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Л.Н. Гумилев атындағы Еуразия ұлттық университетiнiң

ХАБАРШЫСЫ BULLETIN

of L.N. Gumilyov Eurasian National University

ВЕСТНИК

Евразийского национального университета имени Л.Н. Гумилева

МАТЕМАТИКА. КОМПЬЮТЕРЛIК ҒЫЛЫМДАР. МЕХАНИКАсериясы

MATHEMATICS. COMPUTER SCIENCE. MECHANICS Series

СерияМАТЕМАТИКА. КОМПЬЮТЕРНЫЕ НАУКИ. МЕХАНИКА

№2(131)/2020

1995 жылдан бастап шығады Founded in 1995 Издается с 1995 года

Жылына 4 рет шығады Published 4 times a year Выходит 4 раза в год

Нұр-Сұлтан, 2020 Nur-Sultan, 2020 Нур-Султан, 2020

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Бас редактордың орынбасары Жұбанышева А.Ж.

PhD, Л.Н. Гумилев ат. ЕҰУ, Нұр-Сұлтан, Қазақстан Бас редактордың орынбасары Наурызбаев Н.Ж.

PhD, Л.Н. Гумилев ат. ЕҰУ, Нұр-Сұлтан, Қазақстан Редакция алқасы

Абакумов Е.В. PhD, проф., Париж-Эст университетi, Марн-Ла-Вале, Париж, Франция Алексеева Л.А. ф.-м.ғ.д., проф., ҚР БжҒМ Математика және математикалық модельдеу

институты, Алматы, Қазақстан

Алимхан Килан PhD, проф., Л.Н. Гумилев ат. ЕҰУ, Нұр-Сұлтан, Қазақстан Бекжан Турдыбек PhD, проф., ҚХР Шынжан университетi, Шынжан, КНР

Бекенов М.И. ф.-м.ғ.к., доцент,Л.Н. Гумилев ат. ЕҰУ, Нұр-Сұлтан, Қазақстан Гогинава У. ф.-м.ғ.д., проф., Ив. Джавахишвили Тбилиси мемлекеттiк университетi,

Тбилиси, Грузия

Голубов Б.И. ф.-м.ғ.д., проф., Мәскеу физика-техника институты (мемлекеттiк университет) Долгопрудный, Ресей

Зунг Динь ф.-м.ғ.д., проф., Информатикалық технологиялар институты, Вьетнам ұлттық университетi, Ханой, Вьетнам

Ибраев А.Г. ф.-м.ғ.д., проф., Л.Н. Гумилев ат. ЕҰУ, Нұр-Сұлтан, Қазақстан Иванов В.И. ф.-м.ғ.д., проф., Тула мемлекеттiк университетi, Тула, Ресей Иосевич А. PhD, проф., Рочестер университетi, Нью-Йорк, АҚШ

Кобельков Г.М. ф.-м.ғ.д., проф., М.В. Ломоносов атындағы Мәскеу мемлекеттiк университетi, Мәскеу, Ресей

Курина Г.А. ф.-м.ғ.д., проф., Воронеж мемлекеттiк университетi, Воронеж, Ресей Марков В.В. ф.-м.ғ.д., проф., РҒА В.А. Стеклов атындағы Мәскеу мемлекеттiк

институты, Мәскеу, Ресей

Мейрманов А.М. ф.-м.ғ.д., проф., Байланыс және информатика Мәскеу техникалық университетi, Мәскеу, Ресей

Смелянский Р.Л. ф.-м.ғ.д., проф., М.В. Ломоносов атындағы Мәскеу мемлекеттiк университетъi, Мәскеу, Ресей

Умирбаев У.У. ф.-м.ғ.д., проф., Уейна мемлекеттiк университетi, Детройт, АҚШ Холщевникова Н.Н. ф.-м.ғ.д., проф., "Станкин" Мәскеу мемлекеттiк техникалық

университетi, Мәскеу, Ресей

Шмайссер Ханс-Юрген Хабилит. докторы, проф., Фридрих-Шиллер университетi, Йена, Германия

Редакцияның мекенжайы: 010008, Қазақстан, Нұр-Сұлтан қ., Сәтпаев к-сi, 2, 402 бөлме.

Тел: +7 (7172) 709-500 (iшкi 31-410). E-mail: vest_math@enu.kz

Жауапты редактор: А.Ж. Жұбанышева

Л.Н. Гумилев атындағы Еуразия ұлттық университетiнiң хабаршысы.

МАТЕМАТИКА. КОМПЬЮТЕРЛIК ҒЫЛЫМДАР. МЕХАНИКА сериясы

Меншiктенушi: ҚР БжҒМ "Л.Н. Гумилев атындағы Еуразия ұлттық университетi" ШЖҚ РМК Мерзiмдiлiгi: жылына 4 рет.

Қазақстан Республикасыңың Ақпарат және коммуникациялар министрлiгiнде тiркелген.

27.03.2018ж. № 17000-ж тiркеу куәлiгi.

Типографияның мекенжайы: 010008, Қазақстан, Нұр-Сұлтан қ., Қажымұқан к-сi ,12/1, тел: +7 (7172)709-500 (iшкi 31-410).

c Л.Н. Гумилев атындағы Еуразия ұлттық университетi

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Prof., Doctor of Phys.-Math. Sciences, L.N.Gumilyov ENU, Nur-Sultan, Kazakhstan Deputy Editor-in-Chief Aksaule Zhubanysheva

PhD, L.N.Gumilyov ENU, Nur-Sultan, Kazakhstan Deputy Editor-in-Chief Nurlan Nauryzbayev

PhD, L.N.Gumilyov ENU, Nur-Sultan, Kazakhstan Editorial board:

Evgueni Abakumov PhD, Prof., University Paris-Est, Marne-la-Vallee Paris, France

Lyudmila Alexeyeva Doctor of Phys.-Math. Sci., Prof., Institute of Mathematics and Math- ematical Modeling Ministry of Education

and Science Republic of Kazakhstan, Almaty, Kazakhstan Alexander Iosevich PhD, Prof., University of Rochester, New York, USA Alimhan Keylan PhD, Prof., L.N. Gumilyov ENU, Nur-Sultan, Kazakhstan Bekzhan Turdybek PhD, Prof., Shenzhen University, SZU, Chinese

Makhsut Bekenov Candidate of Phys.-Math. Sci., Assoc.Prof.

