D OI 10.31489/2023M 4/30-40 U DC 517.954
Y u.P . A p a k o v 1,2, D .M . M e liq u z ie v a 2’*
1 V.I. R om anovskii In stitu te o f M athem atics o f the A cadem y o f Sciences o f the Uzbekistan, Tashkent, Uzbekistan;
2 N am angan E ngineering-C onstruction Institu te, N am angan, Uzbekistan (E-mail: yusupjonapakov@ gmail.com, meliquziyevadilshoda@ gmail.com)
On a bound ary problem for th e fourth order eq u ation w ith th e third derivative w ith resp ect to tim e
In th is p aper, we consider a b o undary value problem in a rectan g u lar dom ain for a fourth-order hom ogeneous p a rtia l differential equation containing th e th ird derivative w ith respect to tim e. T he uniqueness of th e solution of th e sta te d problem is proved by th e m ethod of energy integrals. Using th e m eth o d of separation of variables, th e solution of th e considered problem is sought as a m ultiplication of two functions X (x) and Y (y). To determ ine X (x),we o b ta in a fourth-order ordinary differential equation w ith four b o undary conditions a t th e segm ent b o u n d ary [0,p], and for a Y (y) - third -o rd er ordinary differential equation w ith th ree b o undary conditions a t th e b o u n d ary of th e segm ent [0,q]. Im posing conditions on th e given functions, we prove th e existence theorem for a regular solution of th e problem . T he solution of th e problem is constructed in th e form of an infinite series, and th e possibility of term -by-term differentiation of th e series w ith respect to all variables is su b stan tia te d . W hen su b stan tia tin g th e uniform convergence, it is shown th a t th e “sm all denom inator” is different from zero.
Keywords: In itia l b o undary problem , Fourier m ethod, uniqueness, existence, eigenvalue, eigenfunction, functional series, absolute and uniform convergence.
Introduction
Problem s ab o u t th e vibrations of rods, beam s and plates, which are of great im portance in stru c tu ra l m echanics, lead to differential equations w ith a higher order th a n th e string equation.
T h e stu d y of m any problem s of gas dynam ics, th e th eo ry of elasticity, th e th eo ry of plates and shells comes to th e consideration of differential equations w ith higher order p a rtia l derivatives. From th e point of view of physical applications, th e fourth order differential equations are also of great interest (see [1-6 ]).
In th e field of m odern science and technology, initial-b o und ary value problem s for fourth-order equations are of great im portance. For exam ple, aircraft wings, bridge slabs, floor system s, and window panes are m odeled as plates w ith various types of end supports, which are successfully described in term s of fourth-order equations [7- 9 ] .
T h e m onograph by T .D . D zhuraev, A. Sopuev [10] is devoted to th e classification of differential equations w ith p a rtial derivatives of th e fourth order, th e form ulation and solution of b o un dary value problem s for such equations.
In th e pap er [11], a problem w ith b o u nd ary conditions for a non-hom ogeneous fourth-order equation w ith m ultiple characteristics and one lower term was considered.
In [12], a b o un d ary value problem for a fourth-order equ ation of th e form Uxxxx u tt = f { x , t)
was investigated.
‘ C orresponding author.
E-mail: meliquziyevadilshoda@ gmail.com
Buketov
University
In [13], a problem was solved w ith initial and bo u n d ary conditions for th e beam oscillation equation of th e form
a Uxxxx + u tt = 0,
in which a beam of length l is clam ped w ith ends in a massive vise.
In [14], a b o u n d ary value problem for a degenerate higher order equation w ith lower term s was studied.
In [15-1 9 ], th e b o u nd ary value problem s for a th ird -o rd er equation w ith m ultiple characteristics containing second derivatives w ith respect to tim e were discussed.
T h e bo u n d ary value problem s for fourth-order equations w ith th e th ird derivative in tim e have been little studied [2 0,21].
1 Form ulation o f the problem
In th e dom ain D = {(x, y) : 0 < x < p, 0 < y < q} we consider th e equation d 4u d 3u
L[u] = dX4 - W = °- (1)
w here p, q e R.
