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ISSN 15630277 eISSN 26174871

ӘЛ-ФАРАБИ атындағы ҚАЗАҚ ҰЛТТЫҚ УНИВЕРСИТЕТI

ХАБАРШЫ

Математика, механика, информатика сериясы

КАЗАХСКИЙ НАЦИОНАЛЬНЫЙ УНИВЕРСИТЕТ имени АЛЬ-ФАРАБИ

ВЕСТНИК

Серия математика, механика, информатика

AL-FARABI KAZAKH NATIONAL UNIVERSITY

Journal of Mathematics, Mechanics and Computer Science

№4 (116)

Алматы

«Қазақ университетi»

2022

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№766 от 22.04.1992 г.). Язык издания: английский. Выходит 4 раза в год.

Тематическаянаправленность: теоретическаяи прикладнаяматематика, механика, информатика.

Редакционная коллегия

научный редакторБ.Е. Кангужин, д.ф.-м.н., профессор, КазНУ им. аль-Фараби,

заместитель научного редактораД.И. Борисов, д.ф.-м.н., профессор, Институт математики с вычислительным центром Уфимского научного центра РАН, Башкирский государственный педагогический

университет им. М. Акмуллы, Россия

ответственный секретарьС.Е. Айтжанов, к.ф.-м.н., доцент, КазНУ им. аль-Фараби Гиви Пэйман – профессор, Университет Питтсбурга, США

Моханрадж Муругесан – профессор, Инженерный и технологический колледж Хиндустана, Индия Эльбаз Ибрагим Мохаммед Абу Эльмагд – профессор, Национальный исследовательский

институт астрономии и геофизики, Египет

Жакебаев Д.Б. – PhD доктор, профессор, КазНУ им.аль-Фараби, Казахстан Кабанихин С.И. – д.ф.-м.н., профессор, чл.-корр. РАН, Институт вычислительной математики и математической геофизики СО РАН, Россия

Майнке М. – профессор, Департамент Вычислительной гидродинамики Института аэродинамики, Германия

Малышкин В.Э. – д.т.н., профессор, Новосибирский государственный технический университет, Россия

Ракишева З.Б. – к.ф.-м.н., доцент, КазНУ им.аль-Фараби, Казахстан

Ружанский М. – д.ф.-м.н., профессор, Имперский колледж Лондона, Великобритания Сагитов С.М. – д.ф.-м.н., профессор, Университет Гетеборга, Швеция

Сукочев Ф.А. – профессор, академик АН Австралии, Университет Нового Южного Уэльса, Австралия Тайманов И.А. – д.ф.-м.н., профессор, академик РАН, Институт математики

им. С.Л. Соболева СО РАН, Россия

Темляков В.Н. – д.ф.-м.н., профессор, Университет Южной Каролины, США Шиничи Накасука – PhD доктор, профессор, Университет Токио, Япония

Индексируется и участвует:

Научное издание

Вестник КазНУ. Серия “Математика, механика, информатика”, № 4 (116) 2021.

Редактор – С.Е. Айтжанов. Компьютерная верстка – С.Е. Айтжанов ИБ N 15229

Формат60×84 1/8. Бумага офсетная. Печать цифровая. Объем 6,25 п.л.

Заказ N 13463. Издательский дом “Қазақ университетi”

Казахского национального университета им. аль-Фараби. 050040, г. Алматы, пр.аль-Фараби, 71, КазНУ.

Отпечатано в типографии издательского дома “Қазақ университетi”.

c

КазНУ им. аль-Фараби, 2022

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ISSN 1563–0277, eISSN 2617-4871 JMMCS. 4(116). 2022 https://bm.kaznu.kz

1-бөлiм Раздел 1 Section 1

Математика Математика Mathematics

IRSTI 27.31.44 DOI: https://doi.org/10.26577/JMMCS.2022.v116.i4.01

B. O. Derbissaly

Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan e-mail: derbissaly@math.kz

ON GREEN’S FUNCTION OF SECOND DARBOUX PROBLEM FOR HYPERBOLIC EQUATION

A definition and justify a method for constructing the Green’s function of the second Darboux problem for a two-dimensional linear hyperbolic equation of the second order in a characteristic triangle is given. In contrast to the (well-developed) theory of the Green’s function for self- adjoint elliptic problems, this theory has not yet been developed. And for the case of asymmetric boundary value problems such studies have not been carried out. It is shown that the Green’s function for a hyperbolic equation of the general form can be constructed using the Riemann- Green function for some auxiliary hyperbolic equation. The notion of the Green’s function is more completely developed for Sturm-Liouville problems for an ordinary differential equation, for Dirichlet boundary value problems for Poisson equation, for initial boundary value problems for a heat equation. For many particular cases, the Greens’ function has been constructed explicitly. However, many more problems require their consideration. In this paper, the problem of constructing the Green’s function of the second Darboux problem for a hyperbolic equation was investigated. The Green’s function for the hyperbolic problems differs significantly from the Green’s function of problems for equations of elliptic and parabolic types.