L.N. Gumilyov ENU, Nur-Sultan, Kazakhstan Ushangi Goginava Doctor of Phys.-Math. Sci., Prof.

Iv. Javakhishvili Tbilisi State University, Tbilisi, Georgia

Boris Golubov Doctor of Phys.-Math. Sci., Prof., Moscow Institute of Physics and Technology (State University)

Dolgoprudnyi, Russia

ung Dinh Doctor of Phys.-Math. Sci., Prof., Information Technology Institute, Vietnam National University, Hanoi, Vietnam

Askar Ibrayev Doctor of Phys.-Math. Sci., Prof., L.N. Gumilyov ENU Nur-Sultan, Kazakhstan

Valerii Ivanov Doctor of Phys.-Math. Sci., Prof., Tula State University, Tula, Russia Georgii Kobel’kov Doctor of Phys.-Math. Sci., Prof., Lomonosov Moscow State University,

Moscow, Russia

Galina Kurina Doctor of Phys.-Math. Sci., Prof., Voronezh State University, Voronezh, Russia

Vladimir Markov Doctor of Phys.-Math. Sci., Prof., Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Anvarbek Meirmanov Doctor of Phys.-Math. Sci., Prof., Moscow Technical University of Com- munications and Informatics, Moscow, Russia

Ruslan Smelyansky Doctor of Phys.-Math. Sci., Prof., Lomonosov Moscow State University, Moscow, Russia

Ualbay Umirbaev Doctor of Phys.-Math. Sci., Prof., Wayne State University,Detroit, USA

Natalya Kholshchevnikova Doctor of Phys.-Math. Sci., Prof., Moscow State Technological University "Stankin", Moscow, Russia Hans-Juergen Schmeisser Dr. habil., Prof., Friedrich-Shiller University

Jena, Germany

Editorial address:2, Satpayev str., of. 402, Nur-Sultan, Kazakhstan, 010008.

Теl.: +7 (7172) 709-500 (ext. 31-410). E-mail: vest_math@enu.kz Responsible Editor-in-Chief: A.Zh. Zhubanysheva

Bulletin of the L.N. Gumilyov Eurasian National University. MATHEMATICS. COMPUTER SCI- ENCE. MECHANICS Series

Owner: Republican State Enterprise in the capacity of economic conduct "L.N. Gumilyov Eurasian National University"

Ministry of Education and Science of the Republic of Kazakhstan. Periodicity: 4 times a year.

Registered by the Ministry of Information and Communication of the Republic of Kazakhstan. Registration certificate

№17000-ж from 27.03.2018.

Address of printing house: 12/1 Kazhimukan str., Nur-Sultan, Kazakhstan 010008; tel: +7 (7172) 709-500 (ext.31-410).

c

L.N. Gumilyov Eurasian National University

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Зам. главного редактора Жубанышева А.Ж.

PhD, ЕНУ имени Л.Н.Гумилева, Нур-Султан, Казахстан Зам. главного редактора Наурызбаев Н.Ж.

PhD, ЕНУ имени Л.Н.Гумилева, Нур-Султан, Казахстан Редакционная коллегия

Абакумов Е.В. PhD, проф., Университет Париж-Эст, Марн-Ла-Вале, Париж, Франция

Алексеева Л.А. д.ф.-м.н., проф., Институт математики и математического моделирования МОН РК, Алматы, Казахстан

Алимхан Килан PhD, проф., ЕНУ имени Л.Н.Гумилева, Нур-Султан, Казахстан Бекжан Турдыбек PhD, проф., Шынжанский университет КНР, Шынжан, КНР Бекенов М.И к.ф.-м.н., доцент, ЕНУ имени Л.Н.Гумилева, Нур-Султан,

Казахстан

Гогинава У. д.ф.-м.н., проф., Тбилисский государственный университет имени Ив. Джавахишвили, Тбилиси, Грузия

Голубов Б.И. д.ф.-м.н., проф., Московский физико-технический институт (государственный университет), Долгопрудный, Россия

Зунг Динь д.ф.-м.н., проф., Институт информационных технологий, Вьетнамский национальный университет, Ханой, Вьетнам Ибраев А.Г. д.ф.-м.н., проф., ЕНУ имени Л.Н.Гумилева, Нур-Султан,

Казахстан

Иванов В.И. д.ф.-м.н., проф., Тульский государственный университет, Тула, Россия

Иосевич А. PhD, проф., Рочестерский университет, Нью-Йорк, США Кобельков Г.М. д.ф.-м.н., проф., МГУ имени М.В. Ломоносова, Москва, Россия Курина Г.А. д.ф.-м.н., проф., Воронежский государственный университет,

Воронеж, Россия

Марков В.В. д.ф.-м.н., проф., Математический институт им. В.А. Стеклова РАН, Москва, Россия

Мейрманов А.М. д.ф.-м.н., проф., Московский технический университет связи и информатики, Москва, Россия

Смелянский Р.Л. д.ф.-м.н., проф., МГУ имени М.В. Ломоносова, Москва, Россия Умирбаев У.У. д.ф.-м.н., проф., Государственный университет Уейна, Детройт,

США

Холщевникова Н.Н. д.ф.-м.н., проф., Московский государственный технологический университет "Станкин", Москва, Россия

Шмайссер Ханс-Юрген Хабилит. доктор, проф., Университет Фридрих-Шиллера, Йена, Германия

Адрес редакции: 010008, Казахстан, г. Нур-Султан, ул. Сатпаева, 2, каб. 402 Тел: +7 (7172) 709-500 (вн. 31-410). E-mail: vest_math@enu.kz

Ответственный редактор: А.Ж. Жубанышева

Вестник Евразийского национального университета имени Л.Н. Гумилева.