Problem A. F ind a solution to equation (1) in th e dom ain D from th e class u (x ,y ) e C t’3 (D ) П 3 2 (—\
C x,y (D j such th a t satisfies th e following bo u n d ary conditions:
u (0, y ) = u (p, y) = Uxx (0, y) = Uxx ( p , y ) = 0, 0 < y < q, (2) Uy (x, 0) = фг ( x ), Uyy (x, 0) = ^2 ( x ) , Uyy (x, q) = фз ( x ), 0 < x < p, (3) w here фг (x), i = 1,3 are th e given sufficiently sm ooth functions, and
фг (0) = фг (p) = ф'1 (0) = ф'1 (p) = 0, фг (0) = фг (p) = 0, i = 2, 3. (4) 2 The uniqueness o f the solution to the problem A Theorem 1. If th e problem A has a solution, th e n it is unique.
Proof. Let th e problem A have two solutions U1 (x ,y ) and U2 (x ,y ). T h en th e function u (x ,y ) = U1 (x ,y ) — u2 (x ,y ) satisfies eq uation (1) and th e uniform b o un d ary conditions. Let us prove th a t u (x, y) = 0 in D .
In th e dom ain D th e following id en tity is valid:
т- r i d . \ d / 1 2 \ 2 „
u L [u] = d x (u u xxx — u xu xx) — d y ( UUyy — 2UyJ + Uxx = 0 . Integ ratin g th e identity over th e dom ain D , we have
q
I [u (p , y ) u xxx (p ,y ) — u (0, y) u xxx (0, y )]dy 0 q
^ [ux (p, y) Uxx (p, y) — Ux (0, y) Uxx (0, y )ld y — 0
p
— / [u (x, q) u yy (x, q) — u (x, 0) u yy (x, 0)]d x +
01 p p q
+ - / [u2 (x, q) — uy (x, 0) dx + / / uxxdxdy = 0.
2 0 0 0
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University
Taking th e hom ogeneous b o u n dary conditions into consideration we ob tain
p p q
1 J u y q) dx + J j u 2xxd x d y = 0.
0 0 0
From th e second term we ob tain
Uxx = 0 ^ u (x ,y ) = x ■f i (y) + /2 ( y ) , (x, y) e D . A ssum ing x = 0 we get
u (0, y ) = f 2 (y) = 0 ^ f 2 (y ) = 0, and supposing x = p we a tta in
u (p ,y ) = p ■ f i (y) = 0 ^ f i (y) = 0. Hence, u (x, y) = 0, (x, y) e D .
T heorem 1 is proved.
3 Existence o f a solution to the problem A
In order to prove th e existence of th e solution of th e problem A, we will first consider th e following auxiliary problem : find a nontrivial solution of equation (1) such th a t satisfies conditions (2) and can be represented as
u ( x , y ) = X (x) Y ( y ) . (5)
S u b stitu tin g (5) into equation (1) and separating th e variables, we find th e following ordinary differential equations w ith respect to th e functions X (x) and Y (y):
X (4) (x) - A4X (x) = 0, (6)
Y ( y ) - A4Y (y) = 0, (7)
w here A4 is th e split param eter.
C onsidering th e b o un d ary conditions (2), we generate th e following problem for equatio n (6):
X (4) - A4X = 0, (8)
X (0) = X (p) = X " (0) = X " (p) = 0. ( ) A nontrivial solution to problem (8) exists if and only if
АП = ( ПП) 4, n = 1,2, 3 ,.. . .
These num bers are th e eigenvalues of problem (8), and th eir corresponding eigenfunctions have th e following form:
^ , . [2 n n . .
X n (x) = W - s m — x. (9)
p p
A general solution (7) has th e form
Yn (y) = C ie kny + e 1 кпУ | C 2 cos | ^ k n y ) + C3 sin | ^23k n y \ J , (10)
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University
__ / \ 4/3 ___
w here kn =
3
^n = \ pr) , n e N and Ci, i = 1,3 are unknow n co n stan ts for now.According to (9) and (10), it follows from equation (5) th a t th e functions
Un (x, y) = y p ( c i e fcny + e- 2 kny ( c / V 3 , \ ^ . / V 3 , \ \ \ . n n cos I — kny I + C3 sin I — kny I I I sin — x are th e p a rticu la r solutions of equatio n (1), which satisfy hom ogeneous conditions (2).
