Hilbert stochastic problem for bianalytical functions on the common contour line Текст научной статьи по специальности «Математика»

MATHEMATICS

HILBERT STOCHASTIC PROBLEM FOR BIANALYTICAL FUNCTIONS ON THE COMMON CONTOUR LINE

Aleksandr M. Volodchenkov,

Smolensk Branch of Plekhanov Russian University of Economics, Smolensk, Russia, alexmw2012@yandex.ru

Alexey V. Yudenkov, Keywords: Hilbert boundary value problem,

Smolensk Academy of Agricultural Sciences, Smolensk, Russia, bianalytical function, elasticity theory,

aleks-ydenkov@mail.ru characteristic operator.

The article studies one of the major tasks of the theory of complex variable function - Hilbert problem for bianalytical functions. By the present time there has appeared a number of original studies aimed at solving the above problem within the determined and stochastic paradigm. In this article Hilbert stochastic problem for bianalytical functions is for the first time being solved for arbitrary shape different form circumference. The task is all the more urgent because Hilbert stochastic problem in its common form has not been solved yet. Meanwhile the given problem has many applications in the continuum mechanics. Also it summarizes the practically important Dirichlet and Schwarz problems. To solve the stochastic Hilbert problem in its common form, the article offers a mathematical apparatus based on Z-bianalytical functions and their properties. The core idea of the solution is that Hilbert stochastic problem comes down to the equivalent system of two stochastic problems of Dirichlet. Then the solution of the above task for bianalytical functions can be expressed with the terms of BM (Brownian motion). Also the classical theory of boundary value problems for analytical functions has been applied

The basic result of the research work is considered to be a development of the general algorithm for Hilbert stochastic problem for bianalytical functions. Besides, there have been found the conditions for solvability of the task. Also the work presents some examples of the algorithm application to the theory of elasticity of a homogeneous isotropic body as well as a body of lineal anisotropy of the common type.

Information about authors:

Aleksandr M. Volodchenkov, Smolensk Branch of Plekhanov Russian University of Economics, Head of Humanities and Sciences Dep., Candidate of Physics and Mathematics, Smolensk, Russia

Alexey V. Yudenkov, Smolensk Academy of Agricultural Sciences, Head of IT and Higher Mathematics Dep., Doctor of Physics and Mathematics, Professor, Smolensk, Russia

Для цитирования:

Володченков А.М., Юденков А.В. Стохастическая задача гильберта для бианалитических функций на контуре общего вида // T-Comm: Телекоммуникации и транспорт. 2017. Том 11. №8. С. 69-72.

For citation:

Volodchenkov A.M., Yudenkov A.V. (2017). Hilbert stochastic problem for bianalytical functions on the common contour line. T-Comm, vol. 11, no.8, рр. 69-72.

7TT

1. Introduction.

Bi-analytic functions are natural generalization of analytic functions. For the first time boundary value problems for bi-analytic functions were formulated by O.V. Kolosov and N.I. Muskhelishvili. They demonstrated thai these simulate basic problems for the theory of elastic isotropic solids [4]. Systematic study of the boundary problems for bi-analytic functions and their generalization started in the fifties of the XX century after F.D. Gahov formulated three main types of the boundary problems for h-analytic functions [1]. The first type corresponds to the first basic problem of polyharmonie function, the second -to the second basic problem of polyharmonie function and the third generalizes Rit|uier problem.

Today there exist two approaches to the boundary value problems for bi-analytic functions and their generalizations. The first makes use of Cauchy singular operator and its properties. The method allows for studying boundary value problems for bi-analytic functions and their generalizations in the deterministic model [2, 61. At the present time the probabilistic approach to the boundary value problems offered by S. Kakutani is actively developing as well [5]. Using this approach it has become possible to explore boundary problems for bi-analytic functions and their generalizations in the stochastic formulation [8]. Nevertheless, the main task of the boundary value theory - Hilbert problem on the arbitrary shape - has not been analyzed yet.

I'll is can be explained by the fact that Brown i an motion is described by the elliptical differential operator of the second kind. Bi-analytic functions appear to be solutions for the differential equation of the second kind. So, further investigation is needed in order to relate a bi-analytic function with certain parameters of Brownian motion.

Notations and definitions. Let us set up notations and definitions that will be in force throughout the paper.

C - extended plane of a complex variable z.

D - domain (finite or infinite) in the plane C.

L - boundary line of the domain D.

Z = X - ty _ complex conjugate value.

t - fixed point on the boundary L.

i — current point on the boundary.

We assume that the origin of coordinates resides in the finite domain D. Let us refer to the classical determination of a bi-anaiytic function [9].

