1. Introduction
The finite-difference method is a standard numerical method for solving boundary value problems. Recently, considerable attention has been attracted to construct a best (or exact) difference approximation for some ordinary and partial differential equations [1-3]. In this paper a best finite-difference method is developed for Helmholtz equation with general boundary conditions on the rectangular domain in R2. The method proposed here comes out from [4] and is based on separation of variables method and expansion of one-dimensional threepoint difference operators for sufficiently smooth solution. The paper is organized as follows. The statement of problem and the separation of variables method are considered in section 2. A detailed description of the best difference approximation to the Helmholtz equation in rectangular domain is given in section 3.
Section 4 is devoted to derive the best approximation for the given third kind boundary conditions. The method of solution for the obtained difference equations is considered in section 5 and numerical examples are given last section 6.
2. Statement of Problem
Let 
be an open rectangular domain in Euclidean R2 space with boundary given by
. The aim is to determine a function
, satisfying equation
(2.1)
with boundary condition
(2.2)
where C in (2.1) is a given number and 
is the outward normal on
.
It is well known that the stabilized oscillation problems and diffusing processes in gas lead to the so called Helmholtz Equation (2.1) with a positive coefficient 
The diffusing process in the moving field leads to the Equation (2.1) with negative coefficient
. If C = 0 the Equation (2.1) leads to Laplace’s ones. Obviously, the properties of the solution of Equation (2.1) depend essentially upon the sign of the coefficient C in (2.1). We will assume that the problem (2.1), (2.2) has an unique and sufficiently smooth solution.
By virtue of variables method looking for the solution 
of Equation (2.1), (2.2) in the form 
(2.3)
we arrive at equation
which is splitted into two independing equations
(2.4a)
and
(2.4b)
where the unknown separation constant ω is to be found.
By virtue of (2.3) the boundary condition (2.2) is splitted info ones for 
and 
(2.5)
and
(2.6)
The solution of boundary value problem (2.4a), (2.5) is founded in a closed from
(2.7)
where
and
When ω < 0 the functions sh and ch in (2.7) are to be replaced by sin and cos respectively and ω replaced by –ω. Analogously, we can find the solutions of boundary value problem (2.4b) and (2.6) in closed form. Then from (2.3) and (2.7) clear, that the problem consists in determining the separation constant ω.
3. Construction of the Best Finite-Difference Equations
For the numerical solution of problem (2.1), (2.2) is introduced the uniform rectangular grid 
where 
and 
are the mesh sizes in the x and y directions respectively. Usually, the Equation (2.1) is approximated by the five-point difference equation
(3.1)
The local discretization error of the Equation (3.1) is of 
order. Now we describe how to derive the best difference scheme for Equation (2.1). To this end, we consider expression
(3.2)
where
. If we denote by 
the values of 
the values of 
and 
respectively, the using (2.3) the Equation (3.2) may be written as
(3.3)
Due to smoothness assumption of solution 
as well as, functions 
and 
the Taylor series expansion yields
(3.4a)
(3.4b)
Because of (2.4) we have
(3.5)
Taking into account (3.4), (3.5) in (3.3) it follows that
(3.6)
where E is unit operator. The difference Equation (3.6) contains unknown nonzero parameter ω and therefore it may be considered as a nonlinear equation with respect to the parameter ω and 
The series in (3.6) may be expressed through analytical functions depending on the sign of quantities ω and β and thereby the Equation (3.6) can be rewritten as
(3.7)
There are three cases:
1) Let 
Then it is easy to show that
(3.8)
2) Let 
In this case D is given by
(3.9)
3) Let 
In this case D is given by
(3.10)
Thus we obtain the best (or exact) five-point difference Equation (3.7) for the Equation (2.1) (see, for example, Mickens [2] and Agarwal [1]). The function 
in (3.7) can be presented as a sum of two ones, i.e.,
(3.11)
where 
and 
correspond to the first and second terms in (3.8), (3.9) and (3.10) respectively.