Key words: Hyperbolic equation, initial-boundary value problem, second Darboux problem, boundary condition, Green function, a characteristic triangle, Riemann–Green function.

Б. О. Дербiсалы

Математика және математикалық моделдеу институты , Алматы қ., Қазақстан e-mail: derbissaly@math.kz

ГИПЕРБОЛАЛЫҚ ТЕҢДЕУ ҮШIН ЕКIНШI ДАРБУ ЕСЕБIНIҢ ГРИН ФУНКЦИЯСЫ

Сипаттамалық үшбұрышта қарастырылатын екiншi реттi екi өлшемдi сызықтық гипербо- лалық теңдеу үшiн Грин функциясын құру әдiстемесi анықталды және негiзделдi. Өз-өзiне түйiндес эллиптикалық есептер үшiн Грин функциясының(жақсы дамыған) теориясынан айырмашылығы, сипаттамалық шекаралық есептер үшiн бүл теория әлi жетiк әзiрлен- бегендiгiнде. Ал симметриялық емес шекаралық есептер жағдайында мұндай зерттеулер жүргiзiлмеген. Жалпы түрдегi гиперболалық теңдеуге арналған Грин функциясын кейбiр (арнайы жолмен құрылған) көмекшi гиперболалық теңдеу үшiн Риман-Грин функциясын қолдана отырып құруға болатындығы көрсетiлдi. Грин функциясыныңтолығырақ тұжы- рымдамасы қарапайым дифференциалдық теңдеу үшiн Штурм-Лиувиль есептерi үшiн, Пуассон теңдеуi үшiн Дирихле шеткi есептерi үшiн, жылуөткiзгiштiк теңдеуi үшiн бастапқы шекаралық есептер үшiн жасалған. Көптеген дербес жағдайларда Грин функциясы айқын түрде құрылған. Алайда, басқа да көптеген есептер оларды қарастыруды талап етедi. Бұл мақалада гиперболалық теңдеу үшiн екiншi Дарбу есебiнiң Грин функциясын құру мәселесi зерттелдi. Гиперболалық есептер үшiн құрылған Грин функциясы эллиптикалық және параболалық есептер үшiн құрылған Грин функциясынан айтарлықтай ерекшеленедi.

Түйiн сөздер: Гиперболалық теңдеу, бастапқы-шекаралық есеп, екiншi Дарбу есебi, шека- ралық шарт, Грин функциясы, характеристикалық үшбұрыш, Риман–Грин функциясы.

c 2022 Al-Farabi Kazakh National University

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Б. О. Дербисалы

Институт математики и математического моделирования, г. Алматы, Казахстан e-mail: derbissaly@math.kz

О ФУНКЦИИ ГРИНА ВТОРОЙ ЗАДАЧИ ДАРБУ ДЛЯ ГИПЕРБОЛИЧЕСКОГО УРАВНЕНИЯ

Дано определение и обоснована методика построения функции Грина для второй задачи Дарбу для двумерного линейного гиперболического уравнения второго порядка, рассматри- ваемого в характеристическом треугольнике. В отличие от (хорошо разработанной) теории функции Грина для самосопряженных эллиптических задач, для характеристических граничных задач эта теория еще не подробно разработана. А для случая несимметрических граничных задач таких исследований не проводилось. Показано, что функция Грина для гиперболического уравнения общего вида может быть построена с использованием функции Римана-Грина для некоторого (специальным образом построенного) вспомогательного гиперболического уравнения. Наиболее полно понятие функции Грина разработано для задач Штурма-Лиувилля для обыкновенного дифференциального уравнения, для краевых задач Дирихле для уравнения Пуассона, для начально-краевых задач для уравнения тепло- проводности. Для многих частных случаев функция Грина была построена в явном виде.

Однако, еще многие задачи требуют своего рассмотрения. В настоящей статье исследована задача о построении функции Грина для второй задачи Дарбу для гиперболического уравнения. Функция Грина для гиперболических задач существенно отличается от функций Грина задач для уравнений эллиптического и параболического типа.

Ключевые слова: Гиперболическое уравнение, начально-краевая задача, вторая задача Дарбу, граничное условие, функция Грина, характеристический треугольник, функция Ри- мана–Грина.

1 Introduction

InS Rn let us consider some a linear differential equation

Lu(x) =f(x), x∈S, (1)

with homogeneous boundary conditions

Qu(x) = 0, x∈S. (2)

If a solution of this problem exists, is unique and can be represented in the integral form u(x) =

S

GQ(x, y)f(y)dy, (3)

then the kernel of this integral operator (3), that is, the function GQ(x, y), is called the Green’s function of problem (1), (2).

It is also said that the Green’s function for each fixed y ∈S satisfies the equation

LGQ(x, y) =δ(x−y), x∈S, (4)

and the boundary conditions (2). Here δ(x−y) is the Dirac delta function. Equation (4) should be understood in the sense of generalized functions.

