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MATHEMATICS
SYSTEMS OF SINGULAR INTEGRAL EQUATIONS WITH CAUCHY KERNEL IN THE HORIZONTAL ELASTICITY THEORY
Alexander M. Volodchenkov,
Smolensk Branch of RUE, Smolensk, Russia, alexmw2012@yandex.ru
Alexey V. Yudenkov,
Smolensk Academy of Agricultural Sciences, Smolensk, Russia, Keywords: systems of singular equations,
aleks-ydenkov@mail.ru Fredholm equation, Noether operator.
The subject of the study is systems of singular integral equations with Cauchy kernel, the characteristic part of which corresponds to the first major problem of the elasticity theory for isotropic body. On the one hand, the above systems are generalizing the known Sherman equations modeling the stressed state of the elastic solids. On the other hand, the singular systems are equal in a special case to the Riemann boundary value problem for bi-analytical functions.
The research is becoming more relevant as by the present time there have been no studies of the systems of the singular integral equations with Cauchy kernel corresponding to the first major problem of the elasticity theory for isotropic bodies. Additionally, the above-mentioned systems may be well applied for solving the problems of continuum mechanics with either isotropic or anisotropic behavior.
The aim of the study is to develop the general method of reducing the singular equations to the Fredholm equations of the second kind.
The paper offers a precise problem statement, sets a connection between the systems of singular integral equations and the boundary value problems for bi-analytical functions, suggests an original method of the system regularity, also looks into special cases. The main research results incorporate the general method of the equally matched regularity (reducing the systems of singular integral equations to the Fredholm equations). This method is based on the properties of the Cauchy integral and on the general properties of the Noether operators. It utilizes the Carleman-Vekua regularization method in the systems of singular integral equations with non-analytical components. Having used the regularity method, it appeared possible to determine the systems index. Further, the study demonstrates particular cases allowing for solving the system of singular integral equations in an essentially closed form.
Information about authors:
Alexander M. Volodchenkov, Smolensk Branch of RUE, 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. Системы сингулярных интегральных уравнений с ядром коши в плоской теории упругости // T-Comm: Телекоммуникации и транспорт. 2017. Том 11. №12. С. 55-59.
For citation:
Volodchenkov А.М., Yudenkov A.V. (2017). Systems of singular integral equations with cauchy kernel in the horizontal elasticity theory. T-Comm, vol. 1 1, no.12, рр. 55-59.
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1. Introduction
The paper looks inlo the systems of singular integral equations with the Cauchy kernel. The first research works on the singular integral equations and their application to the mechanics belong to A. Poincare and D. Ililbert. The new theory there were relying on the classic results of the Fredholm equations. The further progress of the singular equation theory was seen in the works by T. Carleman, I.N. Vekua, N.P. Vekua, N.I. Mushelish-vili, F.D. Gakhov and other Russian and foreign mathematicians and mechanics [1], [2]. Therein the singular equations were analyzed together with the corresponding boundary value problems. Basically these were the problems for analytical functions and analytical vectors. Recently there have appeared quite a number of original publications where classical boundary value problems and singular equations are being studied on the broader class of functions. For example, on bi-analytica! functions.
Herein we analyze the system of singular integral equations corresponding to the Riemann boundary value problem for bi-analytical function. The given system is key to the more complicated systems with a shift and conjugation and to the multivalued boundary problems for bi-analytical functions. The article contains detailed problem setting, the general method of reducing the system of singular integral equations to the system of Fredholm equations( regularity), special cases for solving in an essentially closed form. .As an example of the results application, there has been solved the first basic problem of the theory of elasticity on the circumference. The main results are formulated in the form of theorems and consequences.
2. Main part
Let us stick to the terminology introduced in the work [1]. In particular, by the index of the system of equation we understand a number equal to difference between the number of solutions to a homogeneous equation system and the number of the solvability conditions of the non-homogeneous system. Let the system be Noether if the index of it is a finite number. The regularity is reduction of the singular equation to the Fredholm equation. Let H<k,(L) be a class of functions satisfying the Helder condition together with its derivatives up to degree k including.
Let the following function be a bi-analytical function on some domain D (finite or infinite)
F(z) = (p()(z)+z(p,(z).
Here (pk(z)(k = 1,2)- analytical functions on 1) or analytical components, z = x - iy [3].
We can say that the bi-analytical function is the solution to the partial differential equation of the form
5"F{z)
8z
■ = 0 ■
The real part of the bi-analytical function is called bi-hanrionic function.