Due to th e linearity and hom ogeneity of (1), th e sum of th e p a rticu la r solutions can also be th e solution of equation (1). Taking th is into account we will seek th e solution of problem A in th e form
V 3 , V 3 ,
U (x ,y ) = \ j 2 ^ ( C ie kny + e 1 kny ( C2cos ( ^ ^ y ) + C3sin ( kny П ) sin — x. (11
n= 1 2 2 P
A ssum ing tem porarily th a t th e series in (11) and its derivatives converge uniform ly and requiring th e function defined by th e series (11) to satisfy th e bo u n d ary conditions (3) we o b tain
, . , , . [2 ^ n n
Uy (x, 0) = V1 (x) = \ ~ L V1n sin — x,
V V P n= 1 P
, . , , . [2 ^ n n
Uyy (x, 0) = V2 (x) = \ ~ ъ V2n sin — x,
VV V P n=1 P
Uyy (x, q) = V3 (x) = * - E V3n sin ™ x ,
V P n=1 P
where
1 V3
kn C 1 — — kn C 2 +--- ^ kn C 3 = 'фln, kn C 1 — - kn C 2 - - kn C 3 = V2n
1 / г X -
knekn qC1 + kne 2 q cos ( - - 3 kn q — —П j C2 + kne 2 ? sin
(12) V23knq — ^ J C3 = V3n.
We can see from (12) th a t th e num bers Vin are th e Fourier coefficients of th e function V (x) when th ey are expanded into th e Fourier series in term s of sines on th e interval (0,p) , i.e.
Г- P
^— (’ n n ___
Vin = \ ~ Vi (£) s i n — £d{, i = 1,3.
P P
0
Let us calculate th e d e term in an t of th e system (12), i.e.
A
kn kn2
- 1 k 2 n _ 1 k2 - kn
kneknq kn; e 2knq cos ( kn q — 2П-
kn
— V 3 k2 kn kne-2knq sin ( V3kn q —
V 3 ,
3
= \/3 k n e knqA , A = - + e 2knq sin —— kn q — — .
Buketov
University
Lem m a. For an a rb itra ry positive q, th e inequality A > 0 holds . Proof. W rite th e d eterm in an t in th e form
A = ^3кП eknqA (x n ), A (xn) = 2 + e- ^ xn sin (x n - 6) ,
where, xn = ^ k n q = - ^ 3 {/A^q > 0, n e N .
Find th e m inim um value of A (x n ). To do this, calculate th e first order derivative d A (£ n) = 2e-y3x„ Sin ( n - Xr
dxn 3
1) W hen 0 < x n < П3 , we have A '(x n ) > 0 .This m eans th a t A (x n ) increases at finite discrete values of x n , b u t it does not reach its m axim um value. T h en th e function A (x n ) takes its m inim um value a t n = 1 and we have th e estim atio n
A (x n ) > 2 + e ^ 3xi sin ( x i - = 5 i> 0,
w here x i = ^ {/A^q.
n 4n
2) If x n > —, th e n th e function A (x n ) takes its first m inim um when th e argum ent is — and we
3 3
achieve
A (xn) > 2 ( 1 - e-4M n) = $2 > 0.
3) For sufficiently large values of x , it is obvious th a t th e function A (x ) tend s to i . From here we
find 2
A > 5 = m in {$i ; 52} > 0.
Considering th e above considerations we conclude th a t A n > 0 . T h e lem m a is proved.
Hence, th e system of equations (12) has a unique solution.
Below, we determ ine all unkown num bers Cj, i = 1,3:
^ in k n e 2knq sin — knq - ^2nkne 2 kn» cos — knq + 7 + -7Г ^3n k 3n
C2 = A ^ in k n ^ e - 2knq sin ^ k n q + 3 j - ^ e kn»^
- ^2nkn ^ e- 1 kn» sin ^ V 3 knq + П j + V3ek" qj + V ^ n ^
^ in k n e kn» 1 - e - 2kn» cos V 3 knq + 3 ^ + ^ n k n e kn» f e - 2kn» cos f k n q + 3 ^ - 2 In w h at follows, th e m axim um value of all found positive known num bers in estim ates will be denoted by M .