4. The Best Finite-Difference Boundary Condition
Now we will derive a best difference boundary condition for (2.5), (2.6). Using (2.4) in the Taylor series expansion
we obtain
(4.1)
If 
in (2.5), then we have
(4.2a)
If 
then finding 
from (2.5) and substituting it in (4.1) we get
(4.2b)
where 
and 
are given by
(4.3a)
and
(4.3b)
Analogously, it is easy to verify that the exact difference boundary condition for 
at point 
is given by
when 
(4.4a)
when 
(4.4b)
where 
and 
are given by
(4.5a)
and
(4.5b)
In the same way, as before, one can construct the best difference boundary conditions for
. We omit the evaluation and present only the final results:
, when 
(4.6a)
when 
(4.6b)
and
when 
(4.7a)
when 
(4.7b)
where 
are defined by
(4.8a)
(4.8b)
and
(4.9a)
(4.9b)
5. Method for Solution of Finite-Difference Equations
In this section we consider a method for solving the finite-difference Equations (3.7). For this purpose we rewrite it in the from
(5.1)
in which we have used (3.11) From this it is clear, that Equation (5.1) will be fulfilled if we choose 
and 
such that
(5.2a)
(5.2b)
The last weakly coupled system of Equation (5.2) is splitted into two equations with corresponding boundary conditions. First, we consider the Equation (5.2a) subject to boundary conditions (4.2) and (4.4).
According to (2.1), (2.2) and (2.3) the function 
will be defermined within an arbitrary multiplicative constant. Therefore the three-point finite-difference Equation (5.2a) can be solved by shooting method starting with
, 
and 
which are required to be known. Thanks to (4.2) it is possible to find 
or 
depending on the
. For example, if 
then 
is determined by (4.2a) and 
and ω to be chosen arbitrary. Otherwise, 
is determined by (4.2b) and 
and ω to be chosen arbitrary.
Note, that when 
one of the boundary conditions (2.5), (2.6) is assumed to be homogeneous. For Laplace’s equation we always can leads to equation with homogeneous boundary conditions by change of variables. The exact value of parameter ω must satisfy
(5.3)
where
, for examples, when 
defined by
(5.4)
The nonlinear Equation (5.3) can be solved by Newton’s method:
(5.5)
The value 
in the dominator of (5.5) is found by differentiating the Equation (5.4) and (5.2a) with respect to ω. The iteration process (5.5) is terminated by criterion
(5.6)
where 
is a reassigned accuracy.
If the evaluation of 
causes some difficulty we can use secant method instead of Newton’s ones. After finding ω the three-point difference equations (5.2b) with boundary conditions (4.6), (4.7) can be solved by elimination method.
6. Numerical Results
We have tested the efficiency and accuracy of finitedifference scheme (3.7) on the several examples.
Example 1.
with boundary condition
The exact solution is given by
In Table 1 we present the computed values of 
(exact values of 
present in brackets) for N = 5 and M = 4. In order to use secant method we need two first approximations 
and 
to ω. The iteration was terminated by criterion (5.6) with
.
Example 2.
with boundary condition
The exact solution is given by
In Table 2 we present the computed values of
(exact values of 
present in brackets) for N = 6 and M = 6. In order to use secant method we need two first approximations 
and 
to ω. In this example choose 
and
. The exact value of ω is 
The convergence of 
was tabulated in Table 3. The iteration was terminated by criterion (5.6) with
.
Example 3.
with boundary condition
The exact solution is given by
In Table 4 we present the computed values of
(exact values of 
present in brackets) for N = 6 and M = 4. In order to use secant
Table 1. Computed values of 
for N = 5 and M = 4.
Table 2. Computed values of 
for N = 6 and M = 6.
Table 4. Computed values of 
for N = 6 and M = 4.
method we need two first approximations 
and 
to ω. In this example we were choose 
and
. The exact value of ω is
. The convergence of 
was tabulated in Table 5. The iteration was terminated by criterion (5.6) with ε = 10−7.