Серия МАТЕМАТИКА. КОМПЬЮТЕРНЫЕ НАУКИ. МЕХАНИКА

Собственник: РГП на ПХВ "Евразийский национальный университет имени Л.Н. Гумилева" МОН РК Периодичность: 4 раза в год.

Зарегистрирован Министерством информации и коммуникаций Республики Казакстан.

Регистрационное свидетельство №17000-ж от 27.03.2018г.

Адрес типографии: 010008, Казахстан, г. Нур-Султан, ул. Кажымукана, 12/1, тел.: +7 (7172)709-500 (вн.31-410).

c

Евразийский национальный университет имени Л.Н. Гумилева

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МЕХАНИКА СЕРИЯСЫ, №2(131)/2020

МАЗМҰНЫ

МАТЕМАТИКА-КОМПЬЮТЕРЛIК ҒЫЛЫМДАР МАТЕМАТИКА-КОМПЬЮТЕРЛIК ҒЫЛЫМДАР

Алексеева Л.А., Ахметжанова M.M. Термосерпiмдi өзек динамикасының шектiк есептерiнiң жалпылама функциялар әдiсi

8 Илолов М., Кучакшоев Х.С. Импульстiк әсерлi абстрактiлi бөлшек интегро- дифференциалдық теңдеулер

28 Провоторов В.В., Мурзабекова Г.Е., Нуртазина К.Б. Графтардағы жадылы жылу теңдеуi үшiн сәйкестендiру мәселесi

35 Муталип Р., Науразбекова А.С. Екi айнымалы өрiлген еркiн ассоциативтi алгебралардың тақ автоморфизмдерi

42

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№2(131)/2020

CONTENTS

MATHEMATICS-COMPUTER SCIENCE MATHEMATICS-COMPUTER SCIENCE

Alexeyeva L.A., M.M. Akhmetzhanova Method of generalized functions in boundary value problems of thermoelastic rod dynamics

8 Ilolov M., Kuchakshoev Kh.S. Abstract Fractional Integro-Differential Equations with Im- pulsive Actionss

28 Provotorov V.V., Murzabekova G.E., Nurtazina K.B.On solving the inverse graph problem for the heat transfer equation with memory

35 Mutalip R., Naurazbekova A.S.Odd automorphisms of two generated braided free associa- tive algebras

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НАУКИ. МЕХАНИКА, №2(131)/2020

СОДЕРЖАНИЕ

МАТЕМАТИКА-компьютерные науки МАТЕМАТИКА-КОМПЬЮТЕРНЫЕ НАУКИ

Алексеева Л.А., Ахметжанова M.M.Метод обобщенных функций в краевых задачах динамики термоупругого стержня

8 Илолов М., Кучакшоев Х.С. Абстрактные дробные интегро-дифференциальные уравнения с импульсными воздействиями

28 Провоторов В.В., Мурзабекова Г.Е., Нуртазина К.Б.О решении обратной задачи на графе для уравнения теплопереноса с памятью

28 Муталип Р., Науразбекова А.С. Нечетные автоморфизмы сплетенных свободных ассоциативных алгебр от двух порождающих

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http://bulmathmc.enu.kz, E-mail: vest_math@enu.kz

МРНТИ:30.19.33, 30.19.21

L.A. Alexeyeva, M.M. Akhmetzhanova

Institute of Mathematics and Mathematical Modelling, Almaty, Kazakhstan (E-mail: alexeeva@math.kz, mariella80@mail.ru)

Method of generalized functions in boundary value problems of thermoelastic rod dynamics

Abstract: The method of generalized functions (GFM) has been developed to solve transient and vibrational boundary value problems of thermoelastic rod dynamics using a model of coupled thermoelasticity. Thermoelastic shock waves arising in such structures under the influence of shock loads and heat flows are considered. Conditions on their fronts were obtained. The singularity of the assigned boundary tasks taking into account shock waves has been proved.

On the basis of GFM, a system of algebraic resolving equations is built for a wide class of boundary problems to determine their analytical solutions. Dynamics of the rod under the action of forces and heat sources of various types, including those described by singular generalized functions, which allow modeling the effect of pulsed concentrated sources, are studied. Computer implementation of solutions of one edge problem at stationary oscillations was carried out, results of numerical experiments of calculation of rod thermodynamics at low and high frequencies are presented. These solutions and algorithms can be used for engineering calculations of rod structures to evaluate their strength properties.

Keywords: thermoplasticity, rod, boundary value problems,stress-strain state, general func- tions method.

DOI: https://doi.org/10.32523/2616-7182/2020-131-2-8-27 Introduction. Rod structures are widely used in mechanical engineering as connecting and transmission links for structural elements of a wide variety of machines and mechanisms. During operation, they are subjected to variable mechanical and thermal stresses that create a complex stress-strain state in structural elements, depending on their temperature, and affecting their strength and reliability. Therefore, the determination of a thermal stress state of rod structures taking into account their mechanical properties (in particular, elasticity) is one of the urgent scientific and technical problems.

When studying thermodynamic processes in structures, equations of uncoupled thermoelas- ticity are usually used. In this model at first the temperature problem is solved for determining the temperature field without taking into account the deformation of medium. This reduces a problem to constructing a solution of boundary value problem (BVP) for the heat parabolic equation. After determining a temperature field, BVP of dynamics of thermo-elastic medium is solved, in which a gradient of known temperature field is introduced as a mass force in motion equations of elastic medium. This model describes thermodynamic processes well at low strain rates and is completely unsuitable for describing high-speed dynamic processes.

