ON ONE BOUNDARY VALUE PROBLEM FOR THE FOURTH-ORDER EQUATION IN PARTIAL DERIVATIVES Текст научной статьи по специальности «Математика»

Vestnik ^AUNC. Fiz.-Mat. nauki. 2022. vol. 39. no. 2. P. 32-41. ISSN 2079-6641

MSC 35J15

Research Article

On one boundary value problem for the fourth-order equation in

partial derivatives

4b University str., Tashkent, 100174, Uzbekistan 2 Bukhara State University, M. Ikbol str. 11, Bukhara, 705018, Uzbekistan E-mail: [email protected]

The initial-boundary problem for the heat conduction equation inside a bounded domain is considered. It is supposed that on the boundary of this domain the heat exchange takes place according to Newton's law. The control parameter is equal to the magnitude of output of hot air and is defined on a givenmpart of the boundary. Then we determined the dependence T(0) on the parameters of the temperature process when 0 is close to critical value.

Key words: boundary value problem; Fourier method; the existence of a solution; the uniqueness of a solution.

For citation.Kilichov O. Sh., Ubaydullaev A.N. On one boundary value problem for the fourth-order equation in partial derivatives. Vestnik KRAUNC. Fiz.-mat. nauki. 2022,39: 2, 32-41. DOI: 10.26117/2079-6641-2022-39-2-32-41

The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Kilichov O. Sh., Ubaydullaev A. N., 2022

Introduction and statement of the Problem

A problem with a high derivative on a part of the domain boundary was studied, for the first time, by A.N. Tikhonov. In [1], he studied the problem of a homogeneous heat equation with the following conditions

O. Sh. Kilichov1, A.N. Ubaydullaev

2

1 V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences,

d DOI: 10.26117/2079-6641-2022-39-2-32-41 Original article submitted: 17.07.2022

Revision submitted: 10.08.2022

in domain (0 < x < oo, t > 0).

Funding. The work was done without financial support.

In [2], in n—dimensional bounded domain D A.V. Bitsadze studied the problem

, ^ . dmu . . A u(x) = 0, dvm = f(x), x e D,

and proved its Fredholm property.

Boundary value problems with boundary conditions, for the Laplace, Poisson and Helmholtz equations in the unit ball, containing high-order derivatives were studied by I.I. Bavrin [3], V.V. Karachik and B.Kh. Turmetov [4], V.V. Karachik [5]-[7], V.B. Sokolovsky [8]. A mixed problem with a high derivative in the initial condition for the heat equation was studied by D. Amanov [9]; for the equation of string vibration, the mixed problem with high derivatives in the initial conditions was studied in [10]-[16]. Authors of [17], [18]-[19] studied boundary value problems for many types of high-order partial differential equations. Mixed problems for fourth-order differential equations were studied in [20]. The difference of this problem from other problems is that in the initial condition the sum of products of higher orders is set. Consider the following equation

9u 34u . ,

at = f(x>l) (1)

in domain O = {(x,t): 0 < x < p, 0 < t < T}, and the boundary value problem with boundary conditions

u(0, t) = u(p, t) = 0, 0 < t < T, (2)

uxx(0,t)= uxx(p,t)= 0,0 < t < T, (3)

and initial conditions

k=0

= p(x), 0 < x < p, (4)

t=0

where k,m— are the fixed natural numbers, f(x,t), cp(x)— are the given functions, continuous in O , and u(x,t) e CX'Jt (O)— is the sought-for function.

Uniqueness of the solution to the problem

Theorem 1. The solution to problem (1)-(4) is unique if it exists.

Proof. Let f(x,t) = 0, p(x) = 0 in O. We show that u(x,t) = 0 in O. Following [21], we consider the following integral

an(t) =

u(x,t)Xn(x)dx, 0 < t < T (5)

where functions

2 nn

Xn(x) ^nAnx, An = —, n = 1,2,... (6)

form a complete orthonormal system in L2(0,p) [22]. Differentiating (5) over t, from homogeneous equation ut + uxxxx = 0 we find

an (t) =

ut(x,t)Xn(x)dx

or

P,4

On (t) =

94u

dX4 (x,t)Xn(x)dx. (7)

0

Integrating four times by parts over x in the integral (7) on the right side, we obtain the following equations:

an (t)+ A^ ■ an(t)= 0. (8)

The general solution to equation (8) is written as

a-n(t) = Cn ■ exp(-Ait). (9)

To find the unknown coefficient Cn , due to conditions (4), we obtain Cn = 0. Then from (9) it follows that an(t) = 0. Since Xn(x) is the complete orthonormal system in L2(0,p), it follows from the completeness of function Xn(x) that u(x,t) = 0 almost everywhere in O. Considering that u(x,t) e Cx,t (O), we obtain u(x,t) = 0 in O. Theorem 1 is proved. □

Existence of a solution to the problem

Solutions of the inhomogeneous differential equation (1) are sought in the form of a Fourier series in sines

/2 00

u(x,t) = W — ^un(t)sin Anx. (10)

V pn=1

Let the functions f(x,t), cp(x) be expanded in a Fourier series

I— 00

f(x,t) = W — Anx, (11)

V —n=1

I— 00

p(x) = W — ^p^nAnx, (12)

V —n=1

where

p

2

fn(t)^/p

f(x,t)sin Anxdx, (13)

Pn =

p

p(x)sin Anxdx.

