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
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
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
If F is singular with a null vector X, then
Definition 1.1 Given
1. Regular when
2. Singular when
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
• e.d. of the type
• e.d. of the type
We assume that
Definition 2.1 Let
From the regularity of
and
The complex Weierstrass form
where the first normal Jordantype element is uniquely defined by the set of the
finite eigenvalues of
The second uniquely defined block
The matrix
and
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
where
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
From the above expressions, we arrive easily at the subsystems
and
The proof is completed. □
Definition 2.3 With
Proposition 2.1The subsystem (7) has the solution
if and only if
where
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
by taking the sum of the above equations and using the fact that
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 pencil
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
or, equivalently,
or, equivalently, by using (10), (12)
The initial value
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
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
or, equivalently,
But the matrix
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
where
and
respectively. The set of minimal indices
• 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
where
where
where
where
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
Let
where
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
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
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
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
or, equivalently,
Since
In this case the Kronecker canonical form of the pencil
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
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
or, equivalently,
But the matrix
The unique solution is then given from (49). The proof is completed. □
4 Numerical examples
Example 1
Assume the system (1) for
and
Then
and
The three finite eigenvalues (
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
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
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
Since the matrices F, G are nonsquare, the matrix pencil
and
The Jordan matrix is
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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