Determination I. We call a bi-analytic function of some domain D the following function

F(z) = (p(1(z) + zq>,{z).

(I)

Where qit(z){k=l, 2) - analytic functions of the domain D or analytic components.

We demonstrate that a bi-analytic function is a solution for the following partial differential equation

B2F(z) dz~

- 0 ■

(2)

The real part of a bi-analytic function is a biharmonic function. We turn to classical Hilbert's problem for the bi-analylie functions [1 ].

Find bi-ana!ytic in the domain D function F(z) subject to the boundary conditions

SF(t)

Re

(a„(S)-ibt(s))

a.v^py'

=c*(0

(3)

Here, bt (s), ck (s) are defined in L real-valued functions of the arc length satisfying a Holder condition together with their derivatives inclusively, besides aj {.v) + 6^(.v) * 0. Also it is

assumed that + = 1.

Considering (I), we can obtain from (3)

Where Gi(t) = _^lÊL, =

m

ak - ibk

(6)

Let us formulate the stochastic variant of the problem (3). For this wc introduce a definition of Z-bi-analytic function. Determination 2. Function

F(z)=U(x, y)W(x, y)^<p0 (z)+m(z) (5)

in the domain D is called Z-bi-analytic if for every ze D and for all open sets w, where we D_ analytical components <pa{z),(pi (z) can be written in the form of

= M{Vt (Xrw, %i ))+iM(Vk (X,a, Ytw )),<* = 0,1) Mere r, = inf{/ > 0,Z(/) g D}— the first time to exit from D of Ito process {Z(/)}; M(*) is mathematical expectation.

The functions Uk and Vk are complementary harmonic functions.

For Z-bi-analytie function the fundamental property (2) is not realized. However, if we make use of the characteristic operator

(7)

M(rK)

then we can verify correctness of the follow ing [8].

Theorem 1. Let F be a Z-bi-analytic function, then a2F = 0,

where \ft:) - [jm ^ -characteristic operator.

A/(rw)

Vice versa, if Fc.C\D) and = 0, then F is a Z-bi-analytic function.

2. Problem setting. Let us formulate Hilbert's stochastic problem for the Z-bi-anaiytic function.

Find a Z-bi-analytic function F(z)=U(x,y) + iV(x,y) in L subject the boundary condition

Re

Re

(8)

i = 1,2,

The boundary conditions (8) are satisfied almost certain.

Here td is the first exit time of bivariate Brownian process from the domain D, ct(t) are real-valued functions satisfying together with their derivatives a Holder condition in L\ ak, l\ (it = 1,2) are real-valued functions satisfying a Holder condition together with their derivatives up to the (3 - k ) -degree.

Considering (6) the boundary conditions (8) can be shown as

Refcrt, (trn ) - ))(<?; + +<?,)]= 2 c, (tJf> ),

Re[(a,.(* ) - ib2(ttB )№ + 7^<pl - <pt )] = 2c,(t,B ).

(9)

3. Solution. Let us change the boundary conditions (9) into

p'oK>+K) = G, (trn )|#(fr,) + g,(/,.)]+ e, ut„ h (10)

<Po K ) - Pi Ctc ) = C2 (ttD )k(/fn)-iM/r„)] + % (tTo >. Where

=-im-G, (tmo+sM

For the problems (10) we find regulating factors pk(i. } by

applying the corresponding stochastic Dirichlet problems for Z-analytical functions.

ly

v; (tIJ = arag^--xargrTa-ak

yja; +b;

Pk{tfiak +ibk)=Xt(t.).

We divide the both parts of the equations (10) by the corresponding regulating factors

%(tTB)=%ittnHQ,(tJ/JCt(tTJ

itn '

LIV tn>

(11)

1 (■'//> (r) dr 1 ry/t ( t) dT

2m

if

T-t

_!_ vt

2m -1

(13)

Theorem 3. The number of linearly independent solutions for a uniform problem I and the number of solvability conditions

for a non-uniform problem n are connected with the indices of the boundary conditions by the correlation

l-n- Ind{ax +ibt) + lnd(a2 +ib2),

i.e. I lilbert's stochastic problem for the Z-bi-analytic function is Noetherian.

Examples. Let us view a few pragmatic tasks which illustrate the work of the above-analyzed method of solving Hilbert's stochastic problem for bi-analytic function.

Kolosov problem Tor isotropic body. Consider an infinite isotropic plane weakened by an elliptical hole. Let the contour of the hole be free from external stresses and let the stress state at infinity be defined by the intensification p which is a random variable.