Let L be simple closed contour c® class, putting an upper
bound to the domain l): . Let the upper infinite domain be D . Analyze the system of the singular equations
(A:,(o1w,)(t)^a1{t)[co1(t) + to,,(t) + o)2(t)] + d,(t) fcol(t)+t<P2'(x) + co2(t)
ill r
T1 J
T-t
dx+ JV11(t,T)col(T)dT +
+ Jxiî(t,T)©2(T)dT = g1(t),
(i:2ùï1ro2)(t) = a,(t)[û)1(t) + tco2'(t)-o)2(t)] +
d2(t) fo)|(t) 4- tcOni(T)~o)n(T)
MtJ r
Tri J
T-t
-dx+ J^.ÎMOra^TjdT-
+ j£M(tfT)coI(T)dT = g3(t), teL
where Kk.„(t, t) e H (L x L); ak(t), dt(t)e H(j " k'(L); g k(t) e H,l)(L) (k, n = 1, 2). Let us assume that the equations of the systems are normalized, viz. ai(t)-dt(t) =1-
The characteristical part of the system (1) is equal to the classical Riemann boundary value problem for bi-analytical function [3]
<'(t)+t|4+,(t)+<p;(t) = Gl(t)[<p;'(t)+ftpp(t)+(p-(t)]+Q1(t), (2) (p;,(t)+fcpi,(t)-<it) = G2{t)[<p01{t)+ftp-,(t)-cp-(t)]+Q2(t),
where G,(t)=Mt)~dk(t)> Qk(t)= g*(t) > <p£(t) -ak(t)+dfc(t) VkW ak(t)+dk(t)
boundary values of the analytical components (p^(z) on the domain D" correspondingly.
The boundary problem (2) is in its setting alike to the Riemann boundary problem for the analytical vector, but the analysis of it is significantly complicated as it has the non-analytical component t. Additionally, the same can be said to the full about (1), generalizing the problem (2).
Let us reduce the system (1) to the system of Fredholm equations of the second kind.
Make the following preliminary transformations:
1 ft <¡>2 \T)CIT I CT&2 \r)dr ^ i r- a>2\T)t
7Tl i T-t 7ii . r-t 7ti • T-t
1 ew>2 \r)dz I d it — m * r-t m £ dr\T-t )
O)2(T)C/T.
Notice that the second summand in the right-hand side of the equation has but only a weak singularity. The first summand represents the boundary value of the singular integral of the Cauchy type.
Considering the above, the system (1) is represented in a form;
(£,0,^X0 = «l(/)[®, (t) + Ta>2 '(0+®2(0] +
rf,(i) f co ( r) + f(02 '(r) + r«>; (r)
ri J
'-dT +
T-t
+ Ja", , (r, r)®, (T)dT + \K'n(t, T)m2{T)dT = g, (i),
i I,
(K2a,m2 )(/) 3 a2 (/)[>, (f ) + Jco2 \t)~ a2 (/)] + d2(t) (63,(7)+T0>2\T)-0)2(t)
(3)
m
F
r-t
-dT +
+ Ja:2i {t, t)®, (T)dr + |a:22 (t, r )<s2(r)dT = g2 (/),
r >
where K' (t,x) = K + f AfUil & <k = i,2). + | ffJV«(/,r)-[«(i,-r) + ■ Ä(/,r)|UjiOrfr =
ju ¿L^Tv'I-t J J i ^ >
MO
7li
Introduce auxiliary functions W,(t) = Li,1(t) + to2'(t) + w2(t), WJ(t) = w1(t)+to,'(t)-cö2(t).
(4)
The characteristic pari of the system (3) bearing in mind the functions (4) is written as:
a1(t)W1(t) +
d,(t) fW,(T)dT
aj(t)W2(t)+M)f^^ = fJ(t),
7TI f T - t
(5)
where
CO,
(t)+to2'(t)+co2(t)+ jNllft,TVo1(x)dT+ Jn12 (t, t)»^ (t)dx=ff (t),
l(t) = Q,(t)+jR(t,T)Q((T)dx+©0(t)»
(8)
where ro()(t) is general solution of the homogeneous equation (7), R(t, t) is a generalized resolvent kernel of the integral equation (7), Considering the above for the function Q,{t) we have
(t)='(t)-®i(t)- J"|r (t,x)jJ<o2(T)dT+f;
+ |R{t,T)f1*(T)dT+o)1)(t).
