvalue ; i.e., .
Definition 1.1 Given and an arbitrary , the matrix pencil is called:
The paper is organized as follows. In Section 2, we study the existence of solutions of the system (1) when its pencil is regular. In Section 3 we study the case of the system (1) with a singular pencil, and Section 3 contains numerical examples.
2 Regular case
In this section, we consider the case of the system (1) with a regular pencil. The class of is characterized by a uniquely defined element, known as complex Weierstrass canonical form, , see [39,41,44,45], specified by the complete set of invariants of . This is the set of elementary divisors (e.d.) obtained by factorizing the invariant polynomials into powers of homogeneous polynomials irreducible over the field ℱ. In the case where is regular, we have e.d. of the following type:
• e.d. of the type are called finite elementary divisors (f.e.d.), where is a finite eigenvalue of algebraic multiplicity ;
• e.d. of the type are called infinite elementary divisors (i.e.d.), where q is the algebraic multiplicity of the infinite eigenvalues.
Definition 2.1 Let be elements of . The direct sum of them denoted by is the .
From the regularity of , there exist nonsingular matrices such that
and
The complex Weierstrass form of the regular pencil is defined by
where the first normal Jordantype element is uniquely defined by the set of the finite eigenvalues of and has the form
The second uniquely defined block corresponds to the infinite eigenvalues of and has the form
The matrix is a nilpotent element of with index , where
For algorithms about the computations of the Jordan matrices, see [39,41,44,45].
Definition 2.2 If for the system (1) with boundary conditions (2) there exists at least one solution, the boundary value problem (1)(2) is said to be consistent.
For the regular matrix pencil of the system (1), there exist nonsingular matrices as applied in (3), (4). Let
where is a matrix with columns p linear independent (generalized) eigenvectors of the p finite eigenvalues of , and is a matrix with columns q linear independent (generalized) eigenvectors of the q infinite eigenvalues of .
Lemma 2.1Consider the system (1) with a regular pencil. Then the system (1) is divided into two subsystems:
and
Proof Consider the transformation
and by substituting (6) into (1), we obtain
or, equivalently,
Whereby multiplying by P, we arrive at
Moreover, let
where , , and by using (3) and (4), we obtain
From the above expressions, we arrive easily at the subsystems
and
The proof is completed. □
Definition 2.3 With we denote the discrete MittagLeffler function with two parameters defined by
Proposition 2.1The subsystem (7) has the solution
if and only if
whereis an induced matrix norm andis the discrete MittagLeffler function with two parameters as defined by Definition 2.3.
Proof From [1012,23,46] the solution of (7) can be calculated and given by the formula
or, equivalently, by
The existence and uniqueness of the above solution depends on the convergence of the matrix power series
or, equivalently, if and only if
or, equivalently,
By using the property
we get
or, equivalently,
The proof is completed. □
Proposition 2.2The subsystem (8) has the unique solution
Proof Let be the index of the nilpotent matrix , i.e., . Then if we obtain the following equations:
by taking the sum of the above equations and using the fact that , we arrive easily at the solution (12). The proof is completed. □
Theorem 2.1Consider the system (1) with a regular pencil and boundary conditions of type (2). Then the boundary value problem (1)(2) is consistent if and only if:
1. The pencilhaspdistinct eigenvalues and all lie within the open disk
2.
Furthermore, when the boundary value problem (1)(2) is consistent, it has a unique solution if and only if:
1.
2.