Here, a problem of determining a thermostressed state of a thermoelastic rod is considered, using a model of coupled thermoelasticity. In this case, a heat equation contains a divergence of a velocity of material points of a medium, and a temperature gradient is included in equations of elasticity. This connects equations into one system of differential equations of mixed type without separating a temperature field and elastic deformations.

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Note that nonstationary BVPs of coupled thermoelastodynamics by plane deformation and in 3D-space were considered by authors [1-7] and others. They elaborated analytical Boundary Integral equations Method and numerical Boundary Elements Method for construction BVP solutions in a space of Laplace or Fourier transformation over time. In [3] BIEM is based on potentials theory. In [7] BIEM was elaborated by use General Functions Method which is essentially convenient for solving hyperbolic and mixed problems of mathematical physics. The base ideas of this method are presented in paper [8].

Here we elaborate this method for solving non-stationary BVPs and stationary vibrations problems of dynamics of a thermoelastic rod under the action of power and heat sources of various types, including those described by singular generalized functions. The latter allows to simulate the impact of pulsed concentrated sources of various types. Thermal shock waves that arise in such structures under action of shock loads and heat fluxes are considered, and conditions at their fronts are obtained. Uniqueness of posed boundary value problem is proved, subject to shock waves. Based on GFM, algebraic resulting equations system for wide class of boundary value problems have been constructed for determination of analytical solutions of BVPs. As example the computer implementation of solutions of one BVP was carried out by stationary oscillations at low and high frequencies The results of some computer experiments have been presented.

1. Statement of non-stationary boundary value problems of connected thermoe- lasticity. A thermoelastic rod of length 2L are considered, which is characterized by a density ρ, rigidity EJ, and thermoelastic constants γ, η and κ [1,2].The movement of the cross sections of the rod and the temperature field of the rod is described by a system of hyperbolic-parabolic equations of the form:

ρc2u,xx−ρu,tt−γθ,x+ρF1 = 0,

θ,xx−κ−1θ,t−ηu,xt+F2= 0. (1.1) Here u(x, t) are the components of the longitudinal displacements, θ(x, t) are the relative tem- perature (θ=T(x, t)−T(x,0)), T are absolute temperature, F1 are a longitudinal component of acting forces; a velocity of thermoelastic waves propagation c =

qEJ

ρ . An action of heat sources describes by the function F2 = (λ0κ)−1W(x, t), where W are amount of released (or absorbed) heat per unit volume per unit time, λ0 is a thermal conductivity coefficient.

We suppose that functions F1(x, t), F2(x, t) belong to a space of generalized functions (dis- tributions) of slow growth S [9], that allows us to simulate thermodynamic processes in rods under action of various types of concentrated heat sources. Hereinafter, we use the notation for partial derivatives: ui,j= ∂ui/∂xj = ∂jui. Thermoelastic stress in the rod is determined by the Duhamel-Neumann relation [1,2]:

σ=ρc2u,x−γθ (1.2)

We consider a number of direct boundary value problems of thermoelasticity whose solutions satisfy the following initial and boundary conditions. Initial conditions (Cauchy conditions): at t= 0 the displacement, velocity and temperature are known:

u(x,0) =u0(x), θ(x,0) =θ0(x), |x| ≤L;

tu(x,0) = ˙u0(x), |x|< L (1.3) Boundary conditions at the rod ends (x=x1 =−L, x=x2=L) depend on BVP type. Here at first we consider four classic BVPS.

BVP I. A displacement and temperature at rod ends are known:

u(xj, t) =wj(t), θ(xj, t) =θj(t); j= 1,2 (1.4) BVP II. Stresses and heat fluxes at rod ends are known:

Bulletin of L.N. Gumilyov ENU. Mathematics. Computer science. Mechanics series, 2020, Vol. 131, №2

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σ(xj, t) =pj(t), θ,x(xj, t) =qj(t); j= 1,2 (1.5) BVP III. A displacement and heat fluxes at rod ends are known:

u(xj, t) =wj(t), θ,x(xj, t) =qj(t); j= 1,2 (1.6) BVP IV. Stresses and temperature at rod ends are known:

σ(xj, t) =pj(t), θ(xj, t) =θj(t); j= 1,2 (1.7) It is assumed that the boundary functions satisfy the following smoothness conditions:

uj(t)∈C(0,∞), θj(t)∈C(0,∞), qj(t)∈L1(0,∞), pj(t)∈L1(0,∞) (1.8) and are regular functions from S0 R1

.

Remark. By η= 0 it is the model of uncoupled thermoelasticity, by γ = 0 the first equation (1.1) is the motion equation of elastic rods.

2. Shock thermoelastic waves as generalized solutions of motion equations. The system of equations (1.1) has mixed hyperbolic-parabolic type. Due to a hyperbolic personality, it’s possible an occurrence of thermoelastic shock waves by cause shock effects at ends of a rod.

To derive shock waves, we consider Eqs (1.1) and their solutions in a space of distributions S0. Let u(x, t), θ(x, t) are classic solution of Eqs(1.1). We consider them as regular distributions, which are differentiable between fronts of shock waves, where there derivatives are discontin- ues . According to the rules of differentiation of such generalized functions [9], Eqs (1.1) for thermoelastic shock waves take the form in S0:

ρc2u,xx−ρu,tt−γθ,x+F1+

ρc2u,x−γθ

νx−ρ[u,tt

δF(x, t)+

+∂x

ρc2u

δF(x, t)−∂t[ρu]δF(x, t) = 0,

θ,xx−κ−1θ,t−ηu,xt+F2+∂x[θ]νxδF + [θ,xxδF

κ−1θ+ηu,x

νtδF −∂t[ηu]νxδF = 0.