(t)+
(9)
Substitute the expression (9) into the second integral equation (6), we have
= | /2« " fi(t) ~ ¡R(t,r)/;(r)dT - eoa(t)L
fl (t) = g, (t) - J*,, (t,T)ffl, (T)dT - ¡K'12 (t, t)Q3 (T)dT,
L I.
f:(t) = g,(t)-}^,(t,T)ro|(T)dT-J^(t,T)0)2(T)dT. L L
The system (5) can be viewed as a system consisting of the two independent characteristic singular equations. Having regularized the system (5) using Carleman method, we have
(6)
C0j (t)+to2 '(t) - e>2 (t) + jN2, (t.-rjco, (i)dT + jNn(t, x)^ (T)ck=fj (t),
1. 1.
where NkXt, x) (k= 1,2;/= 1,2) are the known Fredholm kernels, f'k(t) are the known functions. Let us transform the first
integral equation as follows:
ffl,(t)+jNII(t,T>D1eOdr = Q1(t) <7>
L
where
The equation (10) is a Fredholm equation of the second kind. In case of its solvability let us define the function cu2(t), substitute its value into the equation (9). From the equation (9) we can define the function W|(t).
Theorem 1. The system of Ihc singular integral equations with the Cauchy singularity (1) is matched to the system of the Fredholm singular integral equations (6).
In the proof of the theorem 1 we obtained a method of the equally matched regularity.
Notice that the index of the system (6) is naught. So the index of the system (1) is fully defined by the total index of the characteristic equations (5).
It holds true that:
Consequence 1. The system of the singular equation (1) is Noether. The system index is calculated from the formula
IndW = y Inda* ~dt ■ w at+dt
Let us take a special case of the system (1). Let contour L represent a unite circle. For the unite circle it is true that
- 1 t = -t
(11)
Considering the above, transform the system of the characteristic equations corresponding to the system (1).
Qt(t) = -to2'(t)-tD2(t)-jNli(tJT)«i(T)dT+f;(t)-
I.
Considering the absolute term of the equation (7) acquainted, solve the integral Fredholm equation (7) concerning the unknown function C0|(t).
d,(t) rcot(r)+t ico2'(t) + tco2(t)
iri
i
r-t
dr-
-^±fT-2a>1(T)dT = gl(t),
TT it J
(12)
Tilt
(K2a>^o2 )(f) = a, (i)[ft>, (0 + - <a2'(/) - +
d2(t) ff»,(r) + r"'a>2\t)-(02(r)
-I
r/ J
d T-
T-t
-Ml \T-^(r)dr = Sl(f),
m< 1
Considering the expressions (4) we have:
^(t^J^^t),
Tii * T -1
a2(t)W2(t) + ^p^ = f2(t), ni f t -1
(13)
where
f, =
d,(t) fco , (t)
2 m J T2
-dx + g,(t).
t =
d2(t) rw,(T)
2irit
it ,J T
The system (13) appears to be a system of the integral singular equations with the degenerate kernels. The solution of the such system reduces to the consequent solutions of the two characteristic equations and the system of the degenerated Fredholin equations of the second kind. The system (13) is regularizable if
y - ]ncj ^ Q ■ If any of the system's index is below
A* k
at + dk
naught (for example, the index y 2), then to equally regularize the system it is needed to ask for the fulfillment of the conditions
([ID
J J
K„(t,i)
t1 dt
(T)dt = J-
f2(t) t'-'dt, 0=u-,xfl)-
z2(t) LJZ2(t)
Let us formulate the result.
Theorem 2. In case when contour L represents a circumference of a unite radius, the characteristic system corresponding to the system (1) is solvable in an essential closed farm.
3. Conclusion
The paper reveals the general method of the equally matched regularity (reduction to the Fredholm equations) of the singular integral equations systems corresponding to the Riemann boundary value problem for b¡-analytical functions. The system's index has been evaluated. There have been studied certain particular
cases allowing for the solution of the system in an essentially closed case. The practical importance of the research results lies in the fact that the singular integral equations are «tied» to the certain contour to a smaller degree than the boundary value problems. This contributes to their qualitative modeling of the mechanic processes.
Applying the results of the study, we can also test a model for stability, solvability, availability of critical points. From the theoretical point of view the studied systems can be used as key for solution of multi-element boundary value problems and systems of singular integral equations w ith shifts and complex conjugate functions [4], [5], [6].
References
]. Gahov F.D. {1977). Kraevye zadachL Moscow: Nauka. 640 p.
2. Mushelishvili N.I. (1968). Singuljarnye integral'nye urav-nenija. Moscow: Nauka. 512 p.
3. Gabrinovich V.A., Sokolov LA. (1985). Ob issledovanijah po kraevym zadacham dlja polianaliticheskih funkcij. Nauchnye trudy ju-bilejnogo seminara po kraevym zadacham, posvjashhennogo 65-letiju so dnja rozhdenija akademika AN BSSR F.D.Gahova. Minsk: Izd-vo «Universitetskoe», 1985. Pp. 43-47,
4. Volodchenkov A.M., Judcnkov A.V. (2006). Modelirovanie osnovnyh zadach ploskoj teorii uprugosti odnorodnyh anizotropnyh tel kraevym i zadacham i so sdvigom. Obozrenie prikladnoj i promyshlennoj matematiki. No. 3. Pp. 482.