(14)

Substituting the Fourier series (10) and (11) into the given inhomogeneous differential equation (1), we obtain a first-order ordinary differential equation

un (t)+ An ■ un(t) = fn(t).

m (k)

Its solution satisfies the condition u(nk)( 0) = pn and has the following form

k=0

Un(t) = Wn(t) +

fn(T)e

-A^t-T)

dT.

(15)

Here and hereinafter (... ) = 0 for m < 0,

s=0

-Ait

Wn(t) =

£ (-Atf

s=0

m-1

Pn fnm-1-s)(0^(-A:

nn

s=0

,i=0

Substituting solution (15) into series (10), we find

u(x,t) = W 2 L

vpn=i

Wn(t) +

fn(T)e

-An(t-T)

dT

sinAnx.

Calculating the products of the solution of equation (1), we obtain

(16)

9u 9t

L A'

n=1

-Wn(t)+ A- 4fn(t) -

fn(T)e-An(t-T)dT

sinAnx, (17)

9x4

LA

n=1

Wn(t) +

fn(T)e

-Ai(t—t)

dT

sinAnx,

(18)

9ku

L (-*

n=1

w.

k-1

t(t)+L (-An )s f

s=0

4 s- ( -1-s)

(t)+

fn(T)e

-An(t-T)

'dT

sinAnx. (19)

Let us show that series (16) and (17)-(19) converge absolutely and uniformly. If the series (19) converges then the series (16)-(18) whose terms are less than the corresponding terms of the series (19) converge absolutely and uniformly. Let us show the absolute and uniform convergence of series (19).

k

Lemma 1. If p(x) e W— (O), p(0) = p(p) = 0, then the series

12 f • A ■ Pn ■ e-Ant ,20.

^^smAnX ^-m--(20)

pn= L (-An)s

s—0

converges absolutely and uniformly in O.

Proof. Simplifying series (20), we obtain the following series

L

n=1

AnX

(-Aj)k ■ ■ e-Ant

L (-An)'

s—0

<

L

n—1

-An

k+1

-f-An

(-An)m+1 -1

■ Pn

If we choose the largest value of k (k < m), we have

L

n—1

(-An)m+1 - (-An)1

(-An)m+1 -1

■ Pn

<

Lipni.

n=1

Integrating (14) by parts, we obtain |pn| — a- ■

lpnl, where

pn —

P [x)\l 2cosAnxdx. P

Applying the Cauchy-Bunyakovsky inequality, we find

L iPni—L

n—1 n—1

By virtue of

An

P(1) Pn

Pi11

<

P

oo , \ 1/2 / oo ,N 1/2

Li LIp

nx*— n

(1) n

n—1

n—1

(oo

LK1

1/2

Pn

2

L2(0,p)

, we have

'2^ . . (-An)k■ Pn■ e-Ant

An

X

<

n—1

L (-AJ)'

s—0

V6

Pn

L2(0,p)

Lemma 1 is proved. □

Lemma 2. If f(x,t) G w24k-3,k-1) (Q), f- — f1 — 0, I — , then

the series

An

X

4 )k „-Ai tm-1

-An ) • e

^fnm-1-s)(0) ■(£ (-a:

n—1

-A4n s s—0

s—0

(21)

vi—0

k-1

^sinAnX (-An) f

n—1

s—0

Osfnk-1-s)(t)

(22)

k

1

2

converge absolutely and uniformly in O.

Proof. Let us expand the series (21) and (22); it is easy to see that the term

L Ank-4fn(t)i

(23)

n=1

is the largest of the two series.

If series (23) converges, then series (21) and (22) also converge. Integrating (13) by parts, we obtain

|fn(t)| =

1

A

4k-3

(4k-3,0)

(t)

(24)

where

fn4k-3,0)(t) =

p94k-3f(x,t) /2 . ^n .

9x4k(-3 V p ^ ((4k- 3) 2 + An^ dx.