(2.1) Here, the square brackets denote the jump of functions indicated in them at the fronts of shock waves, δF(x, t) is singular generalized function – a simple layer on characteristic surface F in the set D={(x, τ) :|x|< L, τ < t}, on which derivatives have jumps. As follow from (1.1), the next determinant vanishes on F:

ρ c2νx2−νt2 0 νx2 −ηνtνx

=−ηρνtνx c2νx2−νt2

= 0 (2.2)

where ν = (νx, νt) are the normal to F in D. It follows from (1.7) that the lines x=const and t=const are characteristic surfaces for equations (1.1), and for shock waves (Ft):

νt=−c|νx| (2.3)

Here the wave front Ft has a simple form:

Ft=

(x, t) :x±ct) =x0

It is the point of derivatives discontinuity which moves at a speed c from the point x0, where it is formed, in one direction along the rod or another.

As in a domain of differentiability, shock waves are solutions of Eqs. (1.1), from (2.1), taking into account (1.2), to be generalized solution of (1.1) it’s necessary to perform next equalities:

ρc2u,x−γθ

νx−ρ[u,tt

δF +∂x ρc2u

δF −∂t{[ρu]δF}= 0

x{[θ]νxδF}+ [θ,xx

κ−1θ+ηu,x

νtδF = 0 (2.4)

From (2.4), taking into account (2.3), it follows that at the fronts of shock waves the following conditions for jumps must be satisfied:

[u]F

t = 0, [σ]F

t =−ρc[ ˙u]F

t (2.5)

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[θ]Ft = 0, [θ,x]F

t =η[ ˙u]Ft (2.6)

The first condition (2.5) is continuity of displacements which is necessary to conserve contin- uity of a medium. The second condition describes a stress jump (shock), which leads to a jump in velocity at the wave front. From the first and second conditions (2.6) it follows that the temperature is continuous at the wave fronts but a heat flux has a jump proportional to a jump in displacements velocity at wave front.

From these relations follow that a jump in a heat flux in the rod also forms a thermoelastic shock wave, since it causes a jump in velocities at the front, which leads to a jump in stresses on it. Such thermo-shock waves are always formed at the ends of rod if, until a fixed point in time, it was in a static state, and then non-zero stresses or heat fluxes , applied to it at the ends, create thermoelastic shock waves.

3. Uniqueness of BVP solution subject to shock waves. We show uniqueness of the solution of the initial-boundary value problem in presence of shock waves. It is assumed that at each fixed point in time, the domain of solution determination with respect to x are divided into a finite number of intervals between the fronts of shock waves Ftk at which the solution is continuous and differentiable according to (2.1). Denote an energy density of a rod

E(x, t) = 0,5n

ρ(u,t)2+c2(u,x)2+γ(ηκ)−1θ2o and power of internal forces:

M(x, t) =u,t c2u,x−γθ

+ηγ−1θθ,x.

Further we assume kνk= 1. From (2.2) it follows:ν = (νx, νt) = (1,−c)/√

1 +c2. The following theorem is true.

Theorem 1(law of conservation of energy)

L

Z

−L

(E(x, t)−E(x,0))dx=

t

Z

0

dt

L

Z

−L

u,tF1+ηγ−1θF2 dx+

+ Zt

0

(M(L, t)−M(−L, t))dt−ηγ−1 Zt

0

dt ZL

−L

(θ,x)2dx .

Proof. We fix an arbitrary time t >0. Multiplying the first equation (1.1) in the field of dif- ferentiability by u,t, and the second equation by αθ, after a series of equivalent transformations, we obtain the equalities:

ρc2u,tu,xx−u,tu,tt−γu,tθ,x+ρF1u,t= 0⇒

x u,t ρc2u,x−γθ

−0,5∂t

n

(u,t)2+ρc2(u,x)2 o

+γu,txθ+u,tρF1 = 0;

θθ,xx−κ−1θθ,t−ηθu,xt+θF2= 0⇒

−0,5κ−1tθ2+∂x(θθ,x)−ηθu,xt−(θ,x)2+θF2= 0 Folding them, we have

x u,t ρc2u,x−γθ

+αθθ,x

−0,5∂tn

ρ(u,t)2+c2(u,x)2+ακ−1θ2o

−α(θ,x)2+ +θu,xt(γ−αη) +u,tρF1+αθF2= 0.

where α=η/γ. As a result, we obtain the equality:

tE(x, t)−∂xM(x, t) +ηγ−1(θ,x)2 =u,tF1+ηγ−1θF2 (3.1)

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Lets integrate (3.1) over D with allowance for the division of integration region by the fronts of shock waves Fk(x, t) into subdomains where the solution is differentiable. As a result, using the Ostrogradsky-Gauss theorem, we obtain the following integral equality:

L

R

−L

(E(x, t)−E(x,0))dx+α

t

R

0

dt

L

R

−L

(θ,x)2dx=

=

t

R

0

(M(L, t)−M(−L, t))dt+

t

R

0

dt

L

R

−L

(u,tρF1+αθF2)dx+

+ (

R

Fk

P

k

xM(x, t)−νtE(x, t)]F

kdS(Fk) )

.

