On a singular system of fractional nabla difference equations with boundary conditions
© Dassios and Baleanu; licensee Springer. 2013
Received: 20 February 2013
Accepted: 18 May 2013
Published: 19 June 2013
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 non-square, 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.
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., .
Regular when and .
Singular when or and .
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.
We assume that and .
Definition 2.1 Let be elements of . The direct sum of them denoted by is the .
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.
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 .
The proof is completed. □
where is an induced matrix norm and is the discrete Mittag-Leffler function with two parameters as defined by Definition 2.3.
The proof is completed. □
by taking the sum of the above equations and using the fact that , we arrive easily at the solution (12). The proof is completed. □
- 1.The pencil has p distinct eigenvalues and all lie within the open disk
The unique solution is then given from (16). The proof is completed. □
3 Singular case
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 .
where for .
where , , , and .
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.
The proof is completed. □
which means that the solution of the system (32) is unique and is the zero solution. The proof is completed. □
does not provide any non-zero equations. Hence all its solutions can be taken arbitrarily. The proof is completed. □
We can now state the following theorem.
- 2.the column minimal indices are zero, i.e.,(45)
In any other case the system has infinite solutions.
The unique solution is then given from (49). The proof is completed. □
4 Numerical examples
and thus from Theorem 2.1, and since (13) does not hold, the boundary value problem is not consistent.
for every induced matrix norm, from Theorem 3.1 the boundary value problem (1)-(2) is non-consistent.
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 non-square, 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.
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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