In this case the unique solution is then given by
whereCis the unique solution of the algebraic system
Proof By applying the transformation (6) into the system (1), we get the systems (7), (8) with solutions (10), (12) respectively. Note that from Proposition 2.1 the solution (10) exists if and only if
where is the Jordan matrix related to the p finite eigenvalues of the pencil , which is equivalent to the fact that the finite eigenvalues of the pencil must be distinct and all lie within the unit disk . Based on these results, the solution of (1) can be written as
or, equivalently,
or, equivalently, by using (10), (12)
The initial value of the subsystem (7) is not known and can be replaced by a constant vector
The above solution exists if and only if
or, equivalently,
or, equivalently,
For the above algebraic system, there exists at least one solution if and only if
The algebraic system (17) contains equations and p unknowns. Hence the solution is unique if and only if
and
where C is then the unique solution of (17). This can be proved as follows. If we assume that the algebraic system has two solutions and , then
and
or, equivalently,
But the matrix is left invertible since it is assumed to have p linear independent columns and and hence
The unique solution is then given from (16). The proof is completed. □
3 Singular case
In this section, we consider the case of the system (1) with a singular pencil. The class of in this case is characterized by a uniquely defined element, , known as the complex Kronecker canonical form, see [39,41,44,45], specified by the complete set of invariants of the singular pencil . This is the set of the elementary divisors (e.d.) and the minimal indices (m.i.). Unlike the case of the regular pencils, where the pencil is characterized only from the e.d., the characterization of a singular matrix pencil apart from the set of the determinantal divisors requires the definition of additional sets of invariants, the minimal indices. The distinguishing feature of a singular pencil is that either or and . Let , be the right and the left null space of a matrix respectively. Then the equations
where is the transpose tensor, have solutions in , , which are vectors in the rational vector spaces and respectively. The binary vectors and express dependence relationships among the columns or rows of respectively. Note that and are polynomial vectors. Let and . It is known, see [39,41,44,45], that and as rational vector spaces are spanned by minimal polynomial bases of minimal degrees
and
respectively. The set of minimal indices and are known as column minimal indices (c.m.i.) and row minimal indices (r.m.i.) of respectively. To sum up, in the case of a singular pencil, we have invariants of the following type:
• finite elementary divisors of the type ;
• infinite elementary divisors of the type ;
• column minimal indices of the type ;
• row minimal indices of the type .
The Kronecker canonical form, see [39,41,44,45], is defined by
where , are defined as in Section 2. The matrices , , and are defined by
where for . The matrices , are defined as
For algorithms about the computations of these matrices, see [39,41,44,45].
Following the above given analysis, there exist nonsingular matrices P, Q with , such that
Let
Lemma 3.1The system (1) is divided into five subsystems:
the subsystem
the subsystem
the subsystem
and the subsystem
Proof Consider the transformation
Substituting the previous expression into (1), we obtain
Whereby multiplying by P and using (27), we arrive at
Moreover, let
where , , , and . Taking into account the above expressions, we arrive easily at the subsystems (29), (30), (31), (32), and (33). The proof is completed. □
Solving the system (1) is equivalent to solving subsystems (29), (30), (31), (32) and (33). The solutions of the systems (29), (30) are given by (10) and (12) respectively; see Propositions 2.1 and 2.2.
Proposition 3.1The subsystem (31) has infinite solutions and can be taken arbitrarily
Proof If we set
by using (23), (24), the system (31) can be written as
Then, for the nonzero blocks, a typical equation from (37) can be written as
or, equivalently,
or, equivalently,
or, equivalently,
The system (39) is a regulartype system of difference equations with equations and unknowns. It is clear from the above analysis that in every one of the subsystems one of the coordinates of the solution has to be arbitrary by assigned total. The solution of the system can be assigned arbitrarily
The proof is completed. □
Proposition 3.2The solution of the system (32) is unique and is the zero solution
Proof From (25), (26) the subsystem (32) can be written as
Then for the nonzero blocks, a typical equation from (41) can be written as
or, equivalently,
or, equivalently,
or, equivalently,
We have a system of +1 difference equations and unknowns. Starting from the last equation, we get the solutions
which means that the solution of the system (32) is unique and is the zero solution. The proof is completed. □
Proposition 3.3The subsystem (33) has an infinite number of solutions that can be taken arbitrarily
Proof It is easy to observe that the subsystem
does not provide any nonzero equations. Hence all its solutions can be taken arbitrarily. The proof is completed. □
We can now state the following theorem.
Theorem 3.1Consider the system (1) with a singular pencil and known boundary conditions of type (2). Then the boundary value problem (1)(2) is consistent if and only if:
1.
2. the column minimal indices are zero, i.e.,
3.
Furthermore, when the boundary value problem (1)(2) is consistent, it has a unique solution if and only if
1.
2.
In this case the unique solution is given by the formula
whereCis the unique solution of the algebraic system
In any other case the system has infinite solutions.
