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Volume 2011, Article ID 767024, 23 pagesdoi:10.1155/2011/767024 Research Article Green’s Function for Discrete Second-Order Problems with Nonlocal Boundary Conditions Svetlana Roman and

Volume 2011, Article ID 767024, 23 pages

doi:10.1155/2011/767024

Research Article

Green’s Function for Discrete Second-Order

Problems with Nonlocal Boundary Conditions

Svetlana Roman and Art ¯uras ˇStikonas

Institute of Mathematics and Informatics, Vilnius University, Akademijos 4, Vilnius, LT-08663, Lithuania

Correspondence should be addressed to Svetlana Roman,svetlana.roman@mif.vu.lt

Received 1 June 2010; Revised 24 July 2010; Accepted 9 November 2010

Academic Editor: Gennaro Infante

Copyrightq 2011 S Roman and A ˇStikonas This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited

We investigate a second-order discrete problem with two additional conditions which aredescribed by a pair of linearly independent linear functionals We have found the solution to thisproblem and presented a formula and the existence condition of Green’s function if the generalsolution of a homogeneous equation is known We have obtained the relation between two Green’sfunctions of two nonhomogeneous problems It allows us to find Green’s function for the sameequation but with different additional conditions The obtained results are applied to problemswith nonlocal boundary conditions

1 Introduction

The study of boundary-value problems for linear differential equations was initiated bymany authors The formulae of Green’s functions for many problems with classical boundaryconditions are presented in 1 In this book, Green’s functions are constructed for regularand singular boundary-value problems for ODEs, the Helmholtz equation, and linearnonstationary equations The investigation of semilinear problems with Nonlocal BoundaryConditions NBCs and the existence of their positive solutions are well founded on theinvestigation of Green’s function for linear problems with NBCs 2 7 In 8, Green’sfunction for a differential second-order problem with additional conditions, for example,NBCs, has been investigated

In this paper, we consider a discrete difference equation

a2i u i2 a1

i u i1 a0

where a2, a0/ 0 This equation is analogous to the linear differential equation

b2xux  b1xux  b0xux  fx. 1.2

In order to estimate a solution of a boundary value problem for a difference equation,

it is possible to use the representation of this solution by Green’s function9

In10, Bahvalov et al established the analogy between the finite difference equations

of one discrete variable and the ordinary differential equations Also, they constructed aGreen’s function for a grid boundary-value problem in the simplest caseDirichlet BVP.The direct method for solving difference equations and an iterative method for solvingthe grid equations of a general form and their application to difference equations areconsidered in 11, 12 Various variants of Thomas’ algorithm monotone, nonmonotone,cyclic, etc. for one-dimensional three-pointwise equations are described Also, moderneconomic direct methods for solving Poisson difference equations in a rectangle withboundary conditions of various types are stated

Chung and Yau 13 study discrete Green’s functions and their relationship withdiscrete Laplace equations They discuss several methods for deriving Green’s functions Liu

et al.14 give an application of the estimate to discrete Green’s function with a high accuracyanalysis of the three-dimensional block finite element approximation

In this paper, expressions of Green’s functions for1.1 have been obtained using themethod of variation of parameters12 The advantage of this method is that it is possible

to construct the Green’s function for a nonhomogeneous equation 1.1 with the variablecoefficients a2, a1, a0 and various additional conditions e.g., NBCs The main result ofthis paper is formulated in Theorem 4.1,Lemma 5.3, andTheorem 5.4.Theorem 4.1can beused to get the solution of an equation with a difference operator with any two linearlyindependent additional conditions if the general solution of a homogeneous equation isknown.Theorem 5.4gives an expression for Green’s function and allows us to find Green’sfunction for an equation with two additional conditions if we know Green’s function forthe same equation but with different additional conditions.Lemma 5.3is a partial case ofthis theorem if we know the special Green’s function for the problem with discreteinitialconditions We apply these results to BVPs with NBCs: first, we construct the Green’s functionfor classical BCs, then we can construct Green’s function for a problem with NBCs directly

Lemma 5.3 or via Green’s function for a classical problem Theorem 5.4 Conditions forthe existence of Green’s function were found The results of this paper can be used for theinvestigation of quasilinear problems, conditions for positiveness of Green’s functions, andsolutions with various BCs, for example, NBCs

The structure of the paper is as follows In Section 2, we review the properties offunctional determinants and linear functionals We construct a special basis of the solutions

inSection 3and introduce some functions that are independent of this basis The expression

of the solution to the second-order linear difference equation with two additional conditions

is obtained in Section 4 InSection 5, discrete Green’s function definitions of this problemare considered Then a Green’s function is constructed for the second-order linear differenceequation Applications to problems with NBCs are presented inSection 6

2 Notation

We begin this section with simple properties of determinants Let or  and 1 < n∈

For all a i j , b j i ∈ , i, j  1, 2, the equality

b2

1 b2 2

a2

1 a2 2





is valid The proof follows from the Laplace expansion theorem8

Let X  {0, 1, , n},  X  {0, 1, , n − 2} FX : {u | u : X → } be a linear space

of realcomplex functions Note that FX ∼ n1and functions δ i , i  0, 1, , n, such that

such that un

k0u k δ k If we have the vector-functionu  u1, u2 ∈ F2X, then we consider

the matrix functionu : X2 → M2×2  ∼ 4 and its functional determinant Du ij : X2 →

Letif Wu j2/ 0

Huij: Duj 1,i

Wuj2 

Duj 1,i

Duj 1,j2 , i ∈ X, j  −1, 0, 1, , n − 2. 2.4

We define H i,n−1u  H in u  0, i ∈ X Note that H j 1,j  0, H j 2,j  1 for j ∈  X.