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B.O. Derbissaly 5 It is known that if the operator of problem (1), (2) has eigenfunctions{uk(x)}k=1forming the Riesz basis in L2(S), then the solution of the problem can be represented as

u(x) = k=1

1

λkf, vkL2(S)uk(x), (5)

where·,·L2(S) is a scalar product inL2(S), λk are eigenvalues of the operator,{vk(x)}k=1is a biorthogonal system to{uk(x)}k=1. Formula (5) is called the spectral representation of the solution or the spectral representation of the inverse operator.

Representing the scalar product as an integral, we obtain the integral representation (3) of the solution of the problem, where

GQ(x, y) = k=1

1

λkuk(x)vk(y) (6)

is the Green’s function of problem (1), (2). In the case when problem (1), (2) is self-adjoint, the system of its eigenfunctions forms an orthogonal basis. Therefore, we can choosevk(x) = uk(x). In this case, it is easy to see from (6) that the Green’s function is the symmetric function:GQ(x, y) =GQ(y, x).

For series of characteristic problems for a wave equation and a wave equation with potential (despite the fact that these problems are solved by the method of separation of variables) all eigenvalues and eigenfunctions are constructed in the works of T. Sh.

Kal’menov [1], [2] and M. A. Sadybekov [3]- [5]. Therefore, for these problems the Green’s function can be constructed in the form of series (6). Although the presence of the Green’s function is guaranteed for any self-adjoint problem, and it can be constructed in the form of series (6), the use of infinite series for constructing a solution of the problem is not very convenient. Therefore, the construction of the Green’s function in the form of finite sums is actual.

We are interested in the integral representation of Green’s function of the second Darboux problem for a general hyperbolic equation of the second order, since all the properties of Green’s function of this problem come from the integral representation of Green’s function.

The main difference between this paper and others, that in contrast to the previous works of other authors ( [6]- [15] and others), we conduct the investigation and construction of the Green’s function without the assumption of its symmetry. Also, unlike other authors, in this paper we will give a definition of the Green’s function and a method for constructing it for the case of general coefficients.

2 Formulation of the problem

LetΩ ={(ξ, η) : 0≤ξ 1, ξ ≤η 1}. The following hyperbolic equation is considered in Ω:

2u

∂ξ∂η +a(ξ, η)∂u

∂ξ +b(ξ, η)∂u

∂η +c(ξ, η)u= f(ξ, η), (ξ, η)Ω, (7) with the initial condition

(uξ−uη)(ξ, ξ) =ν(ξ), 0≤ξ 1, (8)

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and the boundary condition

u(0, ξ) =τ(ξ), 0≤ξ 1. (9)

We will assume that a, b, aξ, bη, c, f ∈C Ω

; ν, τ ∈C1([0,1]) and

a(ξ, ξ) =b(ξ, ξ), 0≤ξ 1. (10)

In [16] it was shown that equality (10) we can always get.

Also, we assume

aξ(ξ, ξ) =bη(ξ, ξ), 0≤ξ 1. (11)

3 Green’s function of the problem (7)-(9)

Definition 1 Green’s function of the problem (1)-(3) let us call the function G(ξ, η;ξ1, η1), which for every fixed (ξ1, η1)Ω, satisfies the homogeneous equation

L(ξ,η)G(ξ, η;ξ1, η1) = 0, (ξ, η)Ω, atξ =ξ1, η =η1, η=ξ1, ξ =η1; (12) and the next boundary conditions

(Gξ−Gη)(ξ, ξ;ξ1, η1) = 0, 0≤ξ 1, (ξ1, η1)Ω; (13) G(0, ξ;ξ1, η1) = 0, 0≤ξ 1, (ξ1, η1)Ω; (14) and on the above characteristic lines, the following conditions must be met: the values of the derivatives of the Green function in directions parallel to these characteristics must coincide in adjacent regions; i.e.,

∂G(ξ1+ 0, η;ξ1, η1)

∂η +a(ξ1, η)G(ξ1+ 0, η;ξ1, η1)

= ∂G(ξ10, η;ξ1, η1)

∂η +a(ξ1, η)G(ξ10, η;ξ1, η1), at η =η1, η =ξ1; (15)

∂G(η1+ 0, η;ξ1, η1)

∂η +a(η1, η)G(η1+ 0, η;ξ1, η1)

= ∂G(η10, η;ξ1, η1)

∂η +a(η1, η)G(ξ10, η;ξ1, η1), at η =η1, η =ξ1; (16)

∂G(ξ, η1+ 0;ξ1, η1)

∂ξ +b(ξ, η1)G(ξ, η1+ 0;ξ1, η1)

= ∂G(ξ, η10;ξ1, η1)

∂ξ +b(ξ, η1)G(ξ, η10;ξ1, η1), at ξ =ξ1 ξ = η1; (17)

∂G(ξ, ξ1+ 0;ξ1, η1)

∂ξ +b(ξ, ξ1)G(ξ, ξ1+ 0;ξ1, η1)

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B.O. Derbissaly 7

= ∂G(ξ, ξ10;ξ1, η1)

∂ξ +b(ξ, ξ1)G(ξ, ξ10;ξ1, η1)at ξ =ξ1 ξ =η1; (18) and the "corner condition"

G(ξ10, η10;ξ1, η1)−G(ξ1+ 0, η10;ξ1, η1)

+G(ξ1+ 0, η1+ 0;ξ1, η1)−G(ξ10, η1+ 0;ξ1, η1) = 1. (19) must be satisfied as the regions meet at (ξ, η) = (ξ1, η1).

