CC BY
37
A NEW APPROACH TO THE REDUCTION OF MULTIDIMENSIONAL LINEAR BOUNDARY VALUE PROBLEMS FOR AN EQUIVALENT BOUNDARY INTEGRAL EQUATIONS
Arutyunyan Robert Vladimirovich,
candidate of physical and mathematical sciences, docent, Moscow Technical University of Communications and Informatics (MTUCI), Russia, Moscow, [email protected]
Keywords: differential equation of the system, boundary value problem, reduction, boundary integral equation.
This article describes a new method for the reduction of multidimensional linear boundary value problems for systems of differential equations in partial derivatives of the first order to the equivalent boundary integral equations. The method is effective for large scale systems and allows you to effectively automate the reduction by means of analytical calculations on computer systems. This method can be used in principle for solving the Maxwell equations, and other similar systems and in a case where the coefficients of the constitutive equations depend on the spatial position and time. Formula method to generalize the Huygens ratio of Maxwell's equations, Somilian's for elasticity problems and other such well-known integral representation of solutions of systems of differential equations in partial derivatives. The proposed algorithm consists of the following steps. Being on the first stage of a singular solution is carried out by standard integral transformations and the corresponding algorithm can easily be programmed in the computer algebra systems. Some problem is that many boundary problems are put in the form of systems in which the number of unknowns and the same equations. For this reason, you want to carry out the second stage of the transition process corresponding to the system for a new basic vector of unknowns. Depending on the type of boundary conditions of the original system is reduced to a system of boundary equations of the minimum dimension of the first or second kind. At the third stage, the analysis of the correctness of the resulting system of singular boundary integral equations using the symbol theory. Analysis of a fundamentally important issue of the boundary conditions is greatly simplified and can be performed as follows. Determined by the rank and the basis of the linear span of the vector lines of the differential operator matrix. For systems with piecewise constant coefficients is especially useful analysis of boundary conditions by investigating the minors corresponding matrix, described in the article. Nondegenerate minor of this matrix corresponds to the correct system of boundary integral equations. At the final stage of the algorithm is formulated as a system of linear algebraic equations for direct approximate numerical solution of boundary integral equations. In the future, the boundary integral equations reduce to a system of linear algebraic equations and solved by Gauss or other suitable numerical method. The proposed method has several advantages especially when dealing with new insufficiently investigated multidimensional boundary value problems. In the case of classical systems of utility proposed approach is most useful in the case of non-standard boundary conditions. The proposed integral representation applies in the case where the differential coefficients of the operator system are arbitrary piecewise continuous functions. The main problem in this case is to find the fundamental matrix function solutions.
Для цитирования:
Арутюнян Р.В. Новый подход к редукции многомерных линейных краевых задач к эквивалентным граничным интегральным уравнениям // T-Comm: Телекоммуникации и транспорт. - 2016. - Том 10. - №4. - С. 63-66.
For citation:
Arutyunyan R.V. A new approach to the reduction of multidimensional linear boundary value problems for an equivalent boundary integral equations. T-Comm. 2016. Vol. 10. No.4, рр. 63-66.
Introduction
In the application there is a trend towards the use of increasingly complex mathematical models, whereby there is a practical need to solve already known, or new types of high dimension systems of differential equations [1-14]. Such systems, in particular, the equations of electromagnet ism in the case where the characteristics of the environment are tensor quantities and Maxwell system does not break up into independent equations.
Another example is the system of electrodynamics equations describing piezoelectric phenomena [13]. In the theory of electrodynamics are now the urgent task of calculating the natural frequencies of piezoelectric bodies and a variety of complex shapes.
The need for taking into account different kinds of physical effects constantly leads to complication of the former and the emergence of new types of systems of differential equations in partial derivatives.
To solve such problems by using the method of boundary integral equations integral representations of the general solution can be used [3-8]. The possibilities of this approach are extended using analytical calculations on computer systems [9-11].
