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ISSN	1563 – 0277
eISSN	2617 – 4871
Индекс	75872; 25872

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

№ 1 (105)

Алматы «Қазақ университетi»

2020

Зарегистрирован в Министерстве информации и коммуникаций Республики Казахстан, свидетельство № 16508-Ж от 04.05.2017 г. (Время и номер первичной постановки на учет № 766 от 22.04.1992 г.). Язык издания: казахский, русский, английский. Выходит 4 раза в год.

Тематическая направленность: теоретическая и прикладная математика, механика, информатика.

Редакционная коллегия

научный редактор – Б.Е. Кангужин, д.ф.-м.н., профессор, КазНУ им. аль-Фараби, заместитель научного редактора – Д.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 секретарь – Г.М. Даирбаева, к. ф.-м. н., доцент, КазНУ им. аль-Фараби.

Айсагалиев С.А. – д.т.н., профессор, КазНУ им.аль-Фараби, Казахстан Ахмед-Заки Д.Ж. – д.т.н., Университет международного бизнеса, Казахстан Бадаев С.А. – д.ф.-м.н., профессор, КазНУ им.аль-Фараби, Казахстан

Бектемесов М.А. – д.ф.-м.н., профессор, Казахский национальный педагогический университет имени Абая, Казахстан

Жакебаев Д.Б. – PhD доктор, 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

Кыдырбекулы А.Б. – д.т.н., профессор, КазНУ им.аль-Фараби, Казахстан

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

Сукачев Ф.А. – профессор, академик АН Австралии, Университет Нового Южного Уэльса

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 Токмагамбетов Н.Е. – PhD доктор, КазНУ им.аль-Фараби, Казахстан Шиничи Накасука – PhD доктор, профессор, Университет Токио, Япония

Научное издание

Вестник. Серия математика, механика, информатика, № 1(105) 2020.

Редактор – Г.М. Даирбаева. Компьютерная верстка – Г.М. Даирбаева

ИБ N 13401

Формат 60 × 84 1/8. Бумага офсетная. Печать цифровая. Объем 17,6 п.л. Тираж 500 экз. Заказ N 2159. 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, 2020

ISSN 1563-0277, eISSN 2617-4871	Journal of Mathematics, Mechanics, Computer Science. № 1(105). 2020	3
IRSTI 27.31.17	https://doi.org/10.26577/JMMCS.2020.v105.i1.01

1T.М. Aldibekov, 2M.M. Aldazharova

1Doctor of Physical and Mathematical Sciences, Prof., E-mail: tamash59@mail.ru 2PhD, E-mail: a_maira77@mail.ru

Аl-Farabi Kazakh National University, Almaty, Kazakhstan

ON A DIAGONAL SYSTEM OF THE FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS FROM TWO INDEPENDENT VARIABLES

A diagonal system of three ﬁrst-order partial di erential equations in two independent variables is considered. The equations entering into the diagonal system are independent from each other, therefore, the compatibility condition of the system does not arise. We consider the asymptotic behavior of solutions at an inﬁnitely distant point, with respect to some parameter. The main place in the system is occupied by a nonlinear ﬁrst-order partial di erential equation, the remaining equations are adjoining equations, the solutions of which contain the initial value of one independent variable as a parameter. The attached equations are chosen appropriately, and the solution to the system is already studied, which already has an internal connection. The adjoint equations are linear ﬁrst-order partial di erential equations. Using the fact that the zero solutions of the characteristic equations are asymptotically stable on Lyapunov, the conditions when the set of three di erential equations, considered as a diagonal system of partial di erential equations of the ﬁrst order, has a solution with certain initial values and is an inﬁnitesimal function in the vicinity of an inﬁnitely remote point are described . Methods of the theory of functions and di erential inequalities in the theory of ﬁrst-order di erential equations are used.

Key words:di erential equations, diagonal system, ﬁrst order partial derivatives, asymptotic behavior.