(3.2)

We show that, due to conditions at the fronts of shock waves (2.5)-(2.6), the jumps on the right- hand side of this equality are equal to zero. To do this, we make a series of transformations:

[M(x, t)]F

k = [u,tσ]F

k +α[θθ,x]F

k =u,t[σ]F

k+[u,t]F

k +αθ[θ,x]F

k =

= σ+−ρcu,t+γθ [u,t]F

k =ρc cu,+x −u,t [u,t]F

k

(here the signs in the upper index indicate the values of the corresponding functions on the right or left side of the wave front). Consequently,

p1 +c2xM(x, t)−νtE(x, t)]F

k = [M(x, t) +cE(x, t)]F

k =

=ρc cu,+x −u,t [u,t]F

k−ρc[u,t]F

k(cu,x−u,+t ) =ρc[cu,x+u,t] [u,t]F

k = 0 since, in virtue (2.5),

[cu,x+u,t] = 1 ρc

ρc2u,x−θ

Ft+ρc[ ˙u]Ft

=

= 1

ρc[σ+ρcu]˙ F

t = 0 Therefore, from (3.2) we obtain the formula of the theorem.

Theorem 2. The solutions of BVPs I-IV are unique.

Proof. We carry out the opposite. Let there exist two solutions of the considered BVP from the stated ones. Then their difference, by virtue of linearity, will also be a solution of (1.1) for Fj = 0, j = 1,2, and satisfy zero initial and boundary conditions. We write the energy conservation law for such solution. According to Theorem 1:

L

Z

−L

E(x, t)dx+ηγ−1p

1 +c−2)

t

Z

0

dt

L

Z

−L

(θ,x)2dx=

t

Z

0

(M(L, t)−M(−L, t))dt But

t

Z

0

M(±L, t)dt=

t

Z

0

u,t(±L, t)σ(±L, t) +ηγ−1θ(±L, t)θ,x(±L, t) dt= 0

since by one of the factors in each integrand is equal to zero, due to the zero boundary conditions of any BVP. Therefore

L

Z

−L

E(x, t)dx+ηγ−1

t

Z

0

dt

L

Z

−L

(θ,x)2dx= 0

Due to the zero initial conditions and the positive definiteness of the integrands, we obtain u≡0, θ≡0. Then decisions are coincided. The theorem is proved.

4. Generalized solution of BVP. To determine the solution, we pose a boundary value problem in the space of two-dimensional generalized vector functions

S20(R2) ={fˆ= ( ˆf1(x, t),fˆ2(x, t)), (x, t)∈R2, fˆj ∈S0(R2), j= 1,2}

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Their components are generalized functions which belong to S0(R2) [3]). To do this, we intro- duce a generalized regular vector function (mark them with a hat):

(ˆu1,uˆ2) ={ˆu,θ}ˆ ={u(x, t)H (x)H(t), θ(x, t)H (x)H(t)}

Here (u1, u2) = (u(x, t), θ(x, t)) are the solution of BVP, H(x) are the Heaviside function.

In S20(R2) vector-function (ˆu1,uˆ2) satisfies to the next system:

c21u,ˆxx−ˆutt^γθ,ˆx+ ˆF1 =− {u˙0(x)δ(t) +u0(x)δ0(t)}H(L− |x|)+

+c21H(t){(p1(t)−γθ1(t))δ(x+L)−(p2(t)−γθ2(t))δ(x−L)}+ +c21H(t){u1(t)δ0(x+L)−u2(t)δ0(x−L)},

θ,ˆxx−κ−1θ,ˆt−ηˆu,xt+ ˆF2=

=H(t)δ(L+x) (q1(t)−ηu˙1(t))−H(t)δ(L−x) (q2(t)−ηu˙2(t)) + +

θˆ1(t)H(t)δ0(L+x)

θˆ2(t)H(t)δ0(L−x)

−κ−1θˆ0(x)δ(t)H(L− |x|)−

−ηδ(t)H(L− |x|)∂x0(x)−ηu1(0)δ(t)δ(L+x) +ηu2(0)δ(t)δ(L−x)

(4.1)

Here are δ(t) is singular delta - function, ^γ =γ/ρ.

Using the property of the matrix of fundamental solutions Uˆjk(x, t), the solution of Eqs(4.1) can be written as following tensor-functional convolution:

u(x, t)H(t)H(L− |x|) = ˆF1∗Uˆ11+ ˆF2∗Uˆ12+ +c2

2

P

k=1

(−1)k+1n

(pk(t)−γθk(t))∗

tU11(x+L, t) +uk(t)∗

tU11,x(x+L, t) o

+ +H(t)

2

P

k=1

(−1)k+1(qk(t)−ηu˙k(t))∗

t

12 x−(−1)kL

k(t)H(t)∗

tU12,x(x+L)−

−n

˙ u0(x)∗

x

11(x, t) +u0(x))∗

x

11,t(x, t o

H(L− |x|)−

−η u1(0)U12(L+x, t) +η u2(0)U12(x−L, t)−

−κ−1θ0(x)H(L− |x|)∗

xU12−ηH(L− |x|)∂x0(x)∗

xU12

(4.2)

θ(x, t)H(t)H(L− |x|) = ˆF1∗Uˆ21+ ˆF2∗Uˆ22+ +c2

2

P

k=1

(−1)k+1n

(pk(t)−γ θk(t))∗

tU21(x+L, t) +uk(t)∗

tU21,x(x+L, t)o + +H(t)

2

P

k=1

(−1)k+1(qk(t)−ηu˙k(t))∗

t

22 x−(−1)kL

k(t)H(t)∗

tU22,x(x+L)−

−n

˙ u0(x)∗

x

21(x, t) +u0(x))∗

x

21,t(x, t)o

H(L− |x|)−

−ηu1(0)U22(L+x, t) +ηu2(0)U22(x−L, t)−

−κ−1θ0(x)H(L− |x|)∗

xU22−ηH(L− |x|)∂x0(x)∗

xU22

(4.3)

The matrix of fundamental solutions Uij(x, t) (i, j = 1,2) is solution (1.1) for singular F = (F1, F2) =δijδ(x)δ(t)

δij is Kronecker symbol. The integral record of convolutions (4.2), (4.3) has the next form:

u(x, t)H(|x| −L)H(t) = ˆF1∗Uˆ11+ ˆF2∗Uˆ12+

+c2H(t)