In the article the author developed a fundamentally new universal method for the reduction of multidimensional linear boundary value problems for systems of differential equations in partial derivatives of the first order to an equivalent system of boundary integral equations. This method can be used in principle for solving the Maxwell equations, and other similar systems and in a ease where the coefficients of the constitutive equations depend on the spatial position and time.
Integral equations, which are reduced to the original system of differential equations obtained as a result of the above approaches are generally singular. This fact represents a significant problem in the numerical implementation of the method of boundary integral equations of the real and the more interval methods. An effective way to solve this problem is to transform the system of boundary integral equation method by multiplying by matrix singular operator, called the equivalent regularizator [3-8].
Another problem stems from the fact that some integral equations can be dependent. Its solution is possible by switching to the new dependent variable vector using the method described in the literature [3-8].
Since equivalent regularizator is singular, the resulting transformations after the free terms of the new system boundary integral equation method, as a rule, are singular integrals with known densities. The problem of calculating these integrals in the space of continuous functions is incorrect. This fact creates additional problems in the calculation of interval estimates of absolute terms of boundary integral equations. Its solution is possible in some cases with the help of an analytical evaluation of the integral in the neighborhood of a singular point.
The resulting corresponding transformations regular integral equations can be solved described in [1-12] by numerical methods with particular release. As a specific example, consider a universal algorithm for finding fundamental solutions, reduction and analysis of linear systems of differential equations in partial derivatives and related boundary value problems, easy to program in modern computer algebra systems.
Description of the method
Consider the system of linear partial differential equations of the first order with constant coefficients:
A(5/5A-)h(A-)-^(X), xeQ, (1)
where Q q R" — region with piece wise smooth boundary I', if(x)— a vector function of n variables, u(x)bRv,
F(x)eRm, m>q, A=||A..||,
\ = ,a*»i k = \.....n)+bg, i=\.....m; j = \.....cj.
Let A'- differential operator formally adjoint to the operator A, and G(,v) — the system solution
A*(a/fiv)G(.r) = £(A-), xeR", (2)
where E[x) — S(x)E, E— the identity matrix, delta
function.
From (1) and (2) using Green's formula we obtain an integral representation for the solution of the original system:
ff(.vM*)= \GT(y-x)Fiy)dy- pr{y-x)Q{y)u{y)dr ' n r
(3)
a(x)= \d{y-x)dy, (4>
n
q=N|> Q,=(y)> k=1>~>4 *E r'
/ = 1,.j = \,...,q, (vj _ coordinates of the unit normal, external to the region Q.
Formula (3-4) summarize the relation of Huygens for Maxwell's equations, Somilian's for elasticity problems and other such well-known integral representation of solutions of systems of differential equations in partial derivatives.
Further algorithm consists of the following steps. Being on the first stage of a singular solution G(.v) is performed by standard integral transformations and the corresponding algorithm can easily be programmed in the computer algebra systems REDUCE, MAXIMA, MATH EM AT IC A.
Some problem is that many boundary problems are put in the form of systems in which the number of unknowns and the same equations. For this reason, you want to carty out the second stage of the transition from the appropriate procedure (3-4) to a system of boundary integral equation method for a new basic vector of unknown cr(jc) = C(;c)«{jc), where C(.v)-
the transition matrix from one base to another.
Depending on the type of boundary conditions (I) is reduced to a system boundary integral equations of minimum dimension of the first or second kind.
At the third stage, the analysis of the correctness of the resulting system of singular boundary integral equations using the symbol theory.
Analysis of a fundamentally important issue of the boundary conditions is greatly simplified and can be performed as follows. Determines the rank r and basic linear span of the vector lines of the matrix of the operator A: ier],£fa,...,er, j ■
The integral equation can be formulated for a certain set of bases, for example, by asking on the boundary values <x,,a(7^,1 p<r and solving the system of the method of
boundary integral equations for the remaining <j i|,„1,cf ,
T-Comm Том 10. #4-2016
For systems with piecewise constant coefficients is especially useful analysis of boundary conditions with the help of research minors of
l/2£ + (2*)("-|)/2 ¿(tf>)A, (5)
where L(GTFourier transform G1 (x) on an arbitrary hyperplane.
