This article was downloaded by: [RMIT University] On: 06 September 2013, At: 05:16 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK International Journal of Computer Mathematics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gcom20 Explicit closed form solutions of boundary value problems for systems of difference equations Lucas Jódar a a Department of Applied Mathematics, Polytechnical University of Valencia, P.O. Box 22.012, Valencia7, Spain Published online: 19 Mar 2007. To cite this article: Lucas Jódar (1990) Explicit closed form solutions of boundary value problems for systems of difference equations, International Journal of Computer Mathematics, 33:3-4, 217-224 To link to this article: http://dx.doi.org/10.1080/00207169008803852 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
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This article was downloaded by: [RMIT University]On: 06 September 2013, At: 05:16Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK
International Journal of Computer MathematicsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/gcom20
Explicit closed form solutions of boundary value problems for systemsof difference equationsLucas Jódar aa Department of Applied Mathematics, Polytechnical University of Valencia, P.O. Box 22.012, Valencia7, SpainPublished online: 19 Mar 2007.
To cite this article: Lucas Jódar (1990) Explicit closed form solutions of boundary value problems for systems of difference equations, InternationalJournal of Computer Mathematics, 33:3-4, 217-224
To link to this article: http://dx.doi.org/10.1080/00207169008803852
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications onour platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to theaccuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are theopinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should notbe relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arisingdirectly or indirectly in connection with, in relation to or arising out of the use of the Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions ofaccess and use can be found at http://www.tandfonline.com/page/terms-and-conditions
In this paper boundary value problems for systems of difference equations of the type yk-,,+ A k . , ~ , - k , + . . . + A o . p ) . , = h , . where A,EC,, , and h,, Y,+,E@,. for Osjsk-1, are studied from an algebraic point of view. Existence conditions and closed form solutions are given in terms of co- solulions of the algebraic matrix equation Xk +Ak-, Xk-' + ..- + Ao=O.
KEY WORDS: DilTerence system. boundary value problem, co-solution, matrix equation. closed form solution. generalized inverse.
C.R. CATEGORIES: G.1.8.G.1.3.
1 . INTRODUCTION
This paper deals with boundary value problems for systems of difference equations of the type
where A,, B,,, , for 0 5 h 5 k - 1, 0 5 q 5 s, are complex matrices, elements of C, , ,, and h,, r i and y, + .. are vectors in @,.
For example, such systems appear if we wish to solve approximately boundary value problems for systems of differential equations
when the interval [a, b] has been subdivided into s equal parts by the points
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with
and one replaces the derivatives ~ ' ~ ' ( t , ) of the continuous unknown x( t ) at tn , by A " y ( r n ) / ( A ~ ) h , see [5, p. 2051.
The paper is organized as follows. In Section 2 we introduce the concepts of co- solution and complete set of co-solutions for the algebraic matrix equation
A characterization of the concept of a complete set of co-solutions is given in terms of a block diagonality condition for the companion matrix CL defined by
By means of the concept of a complete set of co-solutions of the algebraic equation (1.3), a desirable expression for the general solution of the non- homogeneous difference system
is obtained in Section 3. This expression for the general solution of (1.5) do not involve an increase of the dimension of the problem and has the advantage that contains k free vector parameters. Thus by imposing the boundary value con- ditions of problem (1.1), this is transformed into an algebraic system that may be explicitly solved using generalized inverses.
I f S is a rectangular matrix, element of @,,,, we represent by S + its
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BOUNDARY VALUE PROBLEMS 219
Moore-Penrose pseudo-inverse, and we recall that interesting procedures for computing S f may be found in [ I , p. 123.
2. CO-SOLUTIONS O F THE MATRIX EQUATION x ~ + A ~ - ~ x ~ - ~ + . - + A ~ = o
In this section we generalize the concept of solution for the matrix equation
as well as the concept of a complete set of solutions of Eq. (2.1) where A, is a matrix in C,.,, for O s h s k - 1 .
DEFINITION 2.1 Let X, T be matrices in C,, ,, we say that the pair ( X , T) is a co-solution of ( 2 . I ) , if X #O and
Exumple I Let I be the identity matrix in @, , ,, then ( I , B) is a co-solution of (2.1), if and only if, B is a solution of (2.1).