2

X

k=1

(−1)k+1

t

Z

0

{(pk(τ)−eγθk(τ))U11(x−(−1)kL, t−τ)+

+uk(τ)U11,x(x−(−1)kL, t−τ) o

dτ+

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+H(t)

2

X

k=1

(−1)k+1 Zt

0

n

(qk(τ)−ηu˙k(τ))U12(x−(−1)kL, t−τ)+ (4.4) +θk(τ)U12,x(x−(−1)kL, t−τ)o

dτ−

−H(L− |x|)

L

Z

−L

˙

u0(y)U11(x−y, t) +u0(y)

U1,t1 (x−y, t)dy−

−ηu1(0)U12(L+x, t) +ηu2(0)U12(x−L, t)−

−H(L− |x|)

L

Z

−L

κ−1U12(x−y, t)θ0(y)−ηU12(x−y, t)∂y0(y) dy.

θ(x, t)H(t)H(L− |x|) = ˆF1∗Uˆ21+ ˆF2∗Uˆ22+

+c2

2

X

k=1

(−1)k+1

t

Z

0

(pk(τ)−eγθk(τ))U21(x+L, t−τ)+

+uk(t)U21,x(x+L, t−τ) dτ+

+H(t)

t

Z

0

( 2 X

k=1

(−1)k+1(qk(τ)−ηu˙k(τ))U22

x−(−1)kL, t−τ +

k(τ)U22,x(x+L, t−τ) dτ− (4.5)

−H(L− |x|)

L

Z

−L

0(y)U21(x−y, t) +u0(y))U21,t(x−y, t)+

+ηu1(0)U22(L+x−y, t) +ηu2(0)U22(x−y−L, t) dy−

−H(L− |x|)

L

Z

−L

κ−1θ0(y)U22(x−y, t) +ηU22(x−y, t)∂y0(y) dy.

For regular functions

j∗Uˆij =H(t)H(|x| −L)

t

Z

0 L

Z

−L

Fj(y, τ)Uij(x−y, t−τ)dydτ

For singular Fˆj , wich are applied in physical applications [10], the definition of convolution should be used [9].

If a rod was at rest and the temperature was constant until the initial time, then the initial conditions are zero and the formulas are simplified.

u(x, t)H(|x| −L)H(t) = ˆF1∗Uˆ11+ ˆF2∗Uˆ12+

+c2H(t)

2

X

k=1

(−1)k+1

t

Z

0

n

(pk(τ)−eγθk(τ))U11(x−(−1)kL, t−τ)+

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+uk(τ)U11,x(x−(−1)kL, t−τ) o

dτ+ (4.6)

+H(t)

2

X

k=1

(−1)k+1

t

Z

0

n

(qk(τ)−ηu˙k(τ))U12

x−(−1)kL, t−τ,

+ +θk(τ)U12,x

x−(−1)kL, t−τo dτ θ(x, t)H(t)H(L− |x|) = ˆF1∗Uˆ21+ ˆF2∗Uˆ22+

+c2

2

X

k=1

(−1)k+1

t

Z

0

(pk(τ)−eγθk(τ))U21(x+L, t−τ)+

+uk(t)U21,x(x+L, t−τ) dτ+ (4.7)

+H(t)

t

Z

0

( 2 X

k=1

(−1)k+1(qk(τ)−ηu˙k(τ))U22

x−(−1)kL, t−τ

+ +θk(τ)U22,x(x+L, t−τ) dτ

Formulas (4.6) and (4.7) determine the displacement and temperature inside the rod from the known displacements, stresses, temperature, and heat fluxes at its ends.

5. The Green matrix and its Fourier transform over time. To construct matrix of fundamental solutions of equations of coupled thermo elastodynamics analytically it’s possible only in Fourier or Laplace transform spaces over time. Fourier transformant over time of Green matrix Ukj(x, t) we constructed in [11]. It is fundamental solution of Eqs (1.1) which satisfied to radiation conditions.

Its components have the form:

1j(x, ω) = δ1jsgn(x) 2(λ1−λ2)

iωκ−1

sinx√ λ2

√λ2

−sinx√ λ1

√λ1

+p

λ1sinxp

λ1−p

λ2sinxp λ2

− (5.1)

−γδ2jsgn(x) 2(λ1−λ2)

cosxp

λ1−cosxp λ2

, j = 1,2 U˜2j(x, ω) = sgn(x)

2(λ1−λ2) n

iωηδj1

cosxp λ1

−cosxp λ2

−ω2

sinx√ λ1

√λ1 −sinx√ λ2

√λ2

δj2+ (5.2)

+c2p

λ1sinxp

λ1−p

λ2sinxp λ2

δ2jo

, j = 1,2 Here

λ1,2(ω) = ω 2c2

(ω+iγη) +ic2k−1± q

(ω+i(γη+c2k−1))2−4iωc2k−1

(5.3) the roots of the characteristic equation of system, quadratic with respect to ξ2:

∆(ξ, ω) = (ξ2−ik−1ω)(c2ξ2−ω2)−iγηξ2ω=c2 ξ2−λ1

ξ2−λ2

They depend on only three thermodynamic parameters of the medium:

c, α =γη, β =c2k−1, dimension [α] = [β] = [ω]. In these options

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λ1,2(ω) = ω 2c2

ω+i(α+β)± q

(ω+i(α−β))2−4αβ

(5.4) Their frequency asymptotic behavior is as follows:

a) at ω→ ∞:

λ1 ∼ ω2

c2, λ2∼ iω β

c2 , (5.5)

b) at ω→0:

λ1sim3iω (α+β)

2c2 , λ2∼ iω(α+β)

2c2 . (5.6)

Riemann surface of the matrix ω are univalent, since the values of the components U˜kj are independent of the choice of the sign of the radicals p

λj(ω).

