Solutions of Higher Order Homogeneous Linear Matrix Differential Equations For Consistent and Non Consistent Initial Conditions
Solving Higher Order Homogeneous Linear Matrix Differential Equations
by Praveen Samadhiya*, Dr. K. K. Jain , Dr. Yogesh Sharma,
- Published in Journal of Advances in Science and Technology, E-ISSN: 2230-9659
Volume 1, Issue No. 1, Feb 2011, Pages 0 - 0 (0)
Published by: Ignited Minds Journals
ABSTRACT
In this article, we study a class of linear matrix differetialequations (regular case) of higher order whose coefficients are square constant matrices.By using matrix penciltheory and the Weierstrass cannonical form of the pencilwe obtain formulas for the solutions and we show that the solution is unique for consistent initial conditions and infinite fornon-consistent initial conditions.Moreover we provide some numerical examples.These kinds of systems are inherent in many physical and engineering phenomena.
KEYWORD
higher order homogeneous linear matrix differential equations, consistent initial conditions, non-consistent initial conditions, matrix pencil theory, Weierstrass canonical form, numerical examples, physical phenomena, engineering phenomena
. Introduction
inear Matrix Differential Equations (LMDEs) are inherent in many physical, engineering, echanical, and financial/actuarial models.Having in mind such applications, for instance in inance, we provide the well-known input-output Leondief model and its several important xtensions, advice [3]. In this article, our long-term purpose is to study the solution of LMDEs of igher order (1.1) into the mainstream of matrix pencil theory. This effort is signifficant, since here are numerous applications. Thus, we consider here , (i.e. the algebra of square matrices with elements in the field ) with . For the sake of simplicity we set . In the sequel we adopt the following notations r in Matrix form here (where ( )T is the transpose tensor) and the oefficient matrices F;G are given by ith corresponding dimension of F;G and Z(t), mn_mn and mn_1, respectively. Matrix pencil heory has been extensively used for the study of Linear Differential Equations (LDEs) with time nvariant coefficients, see for instance [3], [7]-[9]. Systems of type (1.1) are more general, ncluding the special case when An = In, where In is the identity matrix of Mn, since the well- nown class of higher-order linear matrix differential equations of Apostol-Kolodner type is erived straightforwardly, see [1] for n = 2, [2] and [10]. he paper is organized as follows: In Section 2 some notations and the necessary preliminary oncepts from matrix pencil theory are presented. Section 3 contains the case that system (1.1) has onsistent initial conditions. In Section 4 the non consistent initial condition case is fully iscussed. In this case, the arbitrarily chosen initial conditions which have physical meaning for regular) systems, in some sense, can be created or structurally changed at a fixed time t = t0. ence, it is derived that (1.1) should adopt a generalized solution, in the sense of Dirac -solutions. . Mathematical Background and Notation his brief section introduces some preliminary concepts and definitions from matrix pencil theory, hich are being used throughout the paper.Linear systems of type (1.1) are closely related to atrix pencil theory, since the algebraic geometric, and dynamic properties stem from the tructure by the associated pencil . efinition 2.1. Given and an indeterminate , the matrix pencil is called egular when m = n and . In any other case, the pencil will be called singular. efinition 2.2. The pencil is said to be strictly equivalent to the pencil if and only f there exist nonsingular such as n this article, we consider the case that pencil is regular. Thus, the strict equivalence relation can e defined rigorously on the set of regular pencils as follows. ere, we regard (2.1) as the set of pair of nonsingular elements of Mn nd a composition rule defined on g as follows: t can be easily verified that forms a non-abelian group. Furthermore, an action of the roup on the set of regular matrix pencils is defined as such that his group has the following properties: ill be called the orbit of at g. Also N defines an equivalence relation on which is called a strict-equivalence relation and is denoted by So, f and only if where are nonsingular elements of algebra he class of is characterized by a uniquely defined element, known as a omplexWeierstrass canonical form, , see [6], specified by the complete set of invariants f . This is the set of elementary divisors (e.d.) obtained by factorizing the invariant olynomials into powers of homogeneous polynomials irreducible over field F. In the case here is a regular, we have e.d. of the following type: e.d. of the type sp are called zero finite elementary divisors (z. f.e.d.) e.d. of the type are called nonzero finite elementary divisors (nz.f.e.d.) e.d. of the type ^sq are called in_nite elementary divisors (i.e.d.). Let B1;B2; : : : ;Bn be elements of . The direct sum of them denoted by B1 _ B2 _ _ _ _ _ Bn is the block diagfB1;B2; : : : ;Bng. hen, the complex Weierstrass form of the regular pencil is defined by , where the _rst normal Jordan type element is uniquely defined by he set of f.e.d. n the last part of this section, some elements for the analytic computation of are rovided. To perform this computation, many theoretical and numerical methods have been eveloped. Thus, the interesting readers might consult papers [2, 4, 10, 11, 13] and the references herein. In order to have computational formulas, see the following Sections 3 and 4, the following nown results should firstly be mentioned. n this section, the main results for consistent initial conditions are analytically presented for the egular case. The whole discussion extends the existing literature; see for instance [2]. Moreover, t should be stressed out that these results offer the necessary mathematical framework for nteresting applications, see also introduction. Now, in order to obtain a unique solution, we deal ith consistent initial value problem. More analytically, we consider the system he conclusion, i.e. , is obtained by repetitively substitution of each equation in the next ne, and using the fact that heorem 3.4. Consider the system (3.1-3.2).Then the solution is unique if and only if the initial onditions are consistent.Moreover the analytic solution of (3.1-3.2) is given by olution that exists if and only if The columns of Qp are the eigenvectors of the finite elementary divisors (eigenvalues) of the pencil sA-B.Let In that case he system has the unique solution roposition 4.1. Consider the system (3.3).Then for non consistent initial conditions the system has infinite solutions. Proof. Let Qp,Qq be the matrices defined in heorem 3.5.If the initial conditions are non consistent then Moreover This means (3.3) is efined for t 6= t0 because if and which is a ontradiction.Let H(t - t0) be the Heaviside function and
onclusions
n this article, we study the class of linear rectangular matrix differential equations of higher-order hose coefficients are square constant matrices. By taking into consideration that the relevant encil is regular, we get effected by the Weierstrass canonical form in order to decompose ifferential system into two sub-systems (i.e. the slow and the fast sub-system). Afterwards, we rovide analytical formulas for that general class of Apostol-Kolodner type of equations when we ave consistent and non-consistent initial conditions. Moreover, as a further extension of the resent paper, we can discuss the case where the pencil is singular. Thus, the Kronecker canonical orm is required. The non-homogeneous case has also a special interest, since it appears often in pplications. For all these, there is some research in progress.
eferences
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