0

If to apply (24) to (23), due to the Bunyakovsky and Bessel inequalities [23], the series is

I 00 k-1

2 L*mAnxL (-An)Sfnk-1-s)(t) <

n=1

s=0

V6

94k-3f

dx4k-3

A

->4 \ k -a4 t m-1 / s

-An) e Ant p(m-1 -s)

n=1

nx m ln

£ (-An)s s=0

s=0

s)(0) Z(-a

a=0

L2(û) 94k-3f

9x4k-3

L2(0)

Lemma 2 is proved. □

Lemmi the series

Lemma 3. If f(x,t) e W^4k—3,0) (O), = = 0,1 = 0,1,..., (2k—2), then

L sinAnx ■ (-A^ ■ fn(T)e-An(t-T)dT

n=1

(25)

converges absolutely and uniformly in O.

Proof. Introducing representation (24) into series (25), we have

L

n=1

A

4k

fn (T)e-An(t-T)dT

<

L An

n=1

fn4k-3,0)(T) ■ e-An(t-T) dT

<

<

00

L An ■

n=1

2 /1

fn4k-3,0)(T)

dT

-2A^(t-T)

dT I <

ou -, < - y1 ■

2rcz— n

n=1

(4k-3,0)

(T)

dTI .

n

2

e

2

n

oo

L

n=1

This implies the uniform and absolute convergence of series (25) in O. Lemma 3 is proved. □ Theorem 2. Let

i) p(x) e W—(O), p(0) = p(p)= 0

ii) f (x, t) e W(4k-3'k-1) (O), f1 = f1 = 0,1 = 07C—k—2), _

then the series (16)-(19) converge absolutely and uniformly in O. Solution (16) satisfies equation (1) and conditions (2) - (4).

Proof. By virtue of the proved lemmas, it is easy to show that the series (16)-(19) converge absolutely and uniformly. Adding (17) and (18) we make sure that solution (16) satisfies equation (1) in O . Conditions (2) and (3) are satisfied due to the properties of function Xn(x). Theorem 2 is proved. □

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

Acknowledgments. The authors are deeply grateful to the referee for a number of comments that contributed to the improvement of the article.

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4k

fn (T)e-An(t-T)dT

<

2^

9

4k-3

f

9x4k-3

L2(Q)

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Kilichov Oybek Sharafiddinovich- Doctoral student, Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Tashkent, Uzbekistan, ORCID 0000-0002-7673-943X.

Ubaydullaev Alisher NematillayevichÄ - Teacher of the Department of Mathematics of Bukhara State University, Bukhara, Uzbekistan, ©ORCID 0000-0002-4219-5155.

Вестник КРАУНЦ. Физ.-мат. науки. 2022. Т. 39. №. 2. С. 32-41. ISSN 2079-6641

УДК 517.95 Научная статья

Об одной краевой задаче для уравнения четвертого порядка в

частных производных

О. Ш. Киличов1, А.Н. Убайдуллаев2

1 Институт математики имени В. И. Романовского Академии наук Узбекистана, 4б ул. Университетская, г. Ташкент, 100174, Узбекистан

2 Бухарский государственный университет, ул. М. Икбол, 11, г. Бухара, 705018, Узбекистан

E-mail: [email protected]

Рассмотрена начально-краевая задача для уравнения теплопроводности внутри ограниченной области. Предполагается, что на границе этой области происходит теплообмен по закону Ньютона. Параметр управления равен величине выхода горячего воздуха и определяется на заданном участке границы. Затем была определена зависимость T(0) от параметров температурного процесса, когда 0 близко к критическому значению.

Key words: краевая задача; метод Фурье; существование решения; единственность решения.

d DOI: 10.26117/2079-6641-2022-39-2-32-41

Поступила в редакцию: 17.07.2022 В окончательном варианте: 10.08.2022

Для цитирования. Kilichov O. Sh., Ubaydullaev A. N. On one boundary value problem for the fourth-order equation in partial derivatives // Вестник КРАУНЦ. Физ.-мат. науки. 2022. Т. 39. № 2. C. 32-41. DOI: 10.26117/2079-6641-2022-39-2-32-41

Конкурирующие интересы. Авторы заявляют, что конфликтов интересов в отношении авторства и публикации нет.

Авторский вклад и ответственность. Все авторы учавствовали в написании статьи и полностью несут ответственность за предоставление ококнчательной версии статьи в печать. Окончательная форма рукописи была одобрена всеми авторами.

Контент публикуется на условиях лицензии Creative Commons Attribution 4.0 International (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Киличов О.Ш., Убайдуллаев А.Н., 2022

Финансирование. Работа выполнена без финансовой поддержки.

Киличов Ойбек Шарафиддинович^ - докторант, Институт математики Академии наук Республики Узбекистан, Ташкент,

Узбекистан, ОЫСГО 0000-0002-7673-943Х.

Убайдуллаев Алишер Нематиллаевич& - Преподаватель кафедры математики Бухарского государственного университета,

Бухара, Узбекистан, СЖСГО 0000-0002-4219-5155.