Green's Function for Discrete Second-Order Problems with Nonlocal Boundary Conditions
© S. Roman and A. Štikonas. 2011
Received: 1 June 2010
Accepted: 9 November 2010
Published: 24 November 2010
We investigate a second-order discrete problem with two additional conditions which are described by a pair of linearly independent linear functionals. We have found the solution to this problem and presented a formula and the existence condition of Green's function if the general solution of a homogeneous equation is known. We have obtained the relation between two Green's functions of two nonhomogeneous problems. It allows us to find Green's function for the same equation but with different additional conditions. The obtained results are applied to problems with nonlocal boundary conditions.
The study of boundary-value problems for linear differential equations was initiated by many authors. The formulae of Green's functions for many problems with classical boundary conditions are presented in . In this book, Green's functions are constructed for regular and singular boundary-value problems for ODEs, the Helmholtz equation, and linear nonstationary equations. The investigation of semilinear problems with Nonlocal Boundary Conditions (NBCs) and the existence of their positive solutions are well founded on the investigation of Green's function for linear problems with NBCs [2–7]. In , Green's function for a differential second-order problem with additional conditions, for example, NBCs, has been investigated.
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 function .
In , Bahvalov et al. established the analogy between the finite difference equations of one discrete variable and the ordinary differential equations. Also, they constructed a Green's function for a grid boundary-value problem in the simplest case (Dirichlet BVP).
The direct method for solving difference equations and an iterative method for solving the grid equations of a general form and their application to difference equations are considered in [11, 12]. Various variants of Thomas' algorithm (monotone, nonmonotone, cyclic, etc.) for one-dimensional three-pointwise equations are described. Also, modern economic direct methods for solving Poisson difference equations in a rectangle with boundary conditions of various types are stated.
Chung and Yau  study discrete Green's functions and their relationship with discrete Laplace equations. They discuss several methods for deriving Green's functions. Liu et al.  give an application of the estimate to discrete Green's function with a high accuracy analysis of the three-dimensional block finite element approximation.
In this paper, expressions of Green's functions for (1.1) have been obtained using the method of variation of parameters . The advantage of this method is that it is possible to construct the Green's function for a nonhomogeneous equation (1.1) with the variable coefficients , , and various additional conditions (e.g., NBCs). The main result of this paper is formulated in Theorem 4.1, Lemma 5.3, and Theorem 5.4. Theorem 4.1 can be used to get the solution of an equation with a difference operator with any two linearly independent additional conditions if the general solution of a homogeneous equation is known. Theorem 5.4 gives an expression for Green's function and allows us to find Green's function for an equation with two additional conditions if we know Green's function for the same equation but with different additional conditions. Lemma 5.3 is a partial case of this theorem if we know the special Green's function for the problem with discrete (initial) conditions. We apply these results to BVPs with NBCs: first, we construct the Green's function for 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 for the existence of Green's function were found. The results of this paper can be used for the investigation of quasilinear problems, conditions for positiveness of Green's functions, and solutions with various BCs, for example, NBCs.
The structure of the paper is as follows. In Section 2, we review the properties of functional determinants and linear functionals. We construct a special basis of the solutions in Section 3 and 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. In Section 5, discrete Green's function definitions of this problem are considered. Then a Green's function is constructed for the second-order linear difference equation. Applications to problems with NBCs are presented in Section 6.
We begin this section with simple properties of determinants. Let or and .
is valid. The proof follows from the Laplace expansion theorem .
We define , . Note that , for .
If and , then we get . So, the function is invariant with respect to the basis and we write .
If we take , , , , , in (2.1), then we get equality (2.6).
, is valid.
Let the functions be linearly independent.
Functionals , are linearly independent on if and only if .
3. Special Basis in a Two-Dimensional Space of Solutions
But this determinant is zero if and only if . We combine Lemma 2.3 and these results in the following lemma.