1Т.М. Алдибеков, 2М.М. Алдажарова

1ф.-м.ғ.д., профессор, E-mail: tamash59@mail.ru

2PhD, E-mail: a_maira77@mail.ru

Әл-Фараби атындағы Қазақ ұлттық университетi, Алматы қ., Қазақстан

Екi тәуелсiз айнымалы бойынша бiрiншi реттi дербес туындылы дифференциалдық теңдеулердiң диагональдық жүйесi туралы

Екi тәуелсiз айнымалы бойынша үш теңдеуден тұратын бiрiншi реттi дербес туындылы дифференциалдық теңдеулердiң жүйесi қарастырылады. Диагоналдық жүйеге кiретiн теңдеулер бiр-бiрiнен тәуелсiз болғандықтан, жүйенiң үйлесiмдiлiк шартының қажетi болмайды. Қандай да бiр параметр бойынша шексiз алыс нүктедегi шешiмнiң асимптотикалық тәртiбi қарастырылады. Жүйедегi негiзгi орында, бiрiншi реттi дербес туындылы бейсызықты дифференциалдық теңдеу болып табылады, қалған екi теңдеулер шешiмдерi бiр тәуелсiз айнымалының алғашқы мәнiн параметр ретiнде қарастыратын, үлестегi бiрiншi реттi дербес туындылы дифференциалдық теңдеулер болып табылады. Үлестегi бiрiншi реттi дербес туындылы дифференциалдық теңдеулер ыңғайлы түрде таңдап алынады да, ендi iшкi байланыстары бар теңдеулерден тұратын жүйенiң шешiмi оқып зерттеледi. Үлестегi теңдеулер бiрiншi реттi дербес туындылы сызықты дифференциалдық теңдеулер болып табылады. Олардың сәйкес сипаттауыш теңдеулерiнiң нөлдiк шешiмдерiнiң Ляпунов бойынша асимптотикалық орнықтылықтарын пайдаланып, анықталған белгiлi бiр алғашқы мәндерiмен берiлген үш дифференциалдық теңдеудiң жиынтығы, бiрiншi реттi дербес туындылы дифференциалдық теңдеулердiң диагоналдық жүйесi түрiнде қарастырылып, шексiз алыстағы нүктенiң аймағында шешiмi шексiз аз функция болатын шарттар сипатталған. Функциялар теориясының және бiрiншi реттi дербес туындылы дифференциалдық теңдеулер теориясының тәсiлдерi қолданылған.

Түйiн сөздер: дифференциалдық теңдеулер, диагоналдық жүйе, дербес туындылар бiрiншi реттi, асимптотикалық тәртiп.

c 2020 Al-Farabi Kazakh National University

4	Aldibekov T.M., Aldazharova M.M.

1Т.М. Алдибеков , 2M.M. Алдажарова

1д.ф.-м.н., проф., E-mail: tamash59@mail.ru

2PhD, E-mail: a_maira77@mail.ru

Казахский национальный университет имени аль-Фараби, г. Алматы, Казахстан

Об одной диагональной системе дифференциальных уравнений с частными производными первого порядка от двух независимых переменных

Рассматривается диагональная система из трех дифференциальных уравнений с частными производными первого порядка от двух независимых переменных. Уравнения, входящие в диагональную систему друг от друга независимы, поэтому условия совместимости системы не возникает. Рассматривается асимптотическое поведение решений на бесконечно удаленной точке, относительно некоторого параметра. Основное место в системе занимает нелинейное дифференциальное уравнение с частными производными первого порядка, остальные уравнения являются присоединенными уравнениями, решения которых содержат начальное значение одного независимого переменного как параметр. Присоединенные уравнения выбираются подходящим образом, и изучается решение системы, уже имеющее внутреннюю связь. Присоединенные уравнения являются линейными дифференциальными уравнениями с частными производными первого порядка. Используя, что нулевые решения характеристических уравнений являются асимптотически устойчивыми по Ляпунову, описываются условия, когда совокупность трех дифференциальных уравнений, рассматриваемых как диагональная система дифференциальных уравнений с частными производными первого порядка имеет решение с определенными начальными значениями и является бесконечно малой функцией в окрестности бесконечно удаленной точки. Применяются методы теории функций и дифференциальных неравенств в теории

дифференциальных уравнений с частными производными первого порядка.

Ключевые слова: дифференциальные уравнения, диагональная система, частные производные первого порядка, асимптотическое поведение.

1 Introduction

Without pretending to completeness, we note some works, whose interests more or less intersect with questions on the subject of discussion. As known, the problems of a di erential equation with ﬁrst-order partial derivatives in the analytical case were considered in [1-2]. The ﬁrstorder partial di erential equations in the nonanalytic case were considered in [3-7]. There is a guide to ﬁrst-order partial di erential equations [8]. Also note some useful works under the subject of investigation [9-22]. Some information on the subject is available in books [23-29].

The study of system of the ﬁrst-order partial di erential equations in the nonlinear case is a di cult task, since, generally speaking, the characteristic system is not developed. The paper considers a certain diagonal system of the ﬁrst-order partial di erential equations, which arises as a result of di erential inequalities. Using the methods of the theory of functions and di erential inequalities in the theory of the ﬁrst-order partial di erential equations, the behavior of the solution is studied.