The features U˜2j(x, ω) are clearly demonstrated in figure 1, where the calculations of this matrix are presented for the following conditional parameters: γ = 0.1, c= 1, k= 1, η = 1 the real (blue line) and imaginary part (green line) of each component are shown here.

Remark. Matrix U˜2j(x, ω) may be used also by solving BVPs of harmonic vibrations by action of periodic over time external forces and thermo-sources.

6. Laplace transforms over time of Green matrix. To solve non-stationary boundary value problems, we should use the Laplace transform of the fundamental matrix U¯1j(x, p), which is obtained using the connection between the Fourier transform and the Laplace transform in time (p↔ −iω, ω ↔ ip ):

1j(x, p) = δj1sgn(x) 2(λ1−λ2

×

−pκ−1

sinx√ λ2

√λ2 − sinx√ λ1

√λ1

+p

λ1sinxp

λ1−p

λ2sinxp λ2

−γδj2sgn(x) 2(λ1−λ2)

cosxp

λ1−cosxp λ2

, j = 1,2

2j(x, p) =− sgn(x) 2(λ1−λ2

×

pηδj1

cosxp

λ1−cosxp λ2

−p2

sinx√ λ1

√λ1 −sinx√ λ2

√λ2

δ2j+ +c2p

λ1sinxp

λ1−p

λ2sinxp λ2

δ2j

o

, j = 1,2 where

λ1,2(p) =− p 2c2

p+α+β± q

(p+ (α−β))2+ 4αβ

The components U˜kj(x, p) are regular and continuous at the point x= 0:

kj(±0, ω) = ˜Ukj(0, ω) = 0, k, j = 1,2, (6.1) But their derivatives

x1j(x, ω) = (λ1−iωκ−1)

1−λ2)

cosxp

λ1−cosxp λ2

+ cosxp λ2

sgn(x) 2 δ1j

+ γ

2(λ1−λ2)

1sin|x|p

λ1−p

λ2sin|x|p λ2

δ2j

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x2j(x, ω) =−δ1jiωη √

λ1sin|x|√

λ1−√

λ2sin|x|√ λ2

2(λ1−λ2) +

−δj2sgn(x)

ω2−λ1c2 2(λ1−λ2)

cosxp

λ1−cosxp λ2

−c2cosxp λ2

at this point suffers a break of the first kind:

x1j(±0, p) =±0,5δ1j, ∂x2j(±0, p) =±0,5c2δj2, j = 1,2 (6.2) (the upper sign corresponds to the left limit at zero, the lower right).

Remark. By η = 0 matrix U˜kj(x, p) is fundamental for equations of uncoupled thermoelas- todynamics. In this case its original has be constructed in [12].

7. Laplace transform of boundary value problems solution. Here we consider the initial boundary value problem with zero initial conditions. By use the property of Laplace transform of convolution we get Laplace transformants of generalized solution from (4.2)-(4.3):

¯

u(x, p)H(|x| −L) = ¯F1(x, p)∗U¯11(x, p) + ¯F2(x, p)∗U¯12(x, p)+

+c2

2

P

k=1

(−1)k+1

¯

pk^γθ¯k11(x−(−1)kL, p) + ¯uk(τ) ¯U11,x(x−(−1)kL, p) + +

2

P

k=1

(−1)k+1

(¯qk−ηp¯uk) ¯U12 x−(−1)kL, p

+ ¯θk12,x x−(−1)kL, p

(7.1)

θ¯(x, p)H(L− |x|) = ¯F1(x, p)∗U¯21(x, p) + ¯F2(x, p)∗U¯22(x, p)+

+c2 P2 k=1

(−1)k+1

(¯pk−γθk) ¯U21(x+L, p) + ¯uk21,x(x+L, p) + +H(t)

2 P

k=1

(−1)k+1(¯qk−ηp¯uk) ¯U22 x−(−1)kL, p

+ ¯θk22,x(x+L, p)

(7.2)

Here, a dash over a function indicates its Laplace transform.

Using the asymptotic properties of the fundamental matrix U¯ji at zero (6.2), from (7.1)- (7.2) we obtain the system of four linear equations at the boundary points to determine the Laplace transformants of unknown boundary functions, respectively to considered BVP. It has the following form:

0,5¯u(−L, p) = F¯1

x

11+ ¯F2

x

12

x=L+ +c2

2

X

k=1

(−1)k+1n

¯

pk(p)− ^γ θ¯k(p)U¯11(−L−(−1)kL, p)+

+¯uk(p) ¯U11,x(−L−(−1)kL, p)o

+ (7.3)

+

2

X

k=1

(−1)k+1n

(¯qk(p) +iωη¯uk(p)) ¯U12

−L−(−1)kL, p

+ +¯θk(p) ¯U12,x

−L−(−1)kL, p o

.

−0,5¯u(L, p) = F¯1

x

11+ ¯F2

x

12

x=L+ +c2

2

X

k=1

(−1)k+1n

¯

pk(p)−˘γθ¯k(p)U¯11(L−(−1)kL, p)+

+¯uk(p) ¯U11,x(L−(−1)kL, p) o

+ (7.4)

Bulletin of L.N. Gumilyov ENU. Mathematics. Computer science. Mechanics series, 2020, Vol. 131, №2

Сурет

Table 1 presents the maximum amplitudes of displacements and temperature in the consid- consid-ered frequency range
Figure 1. a,b - Components U ˜ k j (x, w) : w = 1 (γ = 0.1, c = 1, k = 1, η = 1)
Figure 3. Amplitude (a) of displacements and their real and imaginary parts (b) along the rod:
Figure 4. The amplitude (a) of temperature fluctuations and its real and imaginary part (b) along the rod: ω =1
+6

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