5. Rimskaja L.P. (2014). Sistemy singuljarnyh integral'nyh urav-ncnij so sdvigom Karlemana v teorii skleivanija upriigih poverhnostej. Vestnik Brjanskogo gosudarstvermogo universiteta. No. 4. Pp. 31-34.
6. Beshenkov S.N., Bereznjak l.S. (2014). Konechno-raznostnyj analiz nestacionamogo otryvnogo deformirovanija kontaktirujushhih jelementov konstrukcij. Sistemy komp'juternoj matematiki i ih priloz-henija. No. 15. Pp. 120-124.
СИСТЕМЫ СИНГУЛЯРНЫХ ИНТЕГРАЛЬНЫХ УРАВНЕНИЙ С ЯДРОМ КОШИ В ПЛОСКОЙ ТЕОРИИ УПРУГОСТИ
Володченков Александр Михайлович, Смоленский филиал РЭУ им. Г.В.Плеханова, г. Смоленск, Россия, alexmw2012@yandex.ru Юденков Алексей Витальевич, Смоленская ГСХА, Российская Федерация, г. Смоленск, Россия, aleks-ydenkov@mail.ru
Дннотация
Предметом исследования являются системы сингулярных интегральных уравнений с ядром Коши, характеристическая часть которых соответствует первой основной задаче теории упругости для изотропного тела. Исследуемые системы с одной стороны обобщают известные уравнения Шермана, моделирующие напряжённое состояние сплошных упругих тел. С другой стороны сингулярные системы в частном случае равносильны краевой задаче Римана для бианалитических функций. Актуальность работы обусловлено тем, что до сих пор не проведено полного исследования систем сингулярных интегральных уравнений с ядром Коши, соответствующих первой основной задаче теории упругости для изотропного тела. Так же указанные системы могут быть применены для решения задач механики сплошных сред как с изотропными, так и анизотропными свойствами.
Целью работы является разработка общего метода сведения сингулярных уравнений к уравнениям Фредгольма 2-го рода. В работе даётся точная постановка задачи, устанавливается связь между системами сингулярных интегральных уравнений и краевыми задачами для бианалитических функций, предлагается оригинальный метод регуляризации системы, рассматриваются частные случаи.
К основным результатам работы можно отнести общий метод равносильной регуляризации (сведения системы сингулярных интегральных уравнений к системе уравнений Фредгольма). Этот метод основан на свойствах интеграла типа Коши и общих свойствах нётеровых операторов. Он обобщает метод регуляризации Карлемана - Векуа на системы сингулярных интегральных уравнений с неаналитическими компонентами. Используя регуляризацию, удалось вычислить индекс системы. Так же в работе выделены случаи, позволяющие получить решение системы сингулярных интегральных уравнений в замкнутой форме.
Ключевые слова: системы сингулярных уравнений, уравнения Фредгольма, нетеровы операторы. Литература
1. Гахов Ф.Д. Краевые задачи. М.: Наука, 1977. 640 с.
2. Мусхелишвили Н.И. Сингулярные интегральные уравнения. М.: Наука, 1968. 512 с.
3. Габринович В.А., Соколов И.А Об исследованиях по краевым задачам для полианалитических функций / Научные труды юбилейного семинара по краевым задачам, посвященного 65-летию со дня рождения академика АН БССР Ф.Д.Гахова. Минск: Изд-во "Университетское", 1985. С. 43-47.
4. 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. № 3. С. 482.
5. Римская Л.П. Системы сингулярных интегральных уравнений со сдвигом Карлемана в теории склеивания упругих поверхностей // Вестник Брянского государственного университета. 2014. № 4. С. 31-34.
6. Бешенков С.Н., Березняк И.С. Конечно-разностный анализ нестационарного отрывного деформирования контактирующих элементов конструкций // Системы компьютерной математики и их приложения. 2014. №15. С. 120-124.
Информация об авторах:
Володченков Александр Михайлович, Смоленский филиал РЭУ им. Г.В.Плеханова, заведующий кафедрой естественнонаучных и гуманитарных дисциплин, к.ф.-м.н., г. Смоленск, Россия
Юденков Алексей Витальевич, Смоленская ГСХА, заведующий кафедрой информационных технологий и высшей математики, д.ф.-м.н., профессор, г. Смоленск, Росси