2 Materials and research methods

On the plane of variables (x, y) R2, the set

S = ((x, y) : 0 ≤ x < +∞, c − Lx ≤ y ≤ d + Lx)

On a diagonal system of the ﬁrst-order partial di erential equations. . .

is deﬁned, where c < 0, L > 0, d > 0. The ﬁrst-order diagonal system of partial di erential equations is considered

∂θ

= (b1(x)y + h1(x, y))

∂θ

;

= (b3(x)y + h3(x, y))

;

∂x

∂y

∂u∂x =

b2(x)y + h2(x, y, u, ∂u∂y )

∂u∂y ;

(1)

∂ν

∂x

∂θ

(E),

∂y

(b1(x)y + h1(x, y)) ∂y

b2(x)y + h2 x, y, u, ∂u∂y

∂u∂y C2(E),

x, y)) ∂ν

(E);

(b3(x)y + h3(

∂y

where

is a convex domain in

containing the point

, the

projection of which onto the (x, y)−plane contains S. Denote by I ≡ [0, +∞). The variable x I.

The diagonal system (1) is considered with initial values

		θ(0, y) = ω1(y), ω1(y) C2
		u(0, y) = ϕ(y), ϕ(y)			C2		(2)
		ν(0, y) = ω2(y), ω2(y)				C2

	y		(c, d)	;
where				;
The equation
	∂θ	= (b1(x)y + h1(x, y))			∂θ		(3)
	∂x	= (b1(x)y + h1(x, y))			∂y		(3)
	∂x				∂y
has a characteristic equation
	dy	= −b1(x)y − h1(x, y)					(4)

	dx

(4) we consider under the initial condition y(x0) = y0, (x0, y0) S, and we assume that the solution y = y(x, x0, y0) is deﬁned on the set I . The equation

∂u

(5)

b2(x)y + h2

x, y, u,

∂x

∂y

consider under the initial condition

u(0, y) = ϕ(y)

(6)

where

ϕ(y)

C2, (0, 0, ϕ(0), ϕ (0))

; as known, problem (5), (6) for small

| has a

unique solution u(x, y) of class C

(S). The equation

∂ν

= (b3(x)y + h3(x, y))

∂ν

(7)

∂y

∂x

has a characteristic equation

−b3(x)y − h3

(8)

(x, y)

6 Aldibekov T.M., Aldazharova M.M.

Considering (8) under the initial condition y(x0) = y¯0(x0, y¯0) S and assume that the solution y = y(x, x0, y¯0) is deﬁned on the set I.

Theorem. If in the diagonal system of ﬁrst-order partial di erential equations (1) the following conditions are satisﬁed:

(A)bi(x) C2(I), x I ≡ [0, +∞), hi(x, y) C2(S), hi(x, 0) = 0; i = (1, 3);

(B)h2(x, y, u, ∂u∂y ) C2(E), h2(0, 0, u, ∂u∂y ) = 0 and satisﬁes the Lipschitz condition with

respect to (u, ∂u∂y ) in the domain E; there is an inequality uxxuyy − uxy2

= 0 and the so-

lution u(x, y, u, ∂u∂y )

> 0

of equation (5), with the initial value (6), can be continued for

x > 0, (x, y) S;

< b3(x), x I;|bi(x)| ≤

Kiψ(x), x I;

(C)

The

inequality

holds b1(x)

< b2(x)

Ki > 0, i = (1, 3);

+∞

ψ(x) > 0,

ψ(x)

C(I)

lim ψ(x) = 0

, 0

ψ(x)dx = +

∞;

x x→+∞

By deﬁnition, q(x) = ψ(τ)dτ;

There are limits

lim

(−bi(τ))

i = 1, 3; β

< 0, β

< 0

x→+∞ q(x)

ψ(τ)

dq(τ) =

∂ui,

;

(D) Inequalities hold: h1(x, y) < h2(x, y, u,

∂y ) < h2(x, y), (x, y) S;

ψ(x)

= 0;

∂y

(E) |

i(x, y)|

δ(x)|y|, i

;

x→+∞

1, 3

δ(x)

C(I), δ(x)

lim

δ(x)

∂u

| |

x→+∞

∂u

(

Kbi(τ))

h2(x, y, u, ∂y )

∂y

= Kδ(x) y , lim

q(x)

−

ψ(τ)

dq(τ) = β2,

and

the

inequalities

β1

β2 < β3 are satisﬁed;

Then there exists a solution θ(x, y), u(x, y), ν(x, y) of system (1), with initial value (2) such

that

θ2(x, y) + u2(x, y) + ν2(x, y) → 0 as x0 → +∞.