Proof First we consider that the system has nonzero column minimal indices and nonzero row minimal indices. By using the transformation (34), the solutions of the subsystems (29), (30), (31), (32) and (33) are given by (10), (12), (36), (40) and (44) respectively. Note that from Proposition 2.1 the solution (10) exists if and only if
Furthermore, if
Since is unknown, it can be replaced with the unknown vector C. Then
or, equivalently,
Since and can be taken arbitrarily, it is clear that the general singular discrete time system for every suitable defined boundary condition has an infinite number of solutions. It is clear that the existence of the column minimal indices is the reason that the systems (31) and consequently (33) exist. These systems as shown in Propositions 3.1 and 3.3 have always infinite solutions. Thus a necessary condition for the system to have a unique solution is not to have any column minimal indices which are equal to
In this case the Kronecker canonical form of the pencil has the following form:
Then the system (1) is divided into three subsystems (29), (30), (32) with solutions (10), (12), (40) respectively. Thus
or, equivalently,
The solution exists if and only if
or, equivalently,
or, equivalently,
For the above algebraic system, there exists at least one solution if and only if
The algebraic system (50) contains equations and p unknowns. Hence the solution is unique if and only if
and
where C is then the unique solution of (50). The uniqueness of C can be proved as follows. If we assume that the algebraic system has two solutions and , then
and
or, equivalently,
But the matrix is left invertible since it is assumed to have p linear independent columns and and hence
The unique solution is then given from (49). The proof is completed. □
4 Numerical examples
Example 1
Assume the system (1) for and . Let
and
Then and the pencil is regular. We assume the boundary conditions (2) with
and
The three finite eigenvalues () of the pencil are , 0, , and the Jordan matrix has the form
It is easy to observe that
By calculating the eigenvectors of the finite eigenvalues, we get the matrix
Moreover,
From (9) we get the MittagLeffler function
or, equivalently,
or, equivalently,
or, equivalently,
and since , by using the sum for , we calculate the sum of the matrix power series , and we get
And since
by using the above expression
or, equivalently,
it is easy to observe that the conditions (13), (14) and (15) are satisfied. Thus from Theorem 2.1 the unique solution of the boundary value problem (1)(2) is given by
or, equivalently, by
where C is the unique solution of the algebraic system
or, equivalently,
and thus the unique solution of the boundary value problem is
Example 2
We assume the system (1) as in Example 1 but with different boundary conditions. Let
and
It is easy to observe that
since
and thus from Theorem 2.1, and since (13) does not hold, the boundary value problem is not consistent.
Example 3
Consider the system (1) and let
Since the matrices F, G are nonsquare, the matrix pencil is singular and has invariants such as the finite elementary divisors , , an infinite elementary divisor of degree 1 and the row minimal indices , . Since the Jordan matrix has the form
with
for every induced matrix norm, from Theorem 3.1 the boundary value problem (1)(2) is nonconsistent.
Example 4
Consider the system (1) for and . Let
and
Since the matrices F, G are nonsquare, the matrix pencil is singular and has invariants such as a finite elementary divisor and the row minimal indices , . We assume the boundary conditions (2) with
and
The Jordan matrix is with for every induced matrix norm. By calculating the matrix , we get
Moreover,
and since
we get
By using (59), (60), it is easy to observe that the conditions (45), (46), (47) and (48) are satisfied and thus from Theorem 3.1 the unique solution of the boundary value problem (1)(2) is given by
or, equivalently, by
where C is the unique solution of the algebraic system
or, equivalently,
and thus the unique solution of the boundary value problem is
5 Conclusions
In this article, we study the boundary value problem of a class of a singular system of fractional nabla difference equations whose coefficients are constant matrices. By taking into consideration the cases that the matrices are square with the leading coefficient singular, square with an identically zero matrix pencil and nonsquare, we study the conditions under which the boundary value problem has unique, infinite and no solutions. Furthermore, we provide a formula for the case of the unique solution. As a further extension of this article, one can study the stability, the behavior under perturbation and possible applications in economics and engineering of singular matrix difference/differential equations of fractional order. For all this, there is already some research in progress.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
IKD wrote the first draft of the manuscript and DB correct it and prepared the final version of it. All authors read and approved the final manuscript.
Acknowledgements
We would like to express our sincere gratitude to Professor GI Kalogeropoulos for his helpful and fruitful discussions that clearly improved this article. Moreover, we are very grateful to the anonymous referees for their valuable suggestions that improved the article.
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