Nondegenerate minors of the matrix (5) corresponds to the correct system of boundary integral equation method.
At the final stage of the algorithm is formulated a system of linear algebraic equations for direct approximate numerical solution of boundary integral equation method:
(x) O K(x,y) (y)dy h{x), <6)
r
in the future, the equation (6) reduces to a system of linear algebraic equations and solved by Gauss or other suitable numerical method.
The proposed method has several advantages especially when dealing with new insufficiently investigated multidimensional boundary value problems.
in the case of classical systems of utility proposed approach is most useful in the case of non-standard boundary conditions.
It is noted that an integral representation (3) is applicable in the case w here the differential coefficients of the operator system are arbitrary piecewise continuous functions. The main problem in this case is to find the fundamental matrix function solutions.
From this display technology boundary integral equations of the user requires only minimal information about the specific properties of the system being solved (1) and (he details of the physical picture of the object being modeled.
The previously existing restrictions on the complexity of the boundary value problems are now largely removed by the existence of public symbolic computation systems.
As an example, consider the use of this technique for the reduction of the Maxwell system to a system of boundary integral equation method for the case of piece wise-homogeneous media.
Maxwell system with constant coefficients after the Laplace transform with respect to time can be written in terms of images:
rotH" =(E' + vxB") + spE' +S;(x),
rotH" --p/.iH" + Bt)(x), xeQ, for the case where E— vector of electric intensity (but not the identity matrix), E* — E at the image of the Laplace transform.
We have
u(x) = (H\E')\ F(*) = (<V(x),50(*))r.
Normal matrix can be written as: 0 - > ? , n2
/7, D -n,
Q =
a
о
о ß,
-n2 n, 0
its rank is equal io four, the coordinates of the normal vector has the following expression: m, = cos(?>)sin(.9), =sin(^)sin(5),
n. = cos [&), <p and ,9- Function Point the boundary of T,
Qu = Wo,
C =
Щ o}2 o„ w,
ж =
"c, 0'
,C.=
.ft, Q. ' 1
sinp -COSÇ0COS5 -cos<p-sin^cos<9 0 sin ,9 -cos^cost? -sin^cos<9 sin5 -sinp cos (p 0
Computer implementation in computer system of symbolical calculations
Detail of REDUCE-program to find a fundamental solution is as follows:
MATRIX A, B, GAMMA, MU , ROT, V, ML: gml I : = gml ; gm22: = gml ; gm33: = gm3; mil: = ml; m22: = ml; m33: = m3;
gml2: = gml3: = gm23: =0; m!2: = ml3: = m23: = 0; A: = MAT (
{0, -i * w3, i * w2 , -gm 11, -gm 12, -gm 13), {i * w3, 0, -i * wl, -gml2, -gm22 , -gm23), {-i * w2, i * wl , 0, -gml3, -gm23, -gm33), (p* mll,p* ml2 ,p * ml3,0, -i* w3, i * w2), {p * ml2, p* m22 , p * m23, i * w3, 0, -i * wl), {p * ml3, p * m23 , p * m33, -i * w2, i * wl, 0) );
ROT: = MAT ( (0, -i * w3, i * w2 ), (i * w3, 0, -i * wl), (-i * w2, i * wl ,0) );
GAMMA: = MAT ( {gml I, gml2, gm 13), (gml2, gm22, gm23), {gml3,gm23, gm33) );
MU: = MAT ( {mil, m 12, ml3), (ml2, m22, m23), {ml3, ni23, m33) );
V: = MAT ( {0, -v3, v2), {v3, 0, -vl), (-v2, vl, 0) );
ML: = GAMMA * V * MU ;
FOR i: = 1: 3 DO FOR j: = 1: 3 DO A (1, j): = A (i, j) -ML (i, j); A;
ON GCD;
D: = SUB {1 = -1, Det {Tp (A))); % L: = FACTOR1ZE (D) ; L: = SOLVE {D, vl ); in "g: ml.in"; D:
% B: = D * A A (-1 );
out "g: ml.out"
11: = PART (FIRST (L ), 2)
12: = PART (SECOND (L ), 2)
B;
shut "g: ml.out"; Calculation results: vl : = - o3 * x2 + o2 * x3; v2: =o3 * xl-ol * x3; v3: = -o2 * xl + ol * x2;
PUBLICATIONS IN ENGLISH
Conclusion
A new method for the reduction of linear boundary value problems to boundary integral equations is proposed. The method is effective for large scale systems and allows you to effectively automate the reduction by means of analytical calculations on computer systems.
