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Joseph Evgenievich Andrushkevich, Associate Professor of P. M. Masherov VSU (Vitebsk, Belarus), Candidate of Science in physics and mathematics.
First pro-rector of VSU.


Valery Anatolievich Zhiznevsky, Senior Teacher in the department of physics in VSTU (Vitebsk, Belarus).

Use of generalized Fourier method in solving interior hybrid boundary value problems



Published: 03/22/2004
Original: www.laboratory.ru
© J. E. Andrushkevich, V. A. Zhiznevsky. All rights reserved.
© EDS–Soft, 2004. All rights reserved.


When studying stationary and wave processes of different physical nature (oscillations, heat-conductivity, diffusion, etc), we often come to Laplace and Helmholtz equations:

In electrodynamics, U represents the electric field potential, created by charges or currents provided that there are not volume charges in the region. To illustrate the application of the generalized Fourier method of variable separation (GFM), we consider the internal hybrid boundary problem for these equations in rectangular domain on the plane. The geometry of the task is depicted in Fig. 1.

The problem may be formulated in the following manner: find a function U(x,y), which, in the domain restricted by the contour L, satisfies the equation:

(1)

or the equation:

(1’)

and the boundary conditions at separate sectors of the contour in question:

(2)

We can show, that the solutions for these equations obtained by the classic Fourier method of variable separation, i.e. by representing the function in the form of U(x,y)=X(x)Y(y), cannot satisfy the boundary conditions for the problems in question. When being within the framework of classical Fourier method, we can solve these problems using an artificial technique, namely using the fact the equations (1) и (1’) are linear, we divide the problem into auxiliary sub-problems. For example, for the task with the Helmholtz equation (1’)–(2) auxiliary tasks (3)–(4) and (5)–(6) look as follows:

(3)
(4)
(5)
(6)

The solution to the problem (1’)–(2) is represented as a superposition of solutions to (3)–(4) and (5)–(6), i.e. . Similarly, we can deal with the problem for the Laplace equation (1)-(2).

Let’s illustrate by the boundary value problem for the Helmholtz equation, the most complicated among those considered, the power and possibilities of the generalized Fourier method of variable separation, which allows exclusion of the necessity to consider auxiliary problems. For this purpose, we will look for the particular solutions for the problem (1’)–(2) in the form of GFM-2 [1]

(7)

where the functions as well as the functions are linearly independent. Then the equation (1) can be reduced to the form:

(8)

Acting in accordance with the theory of GFM[1] (GFM-2, r=2), instead of (8) we obtain the system of ordinary differential equations of second order:

(9)

where are the constants of solutions for double-linear functional equations. By using the requirement for the functions (7) to be independent and the hypothesis about the sufficiency of the dimension of functional basis we assume that . The system (9) takes the form:

(10)

where the designations .

The general solutions to the system (10) are given by:

(11)

where — constants of integrating of ODE.

Taking into account (11), the particular solutions of (8) have the form:

(12)

The general solution to the equation (1’) is represented as a superposition of all particular solutions to (12) with different values of constants i.e.

(13)

We will define the values of integration constants in that solution from the boundary conditions (2). The boundary condition along the х–axis, in the form , leads to the expression:

(14)

This expression, under the condition that

(15)

is transformed to:

(16)

If we assume that

(17)

the expression (16) can be considered as an expansion of the function to the Fourier series by cosines at interval (0, b). The coefficients of this expansion are determined as:

(18)

The boundary condition along the х–axis, having the form , taking into account (15) and (17) leads to the expression:

(19)

Under the condition

(20)

the expression (19) is reduced to the Fourier series for :

(21)

The coefficients of this series are given by:

(22)

The boundary condition along the y–axis having the form taking into account the conditions (15), (17) и (20) leads to the expression:

(23)

This expression can be considered as the expansion of the function into the Fourier series by sines at interval . The coefficients of this expansion are determined from:

(24)

The boundary condition along the y–axis, having the form taking into account (15), (17) and (20) leads to the expansion of function into a Fourier series:

(25)

The coefficients of this series are given by:

(26)

Finally, taking into account the conditions (15), (17) and (20), the particular solutions to the boundary value problem (1’)-(2) take the form:

(27)

where and

The expressions, which define the coefficients of these solutions, follow from equations (18), (22), (24), (26):

(28)

By using the described method we can receive the particular solutions for the boundary value problem (1)-(2) with the Laplace equation in the following form:

(29)

where

(30)

The expressions for the coefficients in these solutions are given by:

(31)
(32)
(33)
(34)

The examples presented as an illustration of GFM show obvious advantages of this method over the classic variable separation method. These advantages are most obvious in boundary value problems with the boundary conditions of certain kind. Authors consider the classification of such problems can be the topic of further research.

References

1. Andrushkevich J.E. About a generalization of the Fourier variable separation method. EB &ESU,1998. № 2. (In Russian).

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