Ifuij P · uij, whereP  p m

n  ∈ M2×2 , thendetuij det uij · det P, Wui  Wu i · det P. 2.5

If W u / 0 and P ∈ GL2  : {P ∈ M2×2  : det P / 0}, then we get Hu  Hu So, the

function Hu ijis invariant with respect to the basis{u1, u2} and we write H ij

Lemma 2.1 If w  w1, w2 ∈ F2X, then the equality

Corollary 2.2 If w  w1, w2 ∈ FX2, then the equality

We consider the space F∗X of linear functionals in the space FX, and we use

the notation f, u, f k , u k  for the functional f value of the function u Functionals δ j,

j  0, 1, , n form a dual basis for basis {δ i}n

i0 Thus,δ j , u   u j If f ∈ F∗X, g ∈ F∗Y, where X  {0, 1, , n} and Y  {0, 1, , m}, then we can define the linear functional direct

f, w2

g, w2



 det Mfw. 2.10

Let the functions w1, w2∈ FX be linearly independent.

Lemma 2.3 Functionals f, g are linearly independent on span{w1, w2} ⊂ FX if and only if

D fw / 0.

Proof We can investigate the case where F X  span{w1, w2} The functionals f, g are linearly independent if the equality α1f  α2g  0 is valid only for α1  α2  0 We canrewrite this equality asα1f  α2g, w   0 for all w ∈ span{w1, w2} A system of functions

{w1, w2} is the basis of the span{w1, w2}, and the above-mentioned equality is equivalent tothe condition below

α1



f, w1

α1f  α2g, w2





00

3 Special Basis in a Two-Dimensional Space of Solutions

Let us consider a homogeneous linear difference equation

Lu : a2

i u i2 a1

i u i1 a0

i u i  0, i ∈  X, 3.1

where a2, a0/  0 Let S ⊂ FX a be two-dimensional linear space of solutions, and let {u1, u2}

be a fixed basis of this linear space We investigate additional equations

L1, u   0, L2, u   0, u ∈ S, 3.2

where L1, L2 ∈ S∗ are linearly independent linear functionals, and we use the notationL 

L1, L2 We introduce new functions

v1i : Dδ i , L2u, v2i : DL1, δ i u. 3.3For these functions L m , v n   δ n

m D Lu, m, n  1, 2, that is, v n ∈ Ker L m for m /  n.

So, the function v1 satisfies equationL2, u , and the function v2 satisfies equationL1, u

Components of the functions v1and v2in the basis{u1, u2} are

1 the functionals L1, L2are linearly independent;

2 the functions v1, v2are linearly independent;

The left-hand side of this equality is equal to

Propositions inLemma 3.1are equivalent to the condition Wv / 0.

Corollary 3.3 If functionals L1 , L2are linearly independent, that is, D Lu / 0, and

Remark 3.4 Propositions inLemma 3.1are valid if we take{v1, v2} instead of {v1, v2}

Remark 3.5 If {u1, u2} is another fundamental system and u  Pu, where P ∈ GL2 , then

4 Discrete Difference Equation with Two Additional Conditions

Let{u1, u2} be the solutions of a homogeneous equation

i Wui1  0, and we arrive at the

conclusion that Wu i ≡ 0 the case where {u1, u2} are linearly dependent solutions or

Wui /  0 for all i  1, , n the case of the fundamental system.

In this section, we consider a nonhomogeneous difference equation

where L1, L2are linearly independent functionals

4.1 The Solution to a Nonhomogeneous Problem with Additional

Homogeneous Conditions

A general solution of4.1 is u  C1u1C2u2, where C1, C2are arbitrary constants and{u1, u2}

is the fundamental system of this homogeneous equation We replace the constants C1, C2by

the functions c1, c2 ∈ FX Method of Variation of Parameters 12, respectively Then, bysubstituting

The functions u1and u2are solutions of the homogeneous equation4.1 Consequently,

We can take d −1,i1  0, i  0, , n − 1 Then d 1,i1  f i /a2

i for all i ∈ X, and we obtain the

Since u1, u2are linearly independent, the determinant Wu is not equal to zero and system

4.10 has a unique solution

b 1,i1  c 1;i2 − c 1;i1 − u2i1f i

for i  2, , n We introduce a function H θ ∈ FX ×  X:

i,·, fX  C1u1i  C2u2i We use this formula for the special basis{v1, v2} see

3.11 In this case, we have

u iH i, θ·, f

X  C1v1i  C2v i2, i ∈ X. 4.16Let there be homogeneous conditions

4.2 A Homogeneous Equation with Additional Conditions

Let us consider the homogeneous equation4.1 with the additional conditions 4.4

Lu  0, L1, u   g1, L2, u   g2. 4.20

We can find the solution

u 0;i  g1· v1

i  g2· v2

to this problem if the general solution is inserted into the additional conditions