Kolosov problem in the stochastic setting is equivalent to a mathematical problem for Z-bi-analytic functions subject to the boundary conditions on the given boundary line L

Where

The problems (11) arc the stochastic Dirichlet problems for Z-analytical functions.

The above-stated problems can undoubtedly be solved. Their single solutions we put as [5]:

Re[4\(z)]=M(ft(z)/^(z)).(k=1,2) (12)

Where M(.) is mathematical expectation. It must be pointed out that formulae (12) are determinate. We continue the analytic functions *VAz) to the boundary

line of L applying the Sokhotski-Plemelj formulas: I 1 rMryr

„ ÔF Re—

dx

t

Im—

By

, ieL.

(15)

Let us transfer the boundary conditions (15) from L to the unit circumference I\ We remind that the mapping function is defined by the formula

(16)

then

y/(er) +

1 ( ! lHb a + b

-<P(o)+9i<r)=-

o* +m

-+am

L ' ' ---r

After a short transformation we have

¥l{t)-¥i(t)+ \A(t,T)¥i(r)dT+ = /,(/)"(54)

/ L

Where A(t,x), B(t.x) are Fredholm kernels with a weak singularity if any. f2(t) is the known function satisfying a Holder condition together with its derivative, ^,(7)^^,(2)/X2(z).

The given problem is a part of the linear boundary problems for bi-analytic functions with integrators. Its solution comes down to the settlement of a Fredholm equation of the second kind.

Therefore the following is correct.

Theorem 2, Hilbert's stochastic problem for Z-bi-analytic function comes down to settlement of the two Dirichlet stochastic problems for Z-analytic functions (11) and of the Dirichlet deterministic problem (14) for the analytic integral function.

Estimate the index of the problem (8). Basing on theorem 2, we conclude that the problem index (8) is determined by the possibility to regularize functions G, (i)- Therefore the following

is correct.

where

(\ - \ — +om 'o- J.

<p(a)-(f{a) =

cr +m

o~ +m

f " }

■—hOW

W J, J,

(17)

<p(cr)-<p(cr)

cr +m

Ht) - <p'(co{ï)lv {z) = ^(z),^) = p,(iy(i)).cT€ r. 4 2

((T) = ^ (e 2iam(tr) - w{(7))- j (Fo(cr) - w((7)).

Directly out of (12) we obtain

(18)

(40=

M(P)

2R

<-1

Find the boundary line stresses applying Kolosov-Muskhelishvili formulas.

£T„ = 4 Re

= AHP)

<p\a)

m'(a)

= M(p)Re■

(2e^a) -m)j2 -\ a ma' - 1

()-f»: +2wcos2M(£r) + 2cos(e+ M(«)>)

1 -2m cos 20 + nr

T^FT

Kolosov problem for anisotropic body. Let us consider the case of an infinite anisotropic plate with an elliptical hole free from external stresses.

Mere the first problem of the elasticity theory is equal to the following stochastic vector problem ([3], [7])

% (rB)) + % (a( r„)) + % M rp)) +

(19)

% ((a(TD )) + //2*2<<T(rfl|+

The conditions (19) are being satisfied almost certain, cr is a point on the unit circumference of the plane of the complex variable

If we apply the algorithm, developed in this paper, we have + = Re^T,(<*)(i)]=Wi)- (2°>

From here

¥ (i) = 1 r[ihM(f2) + M[ f,)]da

2 m(^-th)}-

w (|f) 1 A^M(f,) + M(J\)\da

2m(fi,-¿Or

4. Conclusion. The major result of this research appears to be the general algorithm for solving Hilbcrt's stochastic problem for the Z-bi-analytic function. Due to its generalized nature, it can be widely used for solution of the deterministic problems along with other known methods. Besides, this algorithm allows for working under the boundary conditions with random functions. This adds

to the possibility of applying the boundary value problems for bi-analytic functions to the modeling of continuum mechanics.

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3. Volodchenkov A.M., Yudenkov A.V, (2006). Model operation of primal problems of the Hat elastic theory of the homogeneous anisotropic bodies boundary value problems with shill. The Review of the applied and production mathematics, vol. 13, no. 3, pp. 482-483.

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5. Oksendal B, (2003). Stochastic differential equations. Moscow: "World", 300 p.

6. Rasulov K.M. (1993). About one common approach to the solution of classical boundary value problems for polyanalytic functions and their generalizations. Differentia/ equations, vol. 29. no. 2. pp. 320-327.