Proof. Consider the solution u(x, y) of equation (5) with the initial value u(0, y) = ϕ(y). By condition u(x, y), the solution is deﬁned for x I and y (−δ, δ) (c, d), where a su ciently small number δ > 0. Consider equation (4) and solutions y = y(x, x0, y¯0) of equation (4), where (x0, y¯0) S. From conditions A) and E) It follows that the zero solution to equation (4) is asymptotically stable as x → +∞. In fact, in (4), the equation of the ﬁrst approximation has the form

dx = −b1(x)y

From condition A), C) it follows that for solving the equation of the ﬁrst approximation we

have the estimate

|y(x)| ≤ |y(x0)|eβ1[q(x)−q(x0)]

Using this estimate to represent the solution of equation (4) in the form of an integral

equation, found by varying an arbitrary constant, for any ε 0, |β21| we obtain the estimate

|y(x)| ≤ |y(x0)|e(β1+ε)[q(x)−q(x0)]

This implies the asymptotic stability of the zero solution to equation (4). We assume that asymptotic stability occurs for the initial values |y¯0| < δ. Take ω1(y) and the integral

	On a diagonal system of the ﬁrst-order partial di erential equations. . .			7
y¯0	θ(x, y, x0) of equation (4) satisfying the conditions	∂θ	> 0, θ(0, y, x0) = ω1(y), ω1(y) <
		∂y

ϕ(y) and y¯0 (−δ, δ).

Then θ(x, y) = θ(x, y, x0) is a solution of equation (3) satisfying the condition θ(0, y) =

ω1(y), (x, y, θ, θy) E, and θ(x, y, x0) → 0, x0 → +∞.

Consider equation (8) and solutions y = y(x, x0, y¯0) of equation (8), where (x0, y¯0) S. Similarly to equation (4), it is established that condition A) and E) imply that the zero solution to equation (8) is asymptotically stable as x → +∞. We assume that asymptotic

stability occurs for the initial values |

y¯

< δ

. Take

(y)

and the integral

y¯ = ν(x, y, x

)

∂ν

of equation (8) satisfying the conditions

∂y

> 0 and ν(0, y, x0) = ω2(y), ϕ(y) < ω2(y) and

y¯0 (−δ, δ). Then ν(x, y) = ν(x, y, x0) a solution of equation (7) satisfying the condition

ν(0, y0) = ω2(y), (x, y, θ, θy) E, and ν(x, y, x0) → 0, x0 → +∞. By virtue of conditions C) and D), the inequalities hold

(b1

(x)y + h1

(x, y)) ∂y

(x)y + h2

(x, y, u, ∂y )

∂y

< (b3(x)y + h3

(x, y)) ∂y

∂θ

∂u

∂ν

in the domain E. By the choice of ω1(y), ω2(y), the inequalities holds:

ω1(y) < ϕ(y) < ω2(y), y (c, d), (x, y) S;

Consequently,

θ2(x, y) + u2(x, y) + ν2(x, y) → 0 as x0 → +∞. The theorem is proved.

3 Results

The paper considers a diagonal nonlinear system of ﬁrst-order partial di erential equations. The equations in the system are interconnected by di erential inequalities and have initial values, the set of which is the initial values of the system and the initial values of the associated equations are selected appropriately. It is proved that the diagonal system under certain conditions has an inﬁnitesimal solution in a neighborhood of an inﬁnitely distant point.

4 Acknowledgment

The work was carried out with the support of grant ﬁnancing: - Fundamental and applied research in mathematics on the topic ”The investigation of the asymptotic stability of the solution and the development of asymptotic characteristics of a system of ﬁrst-order partial di erential equation”, No. AP05132615.