4 Existence and uniqueness of the Green’s function of the problem (7)-(9)

Figure 1: Splitting the domain Ω.

Theorem 1 The function G(ξ, η;ξ1, η1) that satisfies the conditions (12)-(19) exists and is unique.

Proof. To show that a function G(ξ, η;ξ1, η1), which satisfies the conditions (12)-(19) exists and unique, we divide the domain Ω into several subdomains (see Figure (1)) and consider the following problems sequentially. Let(ξ1, η1)be an arbitrary point of the domain Ω.

In the domain Ω1 ={(ξ, η) : 0< ξ < ξ1, ξ < η < ξ1}we consider the problem

L(ξ,η)G= 0, (ξ, η)Ω1; (20)

(Gξ −Gη)(ξ, ξ;ξ1, η1) = 0, 0≤ξ ≤ξ1; (21) G(0, ξ;ξ1, η1) = 0, 0≤ξ ≤ξ1, (ξ1, η1)Ω2. (22) The problem (20)-(22) is a second Darboux problem and has a unique solution

G(ξ, η;ξ1, η1)0, (ξ, η)Ω1. (23)

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In the domain Ω2 ={(ξ, η) : 0≤ξ ≤ξ1, ξ1 ≤η ≤η}let us consider the problem

L(ξ,η)G = 0, (ξ, η)Ω2; (24)

G(0, ξ;ξ1, η1) = 0, ξ1 ≤ξ ≤η1, (ξ1, η1)Ω2. (25) From (23) we have the next equality

∂G(ξ, ξ1+ 0;ξ1, η1)

∂ξ +b(ξ, ξ1)G(ξ, ξ1+ 0;ξ1, η1) = 0, 0≤ξ ≤ξ1. (26) Integrating (26) by ξ we have

G(ξ, ξ1+ 0;ξ1, η1) = exp

ξ

0

B(t, ξ1)dt

C1(ξ1, η1), 0≤ξ ≤ξ1. (27) Substituting ξ = 0in (27), using condition (14) we have that C1(ξ1, η1)0and

G(ξ, ξ1+ 0;ξ1, η1) = 0, 0≤ξ ≤ξ1. (28) The problem (24),(25),(28) is a Goursat problem and has a unique solution

G(ξ, η;ξ1, η1)0, (ξ, η)Ω2. (29)

Therefore from (29) in the domainΩ3= {(ξ, η) : 0≤ξ ≤ξ1, η1≤η 1}, we get the problem

L(ξ,η)G = 0, (ξ, η)Ω3; (30)

G(0, ξ;ξ1, η1) = 0, η1 ≤ξ 1, (ξ1, η1)Ω3; (31)

∂G(ξ, η1+ 0;ξ1, η1)

∂ξ +b(ξ, η1)·G(ξ, η1+ 0;ξ1, η1) = 0, 0≤ξ ≤ξ1. (32) Integrating (32) by ξ we have

G(ξ, η1+ 0;ξ1, η1) = exp

ξ

0

b(t, η1)dt

C2(ξ1, η1), 0≤ξ ≤ξ1. (33) Substituting ξ = 0in (33), using condition (14) we have that C2(ξ1, η1)0and

G(ξ, η1+ 0;ξ1, η1) = 0, 0≤ξ ≤ξ1. (34) Therefore, the problem (30),(31),(34) is a Goursat problem and has a unique solution

G(ξ, η;ξ1, η1)0, (ξ, η)Ω3. (35)

In the domain Ω4 ={(ξ, η) : 0≤ξ ≤ξ1, ξ ≤η ≤η1}we get the problem

L(ξ,η)G = 0, (ξ, η)Ω4; (36)

(Gξ−Gη)(ξ, ξ;ξ1, η1) = 0, ξ1 ≤ξ ≤η1. (37)

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B.O. Derbissaly 9 From (29) we have

∂G(ξ1+ 0, η;ξ1, η1)

∂η +a(ξ1, η)G(ξ1+ 0, η;ξ1, η1) = 0, ξ1 ≤η ≤η1. (38) Integrating (38) by η we get

G(ξ1+ 0, η;ξ1, η1) = exp

η ξ1

a(ξ1, t)dt

C3(ξ1, η1), ξ1 ≤η≤η1. (39) Substitutingη =ξ1 in (39), using condition (14) we have that C3(ξ1, η1)0and

G(ξ1+ 0, η;ξ1, η1) = 0, ξ1 ≤η ≤η1. (40) This problem (36),(37),(40) is a second Darboux problem and has a unique solution

G(ξ, η;ξ1, η1)0, (ξ, η)Ω4. (41)

Therefore, from (35), (41) in the domain Ω5 = {(ξ, η) : ξ1 ξ η1, η1 η 1} our problem is a Goursat problem

L(ξ,η)G= 0, (ξ, η)Ω5; (42)