Example 2 If i is an eigenvalue of the matrix polynomial L(z) = z k l + ... + A o , then for any nonzero matrix X whose range is contained in
the kernel of L ( i ) , and T = z l , it follows that ( X , z l ) is a co-solution of Eq. (2.1) . In particular, any eigenvalue of the companion matrix C, defined by (1.4) is an eigenvalue of L(z) , see [ 3 , p. 141, thus any matrix equation of the type (2.1) has co-solutions.
Now we introduce the concept of a complete set of co-solutions for equation ( 2 . I ) .
DEFINITION 2.2 Let { ( X i , T ) , I s i s k ) , a set of co-solutions of Eq. (2.1) . Then this set is said to be a complete set of co-solutions of (2.1). if the block matrix W defined by
W =
is invertible in @,,,,,.
Note that if q, is a solution of (2.1), for 1 s i s k, then { ( I , q), 1 g i g k ) is a complete set of co-solutions of (2.1) , if and only if, { T , 1 s i s k ) , is a complete set of solutions of (2.1), see [2.6]. Note result characterizes the existence of a complete set of co-solutions for Eq. (2.1) and provides a procedure for constructing such sets.
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THEOREM 1 Equation (2.1) admits a complete set of co-solutions, if and only if; the compunion mutri .~ C, is similar to a block-diagonal matrix D = [diag ( 7;), 1 5 i 5 k] , where 7 ; ~ C,,,. I f ' M =(Mi/), ,,,,,, is an invertible matrix with Mij E@,~,,
1 i, j 5 k. und sutisfies
then a complete set q f co-solutions of Eq. (2.1) is given by the set {(Mlj, T j ) , 1 S j S k ) .
Proqf' From (2.4) it follows that
k
MksT,= - A j - lMj, and MiST,=Mi+,,, for 1 - 1 (2.5) j= 1
From (2.5) one gets
and from (2.5)-(2.6), we have M,,T:= - - ~ ~ = , Aj- ,MIsT~- ' . Thus (MI,, T,) is a co-solution ofEq. (2.1),for 1 s s 5 k. In order to prove that the set {(MIS, T,),lZs S k),isa complete set of co-solutions of (2.1), note that the matrix W associated to (M,,, T,) by Definition 2.2, coincides with M.
Conversely, if {(Xi, T), 1 S i s k ) , is a complete set of co-solutions of Eq. (2.1), then the matrix Wdefined by (2.3) is invertible and an easy computation yields
3. BOUNDARY VALUE PROBLEMS FOR SYSTEMS O F DIFFERENCE EQUATIONS
Note that if (X, T) is a co-solution of Eq. (2.1), then for any vector c in C,, the sequence jn = XTnc, satisfies the homogeneous difference system
~ k + n + ~ k - l ~ k + n - I +... +AOyn=O, n z O (3.1)
because
THEOREM 2 Let us assume that {(Xi, TJ, 1 S i s k ) is a complete set of co-solutions of Eq. (2.1). Then the general solution of (3.1) is given by the expression
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BOUNDARY VALUE PROBLEMS 22 1
where ci, ,for 1 5 i 5 k , are arbitrary vectors in C,.
Proof' From the previous comment, it is clear that each sequence ynqi defined by
is a solution of (3.1). From linearity, y, defined by (3.2) is a solution of (3.1). In order to prove that (3.2) represents the general solution of (3.1), let (2,) be a solution of (3.1 ), such that z j = d j , for Osjs k - 1.
I f we impose to the expression (3.2) that y j = d j , then it follows that the coefficient vectors ci must satisfy
or equivalently,
where W is the block matrix defined by (2.3), that is invertible from the hypothesis. Solving (3.3). it follows that
Taking these values for vectors ci, it follows that { y , } defined by (3.2) coincides with ( 2 , ) .
Next result that may be regarded as a vectorial discrete analog of Duhamel's principle, provides a particular solution of the non-homogeneous system (1.5).
THEOREM 3 Let us define the sequence {Z , , , } of matrices by the expression
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und let it he a solution of (2 .1) ,for n > m.
Then the sequence ( y , ) defined by I., = 0 for 0 5 n 5 k - 1 . and
is a solution of (1.5) satisfying j, = j , = . . . = j k _ , = O .
Proof From the definition of the sequence {Z , , ,I. it follows that
From (3.5)-(3.7). i t follows that
and thus the resull is established.