References
1. Germain P. Lacour, PL George. F. Pistre, Bezier P. Mathematics and CAD: In 2 books. Bk. 2. World, 1989. - Pp. 264.
2. Pendant JL. Sabonnader J.-C. CAD in elektrotechnics. - M.: Mir, 1988.-208 p.
3. Mikhlin S.G. Linear equations in partial derivatives. M.: Higher, school, 1977.-431 p.
4. Aleksidze M.A. The fundamental function of approximate solutions of boundary problem. - M.: Nauka, 1991, - 352 p.
5. Shapiro Z.Ya. The first boundary value problem for elliptic systems of differential equations // Mat. sb. - 195!. - Vol, 28 (70), No. L-Pp. 55-78.
6. Lopatinsky Ya. A method of reducing boundary problems for systems of differential equations of elliptic type to regular integral equations// Ukr. Mat. zlium. - 1953. - Vol.5, No. 2. - Pp. 131-151.
7. Agranovieh M.S. Elliptic singular integro-differeniial operators // Uspekhi Mat. nauk. - 1965. - Vol. 20, No. 5 (125). - Pp.4-120.
8. Panich OA. Introduction to the general theory of elliptic boundary value problems. Kiev. Vishcha school, 1986. - 128 p.
9. Davenport J., SireJ., Tournier E. Computer Algebra. M.: Mir, 1991,-352 p.
10. Edneral V.F., Kryukov A.P.. Rodionov A. Y. Language analytical calculations REDUCE.-M.: MGU, 1988,-176 p.
11. KUmov D.M., Rudenko V.M. Methods of computer algebra in mechanics. M.; Science Problems, 1989.-215 p.
12. Mikhlin S.G. Multi-dimensional singular integral equations. M.: Fizmatgiz, 1962,-256 p.
13. Shulga N.A.. Bolkisev A.M. Fluctuations piezoelectric bodies. - Naukova Dumka, 1990. - 228 p,
14. Daletskii Y.L, Krein M.G. Stability of solutions of differential equations in a Banach spacc, M.: Nauka, 1970. - 536 p.
НОВЫЙ ПОДХОД К РЕДУКЦИИ МНОГОМЕРНЫХ ЛИНЕЙНЫХ КРАЕВЫХ ЗАДАЧ К ЭКВИВАЛЕНТНЫМ ГРАНИЧНЫМ ИНТЕГРАЛЬНЫМ УРАВНЕНИЯМ
Арутюнян Роберт Владимирович, к.ф-м.н., доцент, МТУСИ, Москва, Россия, [email protected]
Аннотация. В статье описан новый метод редукции многомерных линейных краевых задач для систем дифференциальных уравнений в частных производных первого порядка к эквивалентным граничным интегральным уравнениям. Метод эффективен для систем большой размерности и позволяет эффективно автоматизировать редукцию при помощи систем аналитических вычислений на ЭВМ. Данный метод в принципе может применяться для решения уравнений Максвелла и других подобных систем и в том в случае, когда коэффициенты материальных уравнений зависят от пространственных координат и времени. Формулы метода обобщают соотношения Гюйгенса для уравнений Максвелла, Сомилиана для задач теории упругости и другие аналогичные известные интегральные представления решений систем дифференциальных уравнений в частных производных. Предложенный алгоритм состоит из следующих этапов. Нахождение на первом этапе сингулярного решения осуществляется методами стандартных интегральных преобразований, а соответствующий алгоритм легко программируется в системах компьютерной алгебры. Некоторую проблему представляет то, что многие краевые задачи ставятся в виде систем, в которых количество неизвестных и уравнений не совпадает. По этой причине требуется осуществлять на втором этапе соответствующую процедуру перехода к системе относительно нового вектора базисных неизвестных. В зависимости от типа краевых условий исходная система редуцируется к системе граничных уравнений минимальной