∂G(ξ1+ 0, η;ξ1, η1)

∂η +a(ξ1, η)G(ξ1+ 0, η;ξ1, η1) = 0, η1 ≤η 1; (43)

∂G(ξ, η1+ 0;ξ1, η1)

∂ξ +b(ξ, η1)G(ξ, η1+ 0;ξ1, η1) = 0, ξ1 ≤ξ ≤η1; (44)

G(ξ1+ 0, η1+ 0;ξ1, η1) = 1. (45)

The problem (41)-(45) has a unique solution, and it is easy to see that its solution coincides with the Riemann-Green function, that is,

G(ξ, η;ξ1, η1) =R(ξ, η;ξ1, η1), (ξ, η)Ω5. (46) Therefore from (46) in the domain Ω6 = {(ξ, η) : η1 ξ 1, ξ η 1} we get the problem

L(ξ,η)G= 0, (ξ, η)Ω6; (47)

(Gξ −Gη)(ξ, ξ;ξ1, η1) = 0, η1≤ξ 1; (48)

∂G(η1+ 0, η;ξ1, η1)

∂η +b(η1, η)G(η1+ 0, η;ξ1, η1)

= ∂R(η1, η;ξ1, η1)

∂η +b(η1, η)R(η1, η;ξ1, η1), η1 ≤η ≤ξ1. (49) The problem (47)-(49) is a second Darboux problem and has a unique solution.

Thus, we have shown that for any (ξ1, η1)Ω and (ξ, η) Ω the Green’s function that satisfies the conditions (12)-(19) exists and unique. The theorem is proved.

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5 Construction of the Green’s function of the problem (7)-(9)

As can be seen from the proof of Theorem (1), the Green’s functionG(ξ, η;ξ1, η1) = 0 in the domainsΩ1234. And in the domainΩ5 it coincides with the Riemann function (46).

Let us find a representation of the Green’s function in the domain Ω6. To construct the Green’s functions, we will continue the coefficients of equation (47) in Ω6 = {(ξ, η) : η1 ξ 1, η1≤η ≤ξ} such a way that the following conditions

A(ξ, η) =

a(ξ, η), (ξ, η)Ω6, b(η, ξ), (ξ, η)Ω6, B(ξ, η) =

b(ξ, η), (ξ, η)Ω6, a(η, ξ), (ξ, η)Ω6, C(ξ, η) =

c(ξ, η), (ξ, η)Ω6, c(η, ξ). (ξ, η)Ω6

are met. Actually, show that coefficients of (47) have the following symmetry:

A(ξ, η) =B(η, ξ), C(ξ, η) =C(η, ξ), (ξ, η)Ω6. (50) From (50) we have

A(η, ξ) =

a(η, ξ), (η, ξ)Ω6, b(ξ, η), (η, ξ)Ω6, =

b(ξ, η), (ξ, η)Ω6,

a(η, ξ), (ξ, η)Ω6, =B(ξ, η). If we have chosen (ξ, η) fromΩ6, then(η, ξ)will be from Ω6.

From (4) and (5) we get

A(ξ, ξ) =B(ξ, ξ), Aξ(ξ, ξ) =Bη(ξ, ξ), η1 ≤ξ 1. If the coefficients a, b, aξ, bη, c C

Ω

then in virtue of (50) coefficients A(ξ, η), B(ξ, η), C(ξ, η) in the domain Ω6 = Ω6Ω6 = {(ξ, η) : η1 ξ 1, η1 η 1}

have the following smoothness A, B, Aξ, Bη, C ∈C

Ω6

. (51)

Let(ξ1, η1)be an arbitrary point of the domainΩ. In order to construct the Green function in the domain Ω6, consider the problem:

2G1

∂ξ∂η +A(ξ, η)∂G1

∂ξ +B(ξ, η)∂G1

∂η +C(ξ, η)G1 = 0, (ξ, η)Ω6; (52)

∂G1(η1+ 0, η;ξ1, η1)

∂η +b(η1, η)G1(η1+ 0, η;ξ1, η1)

= ∂R(η1, η;ξ1, η1)

∂η +b(η1, η)R(η1, η;ξ1, η1), η1 ≤η ≤ξ1; (53)

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B.O. Derbissaly 11

∂G1(ξ, η1+ 0;ξ1, η1)

∂ξ +a(ξ, η1)G1(ξ, η1+ 0;ξ1, η1)

= ∂R(ξ, η1;ξ1, η1)

∂ξ +a(ξ, η1)R(ξ, η1;ξ1, η1), η1 ≤ξ ≤ξ1. (54) The problem (52)-(54) is a Goursat problem. Its solution exists and unique. We are interested in the representation of the functionG1(ξ, η;ξ1, η1).

Lemma 1 If the function G1(ξ, η;ξ1, η1) is the solution to the problem (52)-(54), then for any(ξ, η)Ω6 we have G1(ξ, η;ξ1, η1) =G1(η, ξ;ξ1, η1).