7. Redkozubov S.A., Yudenkov A.V., Volodchenkov A.M. (2006). Model operation of process of the linear deformation of a resilient homogeneous body with the help the bianalitichcskikh of functions. Bulletin of the Chuvash state pedagogical university of I. Ya Yakovlev, No, l{48), Cheboksary, pp. 128-137.

8. Yudenkov A.V., Romankov A.V, (2012). A stochastic task ofGilbert for the bianalitichcskikh of functions in the isotropic elastic theory. The Mountain informational and analytical bulletin (the scientific and technical maya/ine), no. 6^ pp. 160-164.

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References

СТОХАСТИЧЕСКАЯ ЗАДАЧА ГИЛЬБЕРТА ДЛЯ БИАНАЛИТИЧЕСКИХ ФУНКЦИЙ

НА КОНТУРЕ ОБЩЕГО ВИДА

Володченков Александр Михайлович, Смоленский филиал РЭУ им. Г.В.Плеханова, г. Смоленск, Россия, alexmw2012@yandex.ru Юденков Алексей Витальевич, Смоленская ГСХА, г. Смоленск, Россия, aleks-ydenkov@mail.ru

Дннотация

Изучается одна из основных краевых задач теории функции комплексного переменного - задача Гильберта на классе бианалитических функций. К настоящему времени вышло достаточно много оригинальных работ, посвящённых решению указанной задачи в детерминированной и стохастической постановке. Основным отличием данной статьи является то, что стохастическая задача Гильберта для бианалитических функций впервые решается в случае произвольного контура отличного от окружности. Актуальность задачи определяется тем, что до настоящего времени стохастическая задача Гильберта в общем виде не была решена. Между тем указанная задача имеет многочисленные приложения в механике сплошных сред. Также задача Гильберта является обобщением практически важных задач Дирихле и Шварца. Для решения стохастической задачи Гильберта в общем виде в работе предлагается математический аппарат, базирующийся на Z-бианалитических функциях и их свойствах.

Основной идеей решения является то, что стохастическая задача Гильберта сводится к равносильной системе, состоящей из двух стохастических задача Дирихле. После этого решение стохастической задачи Гильберта для бианалитической функции можно выразить в терминах броуновского движения. Также при решении использовалась классическая теория краевых задач для аналитических функций. Основным результатом работы можно считать разработанный общий алгоритм решения решение стохастической задачи Гильберта для бианалитической функции. Кроме того, установлены условия разрешимости задачи, рассмотрены примеры применения алгоритма для решения основной задачи теории упругости для однородного изотропного тела, а также для тела, обладающего прямолинейной анизотропией общего вида.

Ключевые слова: краевая задача Гильберта, бианалитическая функция, теория упругости, характеристический оператор. Литература

1. Гахов Ф.Д. Краевые задачи. М.: Наука, 1977. 640 с.

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3. Володченков А.М., Юденков А.В. Моделирование основных задач плоской теории упругости однородных анизотропных тел краевыми задачами со сдвигом // Обозрение прикладной и промышленной математики. 2006. Т.13.№ 3. С 482-483.

4. Мусхелишвили Н.И. Сингулярные интегральные уравнения. М.: Наука, 1968. 512 с.

5. Оксендаль Б. Стохастические дифференциальные уравнения. М.: Мир, 2003. 300 с.

6. Расулов К.М. Об одном общем подходе к решению классических краевых задач для полианалитических функций и их обобщений // Дифференциальные уравнения. 1993. т. 29. № 2. С. 320-327.

7. Редкозубов С.А., Юденков А.В., Володченков А.М. Моделирование процесса линейной деформации упругого однородного тела с помощью бианалитических функций // Вестник Чувашского государственного педагогического университета им. И.Я.Яковлева. №1(48), Чебоксары, 2006. С. 128-137

8. Юденков А.В., Романков А.В. Стохастическая задача Гильберта для бианалитических функций в изотропной теории упругости // Горный информационно-аналитический бюллетень (научно-технический журнал). 2012. № 6. С. 160-164.

9. Balk M.B. Polyanalytie functions // In Complex Analysis: Methools, Trends and Applications. Eds.: E. Lanckau, W. Tutschke. Berlin: Akademie - Verlag, 1983. С. 63-84. Информация об авторах:

Володченков Александр Михайлович, Смоленский филиал РЭУ им. Г.В.Плеханова, г. Смоленск, заведующий кафедрой естественнонаучных и гуманитарных дисциплин, к.ф.-м.н., Смоленск, Россия

Юденков Алексей Витальевич, Смоленская ГСХА, г. Смоленск, заведующий кафедрой информационных технологий и высшей математики, д.ф.-м.н., профессор, Смоленск, Россия