размерности первого или второго рода. На третьем этапе осуществляется анализ корректности получившейся системы сингулярных граничных интегральных уравнений с использованием теории символа. Анализ принципиально важной проблемы краевых условий существенно упрощается и может быть выполнен следующим образом. Определяются ранг и базисные вектора линейной оболочки строк матрицы дифференциального оператора. Для систем с кусочно-постоянными коэффициентами особенно удобен анализ краевых условий при помощи исследования миноров соответствующей матрицы, описанной в статье. Невырожденному минору этой матрицы соответствует корректная система граничных интегральных уравнений. На заключительном этапе алгоритма формулируется система линейных алгебраических уравнений для непосредственного приближенного численного решения системы граничных интегральных уравнений. В дальнейшем граничные интегральные уравнения сводятся к системе линейных алгебраических уравнений и решаются методом Гаусса или другим подходящим численным методом. Предложенная методика обладает рядом преимуществ прежде всего при решении новых недостаточно исследованных многомерных краевых задач. В случае классических систем предложенный подход наиболее полезен для случая нестандартных краевых условий. Предложенное интегральное представление применимо и в случае, когда коэффициенты дифференциального оператора системы являются произвольными кусочно-непрерывными функциями. Основной проблемой в данном случае является отыскание матрицы-функции фундаментального решения.
Ключевые слова: дифференциальные уравнения в частных производных, система, краевая задача, редукция, граничные интегральные уравнения. Литература
1. Жермен -Лакур П., Жорж П.Л., Пистр Ф., Безье П. Математика и САПР. В 2-х книгах. Том. 2. М.: Мир, 1989. 264 с.
2. Ж.-Л. Кулон, Ж.-К. Сабоннадьер. САПР в электротехнике. М.: Мир, 1988. 208 с.
3. Михлин С.Г. Линейные уравнения в частных производных. М.: Высшая школа, 1977. 431 с.
4. Алексидзе М.А. Фундаментальные функции в приближенных решениях граничных задач. М.: Наука, 1991. 352 с.
5. Шапиро З.Я. Первая краевая задача для эллиптических систем дифференциальных уравнений // Матем. сб. 1951. Том 28(70), №1. С. 55-78.
6. Лопатинский Я.Б. Об одном способе приведения граничных задач системы дифференциальных уравнений эллиптического типа к регулярным интегральным уравнениям // Укр. Мат. журн. 1953. В. 5, № 2. С. 131-151.
7. Агранович М.С. Эллиптические сингулярные интегро-дифференциальные операторы // УМН. 1965. Том 20, выпуск 5(125). С. 3-120.
8. Панич О.И. Введение в общую теорию эллиптических краевых задач. Киев.: Вища школа, 1986. 128 с.
9. Дэвенпорт Дж., Сирэ И., Турнье Э. Компьютерная алгебра. М.: Мир, 1991. 352 с.
10. Еднерал В.Ф., Крюков А.П., Родионов А.Я. Язык аналитических вычислений REDUCE. М.: МГУ, 1988. 176 с.
11. Климов Д.М., Руденко В.М. методов компьютерной алгебры в механике. М.: Наука, 1989. 215 с.
12. Михлин С.Г. Многомерные сингулярные интегралы и интегральные уравнения. М.: Физматгиз, 1962. 256 с.
13. Шульга Н.А., Болкисев А.М. Колебания пьезоэлектрических тел. Киев: Наукова думка, 1990. 228 с.
14. Далецкий ЮЛ., Крейн М.Г. Устойчивость решений дифференциальных уравнений в банаховом пространстве. М.: Наука, 1970. 536 с.
T-Comm ^м 10. #4-2016