To show that the function G1(η, ξ;ξ1, η1) satisfies the equation (52), in (52) replace ξ = η2, η =ξ2,(η2, ξ2)Ω6 and after using the symmetry conditions of coefficients, we get that G1(η, ξ;ξ1, η1)satisfies the equation (52).

Also doing the substitution of ξ = η2 in (53) and using the symmetry conditions of coefficients, we get the condition (54). Similarly, by replacing η = ξ2 in (54) and using the symmetry conditions of coefficients, we get the condition (53).

Thus, we have shown that the function G1(η, ξ;ξ1, η1) is also a solution to the problem (52)-(54). Since the solution to problem (52)-(54) is unique, then

G1(ξ, η;ξ1, η1) =G1(η, ξ;ξ1, η1), (ξ, η)Ω6.

Solution of the problem (52)-(54) we search in the following form G1(ξ, η;ξ1, η1) =R(ξ, η;ξ1, η1) +g(ξ, η;ξ1, η1),(ξ, η)Ω6. Then we get the following problem

2g

∂ξ∂η +A(ξ, η)∂g

∂ξ +B(ξ, η)∂g

∂η +C(ξ, η)g= 0, (ξ, η)Ω6; (55)

∂g(η1, η;ξ1, η1)

∂η +b(η1, η)g(η1, η;ξ1, η1) = 0, η1≤η ≤ξ1; (56)

∂g(ξ, η1;ξ1, η1)

∂ξ +a(ξ, η1)g(ξ, η1;ξ1, η1) = 0, η1 ≤ξ ≤ξ1. (57) It is easy to see that the solution to the problem (55)-(57) has the form

g(ξ, η;ξ1, η1) =R(η, ξ;ξ1, η1), (ξ, η)Ω6.

Lemma 2 Let (ξ, η)be an arbitrary point of the domain Ω. By internal variables (ξ1, η1) the Green’s function of the problem (7)-(9) has the following properties:

L(ξ11)G(ξ, η;ξ1, η1) = 0, (ξ1, η1)Ω, at ξ1 =ξ, ξ1 =η, η1 =ξ; (58) (Gξ1−Gη1)(ξ, η;ξ1, ξ1) + (a−b)(ξ1, ξ1)G(ξ, η;ξ1, ξ1) = 0, 0≤ξ1 1; (59)

G(ξ, η; 0, η1) = 0, 0≤η11; (60)

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∂G(ξ, η;ξ−0, η1)

∂η1 −a(ξ, η1)G(ξ, η;ξ−0, η1) = 0, at η1= η, η1 =ξ; (61)

∂G(ξ, η;ξ1, η−0)

∂ξ1 −b(ξ1, η)G(ξ, η;ξ1, η−0) = 0, at ξ1 =ξ; (62)

∂G(ξ, η;ξ1, ξ−0)

∂ξ1 −b(ξ1, ξ) G(ξ, η;ξ1, ξ−0)

= ∂G(ξ, η;ξ1, ξ+ 0)

∂ξ1 −b(ξ1, ξ)G(ξ, η;ξ1, ξ+ 0); (63) G(ξ, η;ξ−0, η−0)−G(ξ, η;ξ+ 0, η−0)

+G(ξ, η;ξ+ 0, η+ 0)−G(ξ, η;ξ 0, η+ 0) = 1; (64) G(ξ, η;ξ, ξ−0)−G(ξ, η;ξ, ξ+ 0)−G(ξ, η;ξ−0, ξ) = 0. (65) Properties (58)-(65) are easy to get out of the construction of the Green’s function of problem (7)-(9). Under these conditions (58)-(65) it is possible to uniquely restore the Green’s function of problem (7)-(9).

Using properties (58)-(65) we can use it to write the integral representation of the solution to problem (7)-(9). To do this, we consider the following integral

Ω(ξη)

G(ξ, η;ξ1, η1)f(ξ1, η1)11

=

Ω(ξη)

G(ξ, η;ξ1, η1)

2u

∂ξ1∂η1 +a∂u

∂ξ1 +b∂u

∂η1 +cu

11. (66)

Applying Green’s theorem in a plane [17] and using the conditions (8), (9) properties of Green’s function (58)-(65), from (66) we get the following representation of the solution to problem (7)-(9) in the domainΩ(ξη)= Ω5Ω6:

u(ξ, η) = 1

2(G(ξ, η; 0, ξ+ 0)−G(ξ, η; 0, ξ−0))τ(ξ) + 1

2G(ξ, η; 0, η−0)τ(η) +1

2 ξ

0

G(ξ, η;ξ1, η1)ν(ξ1)1+

Ω(ξη)

G(ξ, η;ξ1, η1)f(ξ1, η1)11.

6 Conclusion

In this paper, an integral representation of the Green function for a general second-order hyperbolic equation for the second Darboux problem is constructed, since all the properties of the Green function of this problem follow from the integral representation of the Green function. It is shown that the main difference between this work and other previous works by other authors, we conduct research and build a function Green’s solution of this problem without using the symmetry conditions of the lower coefficients. In addition, unlike other authors, it is in the article that we will give a definition of the Green function and a method for constructing it for cases of general coefficients.