Let be the basis of the linear space . Then the following propositions are equivalent:
(1)the functionals , are linearly independent;
(2)the functions , are linearly independent;
Propositions in Lemma 3.1 are equivalent to the condition .
Propositions in Lemma 3.1 are valid if we take instead of .
(see (2.15)). So, the definition of is invariant with respect to the basis : , .
4. Discrete Difference Equation with Two Additional Conditions
For , this equality shows that , and we arrive at the conclusion that (the case where are linearly dependent solutions) or for all (the case of the fundamental system).
where , are linearly independent functionals.
4.1. The Solution to a Nonhomogeneous Problem with Additional Homogeneous Conditions
where , , , , , .
4.2. A Homogeneous Equation with Additional Conditions
to this problem if the general solution is inserted into the additional conditions.
The solution of nonhomogeneous problems is of the form (see (4.19) and (4.21)). Thus, we get a simple formula for solving problem (4.3)-(4.4).
Formula (4.22) can be effectively employed to get the solutions to the linear difference equation, with various , , , any right-hand side function , and any functionals , and any , , provided that the general solution of the homogeneous equation is known. In this paper, we also use (4.22) to get formulae for Green's function.
4.3. Relation between Two Solutions
and we can express the solution of the second problem (4.23) via the solution of the first problem.
between the two solutions of problems (4.23) is valid.
If we expand the determinant in (4.27) according to the last row, then we get formula (4.26).
Note that in this formula the function is in the first term only and is invariant with regard to the basis .
5. Green's Functions
5.1. Definitions of Discrete Green's Functions
then is called Green's function of operator with the additional condition . Green's function exists if . This condition is equivalent to for . 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 . If , then for , and , , where (or , , , , . So, is a unique solution of problem (5.4) with , .
The function is an example of Green's function for (4.3) with discrete (initial) conditions . In the case , formula (5.6) is the same as (4.15), .
So, if we know the function , then we can calculate , and vice versa. If ( ), then coincides with .
5.2. Green's Functions for a Linear Difference Equation with Additional Conditions
Let us consider the nonhomogeneous equation (4.3) with the operator: , where additional homogeneous conditions define the subspace .
too. If we expand this determinant according to the last row and divide by , then we get the right-hand side of (5.18). The lemma is proved.
If , where , then we get that Green's function , that is, it is invariant with respect to the basis .
with the same , is useful.
A further proof of this theorem repeats the proof of Lemma 5.3 (we have instead of ).
can be used to get the solutions of the equations with a difference operator with any two linear additional (initial or boundary or nonlocal boundary) conditions if the general solution of a homogeneous equation is known.
6. Applications to Problems with NBC
We can write many problems with nonlocal boundary conditions (NBC) in this form, where , , is a classical part and , , is a nonlocal part of boundary conditions.
We suppose that the points , are coincident with the grid points, that is, , .
Note that the index of on the right-hand side of (6.9) is shifted (cf. (6.1)).
for differential problem (6.6) at grid points in the case .
where are approximations of the weight functions in integral boundary conditions, is a quadrature formula for the integral approximation (e.g., trapezoidal formula ).
where is defined by (6.15).
if , where is defined by formula (6.17).
We could substitute (6.15) into (6.25) and obtain an explicit expression of Green's function. However, it would be quite complicated, and we will not write it out. Note that, if , , then discrete problem (6.23) is the same as (6.7)-(6.8). For example, it happens if a trapezoidal formula is used for the approximation , and we take . It is easy to see that we could obtain the same expression for Green's function (6.19) in this case.
All the cases yield . Consequently, Green's function for the three problems does not exist.
Green's function for problems with additional conditions is related with Green's function of a similar problem, and this relation is expressed by formulae (5.26). Green's function exists if . If we know Green's function for the problem with additional conditions and the fundamental basis of a homogeneous difference equation, then we can obtain Green's function for a problem with the same equation but with other additional conditions. It is shown by a few examples for problems with NBCs that but formulae (5.26) can be applied to a very wide class of problems with various boundary conditions as well as additional conditions.
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