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[29]Yanenko N.N. and Rojdestvensky B.L., Sistemy kvazilineinyh uravneenii i ih prilozhenie k gazovoi dinamike [Systems of quaziliniear di erential equations and their application to gas dynamics] VI.7-9, (М., 1978), 676

10 ISSN 1563-0277, eISSN 2617-4871

Journal of Mathematics, Mechanics, Computer Science. № 1(105). 2020

IRSTI 27.27.17	https://doi.org/10.26577/JMMCS.2020.v105.i1.02
1U.R. Kusherbaeva	, 2G.E. Abduakhitova , 3A. Assadi

1Cand. Sc. (Phys.-Math.), senior lecturer, E-mail: ulbi-70@mail.ru

2Cand. Sc. (Phys.-Math.), associate professor, E-mail: gulzhanae@gmail.com Al-Farabi Kazakh National University, Almaty, Kazakhstan

3Professor, E-mail: amirassadi@gmail.com, University of Wisconsin, Madison, USA

ON CONTINUOUS SOLUTIONS OF THE MODEL HOMOGENEOUS BELTRAMI EQUATION WITH A POLAR SINGULARITY

This paper consists of two parts. The ﬁrst part is devoted to the study of the Beltrami model equation with a polar singularity in a circle centered at the origin, with a cut along the positive semiaxis. The coe cients of the equation have a ﬁrst-order pole at the origin and do not even belong to the class L2(G). For this reason, despite its speciﬁc form, this equation is not covered by the analytical apparatus of I.N. Vekua [1] and needs to be independently studied. Using the technique developed by A.B. Tungatarov [2] in combination with the methods of the theory of functions of a complex variable [3] and functional analysis [4], manifolds of continuous solutions of the Beltrami model equation with a polar singularity are obtained. The theory of these equations has numerous applications in mechanics and physics. In the second part of the article, the coe cients of the equation are chosen so that the resulting solutions are continuous in a circle without a cut [5]. These results can be used in the theory of inﬁnitesimal bendings of surfaces of positive curvature with a ﬂat point and in constructing a conjugate isometric coordinate system on a surface of positive curvature with a planar point [6].

Key words: Beltrami equation, equation with a polar singularity.

1Ұ.Р. Көшербаева, 2Г.Е. Абдуахитова, 3А. Асади

1физ.-мат.ғ.к., аға оқытушы, E-mail: ulbi-70@mail.ru 2физ.-мат.ғ.к., доцент, E-mail: gulzhanae@gmail.com

Әл-Фараби атындағы Қазақ ұлттқ университетi, Алматы қ, Қазақстан

3профессор, E-mail: amirassadi@gmail.com, Висконсин университетi, Мэдисон, АҚШ

Полярлық ерекшелiгi бар модельдi Бельтрами теңдеуiнiң үзiлiссiз шешiмдерi

Қарастырып отырған еңбек екi бөлiмнен тұрады. Бiрiншi бөлiмi центрi координаталар бас нүктесiнде орналасқан оң жақ жарты осьсiз дөңгелекте берiлген полярлық ерекшелiгi бар модельдi Бельтрами теңдеуiн зерттеуге арналған. Қарастырылған теңдеудiң коэффициентерiнiң координаталар бас нүктесiнде бiрiншi реттi полюстерi бар және олар L2(G) класында жатпайды. Сондықтан И.Н.Векуаның [1] аналитикалық аппаратымен қамтылмаған және жеке зерттеудi қажет етедi. А.Б. Тунгатаров [2] жасаған күрделi әдiстi [3] және функционалдық талдаудың [4] теориясы әдiстерiмен үйлестiре отырып, Белтрами моделдi теңдеуiнiң полярлық ерекшелiгi бар үздiксiз шешiмдерiнiң түрлерi алынды. Бұл теңдеулер теориясының механика мен физикада көптеген қолданыстары бар. Жұмыстың екiншi бөлiмiнде теңдеудiң шешiмi қарастырылған дөңгелекте үзiлiссiз болатындай етiп коэффициенттер таңдалады [5]. Бұл нәтижелердi тығыздық нүктесi бар қисықтығы оң ақырсыз аз иiлетiн беттер теориясында және тығыздық нүктесi бар қисықтығы оң беттерде изометриялы түйiндес координаттарды құрастыру үшiн қолдануға болады [6].

Түйiн сөздер: Бельтрами теңдеуi, полярлық ерекшелiгi бар теңдеу.

1У.Р. Кушербаева, 2Г.Е. Абдуахитова, 3А. Асади

1канд. физ.-мат. наук, старший преподаватель, E-mail: ulbi-70@mail.ru 2канд. физ.-мат. наук, доцент, E-mail: gulzhanae@gmail.com

Казахский национальный университет им. аль-Фараби, г. Алматы, Казахстан

3профессор, E-mail: amirassadi@gmail.com, Висконсинский университет, Мэдисон, США

О непрерывных решениях модельного однородного уравнения Бельтрами с полярной особенностью

c 2020 Al-Farabi Kazakh National University

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