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B.O. Derbissaly 13

References

[1] Kalmenov T. Sh. On the Spectrum of a Self-Adjoint Problem for the Wave Equation // Vestnik Akad. Nauk. Kazakh.

SSR. - 1982. - V. 2. - P. 63-66. [in Russian]

[2] Kal’menov T. Sh. Spectrum of a boundary - value problem with translation for the wave equation // Differential equations.

- 1983. - V. 19, No. 1. - P. 64 - 66. [in Russian]

[3] Sadybekov M. A., Orynbasarov E. M. Baseness of the system of the eigenfunctions and associated functions with displacement of Lavrentev-Bitsadze equation // Doklady Mathematics. - 1992. - V. 324, No. 6. - P. 1152-1154. [in Russian]

[4] Orynbasarov E. M., Sadybekov M. A. The basis property of the system of eigen- and associated functions of a boundary value problem with shift for the wave equation // Math. Notes. - 1992. - V. 51, No. 5. - P. 482-484. [in Russian]

[5] Yessirkegenov N. A., Sadybekov M. A. Spectral properties of boundary-value problem with a shift for wave equation //

Russian Math. (Iz. VUZ). - 2016. - V. 60, No. 3. - P. 41-46.

[6] Kreith K. Symmetric Green’s functions for a class of CIV boundary value problems // Canad. Math. Bull. - 1988. - V.

31. - P. 272-279.

[7] Kreith K. Establishing hyperbolic Green’s functions via Leibniz’s rule // SIAM Rev. - 1991. - V. 33. - P. 101-105.

[8] Kreith K. A self-adjoint problem for the wave equation in higher dimensions // Comput. Math. Appl. - 1991. - V. 21. - P. 129-132.

[9] Kreith K. Mixed selfadjoint boundary conditions for the wave equation // Differential equations and its applications (Budapest), Colloq. Math. Soc. Janos Bolyai, 62, North-Holland, Amsterdam. - 1991. - P. 219-226.

[10] Iraniparast N. A method of solving a class of CIV boundary value problems // Canad. Math. Bull. - 1992. - V. 35, No.

3. - P. 371-375.

[11] Iraniparast N. A boundary value problem for the wave equation // Int. J. Math. Math. Sci. - 1999. - V. 22, No. 4. - P.

835-845.

[12] Iraniparast N. A CIV boundary value problem for the wave equation // Appl. Anal. - 2000. - V. 76, No. 3-4. - P. 261-271.

[13] Haws L. Symmetric Green’s functions for certain hyperbolic problems // Comput. Math. Appl. - 1991. - V. 21, No. 5. - P. 65-78.

[14] Iraniparast N. Boundary value problems for a two-dimensional wave equation // Journal of Computational and Applied Mathematics. - 1994. - V. 55. - P. 349-356.

[15] Iraniparast N. A selfadjoint hyperbolic boundary-value problem // Electronic Journal of Differential Equations, Conference. - 2003. - V. 10. - P. 153-161.

[16] Derbissaly B. O, Sadybekov M. A. On Green’s function of Darboux problem for hyperbolic equation // Bulletin of KazNU.

Series of mathematics, mechanics, computer science. - 2021. - V. 111, No. 3. - P. 79-94.

[17] Riley K.F, Hobson M.P., Bence S.J. Mathematical methods for physics and engineering // Cambridge University Press, 2010.

Список литературы

[1] Кальменов Т. Ш. О спектре самосопряженной задачи для волнового уравнения // Вестник АН КазССР. - 1982. - Т. 2. - С. 63-66.

[2] Кальменов Т. Ш. Спектр краевой задачи со смещением для волнового уравнения // Дифференц. уравнения. - 1983.

- Т. 19, Н. 1. -С. 64 - 66.

[3] Садыбеков М. А., Орынбасаровa Е. М. Базисность системы корневых функций краевой задачи со смещением для уравнения Лаврентьева–Бицадзе // Докл. РАН. - 1992. - Т. 324, Н. 6. - С. 1152-1154.

[4] Орынбасаров Е. М., Садыбеков М. А: Базисность системы собственных и присоединенных функций краевой задачи со смещением для волнового уравнения // Матем. заметки. - 1992. - Т. 51, Н. 5. - С. 86-89.

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[5] Yessirkegenov N. A., Sadybekov M. A. Spectral properties of boundary-value problem with a shift for wave equation //

Russian Math. (Iz. VUZ). - 2016. - V. 60, No. 3. - P. 41-46.

[6] Kreith K. Symmetric Green’s functions for a class of CIV boundary value problems // Canad. Math. Bull. - 1988. - V.

31. - P. 272-279.

[7] Kreith K. Establishing hyperbolic Green’s functions via Leibniz’s rule // SIAM Rev. - 1991. - V. 33. - P. 101-105.

[8] Kreith K. A self-adjoint problem for the wave equation in higher dimensions // Comput. Math. Appl. - 1991. - V. 21. - P. 129-132.