The solution of nonhomogeneous problems is of the form u i  u f ;i u 0;isee 4.19 and

4.21 Thus, we get a simple formula for solving problem 4.3-4.4

Theorem 4.1 The solution of problem 4.3-4.4 can be expressed by the formula

u i δ i k− Lkvi , H k, θ· , f

X  g1· v1

i  g2· v2

i , i ∈ X. 4.22

Formula4.22 can be effectively employed to get the solutions to the linear difference

equation, with various a0, a1, a2, any right-hand side function f, and any functionals L1, L2

and any g1, g2, provided that the general solution of the homogeneous equation is known Inthis paper, we also use4.22 to get formulae for Green’s function

4.3 Relation between Two Solutions

Next, let us consider two problems with the same nonhomogeneous difference equation with

a difference operator as in the previous subsection

Lu  f, Lv  f,

l m , u   f m , m  1, 2, L m , v   F m , m  1, 2, 4.23and D L / 0 The difference w  v − u satisfies the problem

Corollary 4.2 The relation

between the two solutions of problems4.23 is valid.

Proof If we expand the determinant in4.27 according to the last row, then we get formula

5.1 Definitions of Discrete Green’s Functions

We propose a definition of Green’s functionsee 9,12 In this section, we suppose that

 and X n : X  {0, 1, , n} Let A : FXn  → FX n −m   Im A be a linear operator,

0 ≤ m ≤ n Consider an operator equation Au  f, where u ∈ FX n is unknown and

f ∈ FX n −m is given This operator equation, in a discrete case, is equivalent to the system oflinear equations

whereB  b ji  ∈ M n1×M−nm, rank B  M − n  m, and denote

We have a system of linear equations Au  f, where f   f j  ∈ M n1×1, A 

a ji  ∈ M n1×M1 The necessary condition for a unique solution is M ≥ n Additional

equations 5.2 define the linear operator B : FX n  → FX M −nm and the additional

operator equation Bu 0, and we have the following problem:

then G ∈ FX n × X n −m  is called Green’s function of operator A with the additional condition

Bu  0 Green’s function exists if Ker A ∩Ker B  {0} This condition is equivalent to det A / 0

for M  n In this case, we can easily get an expression for Green’s function in representation

5.5 from the Kramer formula or from the formula for u  A−1f If A−1  g ij , then G ij 

g ij for i ∈ X n , j ∈ X n −m andAG  E, BG  O, where G  G ij  ∈ M n1×n−m1 or

n

k0a ik G kj  δ i

j , i ∈ X n −m,n

k0b ik G kj  0, i ∈ X m , j ∈ X n −m  So, G 0j , , G nj is a uniquesolution of problem5.4 with f j  δ0

The function H θ ∈ FX ×  X is an example of Green’s function for 4.3 with discrete initial

conditions u0 u1 0 In the case m  2, formula 5.6 is the same as 4.15, X  X n−2

Remark 5.2 Let us consider the case m  2 If f i  f i1, where the function f is defined on

X :  {1, 2, , n − 1}, then we use the shifted Green’s function G ∈ FX × X

u in−1

j1

G ij f j , G ij: Gi,j−1, i ∈ X n 5.7

For finite-difference schemes, discrete functions are defined in points xi ∈ 0, L and

f i  fx i In this paper, we introduce meshes

Note that the Wronskian determinant can be defined by the following formulasee

Lemma 5.3 Green’s function for problem 4.3 with the homogeneous additional conditions

L1, u   0, L2, u   0, where functionals L1and L2are linearly independent, is equal to

too If we expand this determinant according to the last row and divide by DLu, then we

get the right-hand side of5.18 The lemma is proved

Ifu  Pu, where P ∈ GL2, then we get that Green’s function G ij  Gu ij  Gu ij,that is, it is invariant with respect to the basis{u1, u2}

For the theoretical investigation of problems with NBCs, the next result about the

relations between Green’s functions G u ij and G v ijof two nonhomogeneous problems

Lu  f, Lv  f,

l m , u   0, m  1, 2, L m , v   0, m  1, 2, 5.20with the same f, is useful.

Theorem 5.4 If Green’s function G u exists and the functionals L1and L2are linearly independent, then

A further proof of this theorem repeats the proof ofLemma 5.3we have G u instead of H θ

Remark 5.5 Instead of formula5.18, we have

The function H θ ∈ FX ×  X is an example of Green’s function for 4.3 with discrete initial

conditions. .. get formulae for Green’s function

4.3 Relation between Two Solutions

Next, let us consider two problems with the same nonhomogeneous difference equation with. ..

Bu  Green’s function exists if Ker A ∩Ker B  {0} This condition is equivalent to det A / 0

for M  n In this case, we can easily get an expression for Green’s function