The following corollary provides the general solution of (1.5).
COROLLARY 1 Let us suppose that the companion matrix C, giuen by (1.4) satisfies the condition of' Theorenz I , and let { ( X i , 7;), 1 5 i 5 k ) be a complete set of cv-solutions q f Eq. (2 .1) . I f ( j n ) is the sequence defined by (3.6), then the general solution of the non-homogeneous system (1.3) is gicen by the expression
lvhere c,. for 1 5 i 5 k , is any cector in C,.
Proof' The result is an easy consequence of Theorems 2 and 3, because the system ( 1.5) is linear.
Now, we are in a good position in order to study the boundary value problem ( 1 . 1 ). Let us assume that Eq. (2.1) has a complete set of co-solutions {(Xi, ?;), I 5 i s k ) . Then, if we impose that the sequence ( y , ) given by (3.8), satisfies the boundary value conditions of (1.1), it follows that the free vectors ci , 1 S i s k , must verify
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B O U N D A R Y V A L U E PROBLEMS
or.
wherc
because j, = 0. for 0 5 q 5 k - 1. Thus there exist solutions of the boundary value problem (1.11, if and only if,
there exist vectors c,, for I siz k, such that
wherc S = (5, , J, I ( i j m I s,i<li, is the block partitioned matrix, whose entries are defined by
and R, , for I ( r 5 171, are defined by (3.1 I ) , (3.6). Now, from Theorem 2.3.2 of 181, it follows that
only if,
and in this case, the general solution of (3.12) takes
(3.13)
system (3.12) is solvable if and
the form
wherc 1 denotes the identity matrix in C,,.,, and Z is an arbitrary vector in C,, Summarizing the following result has been proved:
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to u block diagonal matrix D=[diag(7J, 1 S i S k] , with 7 ; . ~ @ , , , , and let ( (Xi , 7;.), I s i s k ) , be a complete set of co-solutions of Eq. (2.1). Let S=(SiSj) , the block matrix in C,, , ,,, defined by (3.13) and let Ri , for 1 6 i s m , be the matrices dcfinrd h j (3. / I ) , (3.6).
i ) The boundary d u e problem (1.1) is solvable, if and only if, the condition (3.14) is sat isfied.
ii) lf(3.14) is satisfied, then the general solution of Problem (1.1) is given by the expression (3.8), where { j , ) is defined by (3.5)-(3.6), and vectors c , , . . . , c, are given hy (3.15).
Remark The general solution of the difference system (1.5) has been studied by Lancaster in [ 7 ] , where a closed form expression of the general solution of (1.5) has been proposed by using spectral pairs of the matrix polynomial L(z) = zk l+Ak- , zk - '+ . " + A , . These results are extended in [2] to non-monic equa- tions, but these general solutions seem not to be immediately useful.
Another expression for the general solution of (1.5) has been recently proposed in [ 4 ] , in terms of a solvent chain of Eq. (2.1). but such representation of the general solution seems not to be useful for the study of boundary value problems with complicated boundary value conditions, such as those of problem (1.1).
This paper has been partially supported by a grant from the D.G.I.C.Y.T.. project PS87-0064.
[ I ] S. L. Campbell and C. D. Meyer, Jr., Generalized Inverses of Linear Transformations, Pitman, London. 1979.
[2] J. E. Dennis. J. F. Traub and R. P. Weber, The algebraic theory of matrix polynomials. SlAM J. Numer. And. 13 ( 1 976), 83 1-845.
[3] I. Gohberg, P. Lancaster and L. Rodman. Matrix Polynomials, Academic, New York, 1982. [4] V. Hernandez and F. Incertis. A block bidiagonal form for block companion matrices, Linear
Algebra and its Appl. 75 (1986), 241 -256. [5] E. L. Ince, Ordinary Diffcwntial Equations, Dover, 1956. [6] L. Jodar. Algebraic and differential operator equations, Linear Algebra and its Appl. 102 (1988),
35-53. [7] P. Lancaster. A fundamental theorem on lambda matrices, 11. Difference equations with constant
coefficients. Linear Algehra Appl. 18 (1977). 213-222. 181 C. R. Rao and S. K. Mitra, Generalized Inverse of Matricies and its Applications. John Wiley. 1971.