[9] Kreith K. Mixed selfadjoint boundary conditions for the wave equation // Differential equations and its applications (Budapest), Colloq. Math. Soc. Janos Bolyai, 62, North-Holland, Amsterdam. - 1991. - P. 219-226.

[10] Iraniparast N. A method of solving a class of CIV boundary value problems // Canad. Math. Bull. - 1992. - V. 35, No.

3. - P. 371-375.

[11] Iraniparast N. A boundary value problem for the wave equation // Int. J. Math. Math. Sci. - 1999. - V. 22, No. 4. - P.

835-845.

[12] Iraniparast N. A CIV boundary value problem for the wave equation // Appl. Anal. - 2000. - V. 76, No. 3-4. - P. 261-271.

[13] Haws L. Symmetric Green’s functions for certain hyperbolic problems // Comput. Math. Appl. - 1991. - V. 21, No. 5. - P. 65-78.

[14] Iraniparast N. Boundary value problems for a two-dimensional wave equation // Journal of Computational and Applied Mathematics. - 1994. - V. 55. - P. 349-356.

[15] Iraniparast N. A selfadjoint hyperbolic boundary-value problem // Electronic Journal of Differential Equations, Conference. - 2003. - V. 10. - P. 153-161.

[16] Derbissaly B. O, Sadybekov M. A. On Green’s function of Darboux problem for hyperbolic equation // Bulletin of KazNU.

Series of mathematics, mechanics, computer science. - 2021. - V. 111, No. 3. - P. 79-94.

[17] Riley K.F, Hobson M.P., Bence S.J. Mathematical methods for physics and engineering // Cambridge University Press, 2010.

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ISSN 1563–0277, eISSN 2617-4871 JMMCS. №4(116). 2022 https://bm.kaznu.kz

IRSTI 27.31.17 DOI: https://doi.org/10.26577/JMMCS.2022.v116.i4.02

J. Ferreira1 , M. Shahrouzi2 Sebasti˜ao Cordeiro3 Daniel V. Rocha4

1Federal Fluminense University, Volta Redonda-RJ, Brazil

2Jahrom University, Jahrom, Iran

3Federal University of Par´a, Abaetetuba, Brazil

4Federal University of Par´a (UFPA), Belem, Brazil e-mail:jorge_ferreira@id.uff.br

BLOW UP OF SOLUTION FOR A NONLINEAR VISCOELASTIC PROBLEM WITH INTERNAL DAMPING AND LOGARITHMIC SOURCE

TERM

This paper is concerned with blow up of weak solutions of the following nonlinear viscoelastic problem with internal damping and logarithmic source term

|ut|ρutt+M(u2)(Δu)Δutt+ t 0

g(tsu(s)ds+ut=u|u|p−2R ln|u|kR

with Dirichlet boundary initial conditions in a bounded domain Ω Rn. In the physical point of view, this is a type of problems that usually arises in viscoelasticity. It has been considered with power source term first by Dafermos [3], in 1970, where the general decay was discussed. We establish conditions ofp,ρand the relaxation functiong, for that the solutions blow up in finite time for positive and nonpositive initial energy. We extend the result in [15] where is considered M = 1and external force type |u|p−2u in it. Further we estate and sketch the proof of a result of local existence of weak solution that is used in the proof of the theorem on blow up. The idea underlying the proof of local existence of solution is based on Faedo-Galerkin method combined with the Banach fixed point method.

Key words: Nonlinear Viscoelastic Equation, Logarithmic Source, Blow Up, Local existence.

Ж. Феррейра1, M. Шахрузи2, Себастьяо Кордейро3, Даниел В. Роча4

1Флуминенсе Федералдық университетi, Вольта Редонда-РЖ қ., Бразилия

2Джахром университетi, Джахром қ., Иран

3Пара федералды университетi, Абаэтетуба қ., Бразилия

4Пара Федералдық университетi, Белем қ., Бразилия e-mail:jorge_ferreira@id.uff.br

Iшкi демпферлiк және логарифмдiк көздi сызықты емес тұтқыр серпiмдi есеп шешiмiнiң қирауы

Бұл жұмысΩRnшектелген облыста бастапқы және Дирихле шартымен қойылған тұтқыр- серпiмдi iшкi демпфiрлiк және логарифмдiк сызықты емес мүшелерi бар

|ut|ρutt+M(u2)(Δu)Δutt+ t

0

g(tsu(s)ds+ut=u|u|p−2R ln|u|kR

c 2022 Al-Farabi Kazakh National University

Сурет

Figure 1: Towards the calculation of the cantilever bar for free oscillations: a) – the calculated scheme; b) – diagram M 5 ; c) – diagram M 4 ; d) – diagram M 3 ; e) – diagram M 2 ; f) – diagram M 1
Figure 2: Towards the calculation of the cantilever bar (example): a) – the preset scheme; b) - diagram M 5 ; c) - diagram M 4 ; d) – diagram M 3 ; e) – diagram M 2 ; f) – diagram M 1
Figure 4: The oscillation frequency response dependence on the relative stiffness value of the cantilever bar
Figure 2: Geometric interpretation of the equation of connection of the problem of synthesis of a four-link mechanism.
+7

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