Taking account condition (4), we in tegrate ^ i (x) by p a rts four tim es, and ^ (x), i = 2 ,3 by p a rts two tim es we get th e following estim ates:
Ы < M , i^ii < M i - p l , i = 2, 3, (13)
Buketov
University
where
p p
Ф1? = у ф (4) (e) х ? ( о de, Фгга = у ф" ( о х ? ( о de, i = 2, з.
0 0
For Сг, i = 1, 3 we can w rite th e following estim ations:
|С г| eknq < M ( e-2knq + 1ф2?1e-2knq + ^^ЗР1 ) < M
V k? k? k? /
IC 21 < M f ( ^ ^ ) f e -3knq + ^ j + e-knqj
|Фгп| . |Ф2п| . |Фзп|
16 + 14 + 14 n 3 n 3 П 3
k?
|C3| < M
16 1 14 1 14
n 3 n 3 n 3 П ф ? 1 f 1 + e - 2Ч + M f d ( 1 + e -2k .M I < M
V k? V2
|Фг?| . |Ф2?|
16 + 14 n 3 n 3
Theorem 2. If ф г^ ) e C4 [0,p ], фг (x) e C2 [0,p], i = 2 ,3 and th e corresponding conditions (4) are satisfied, th en th e solution of th e problem A exists and it is represented by th e series (11).
Proof. If th e series (11) and its derivatives u xxxx, u yyy converge uniform ly in th e region D , th en th e function u (x, y) defined by th is series will be th e solution of th e problem A.
From (11) we have
те
|u (x ,y )| < £ ( |C1| eknq + IC2I + |C 3 ^ |X ? (x)|.
?=1
(14)
T hen, tak in g account of (13) it is obtained from (14) th a t
|U ( x , ;</)!< M f £__ ! % i + £16 1 / y I% 114 + £1 / у |Фз" '14 i 1 n 3 1 n 3 1 n 3
V?=1 ?=1 ?=1
< .
T his implies th a t th e series (11) converges absolutely and uniformly.
Now let us prove th a t th e p a rtial derivatives of th e series (11) w ith respect to b o th variables included in th e equation also converge absolutely and uniform ly in th e region D . C alculating th e derivatives w ith respect to y, it follows from (11) we ob tain
d 3u
d y 3 £ k
?=1
C1ekny + e 1 knV I C2 cos I ^ k ? y ] + C3 sin | -^ 3 k?y X ? (x) . (15)
From (15) we determ ine th e estim atio n d 3u
dy < £ k? ( I C1I ek~» + | C'2 | + | C3 | ) | X ? (x) | < M ( £ ^ + £ ^ + £ ^
?=1 v 7 V?=1 n 3 ?=1 n 3 ?=1 n 3 ,
Using th e Cauchy-Bunyakovsky and Bessel inequality we a tta in
(16)
д3гdy3
(
Е ^т е . . + , / Е |Фте 2? | % / Е -V + , / Е |Ф 3 „ |\ / Е -V /те /те /те \ <?=1 ? 3 у ?=1 у ?=1 ? 3 у ?=1 у ?=1 ? 3 у
< m ( Е Щт1 + ||Ф2?У / Е Л + ||Ф3?И < ~ ,
\ ?=1 ? 3 у ?=1? 3 у ?=1? 3 /
Buketov
University
where
^ 2 2
^ |^ 1 n |2 < v(4)(x) _ , ^ l^ in l2 < Vi(2)(x)
l 2 [°;p] —r , i = 2, 3.
L2 [°,p]
Therefore, th e series (16) converges absolutely and uniformly. T h e absolute and uniform convergence d 4u d 3u of th e p a rtial derivative of th e fourth order in x series (11) follows from th e equality ^ —4 = -^-3 .
T heorem 2 is proved.
Conclusion
T h e article considers an initial b o u n d ary value problem for a fourth-order equation containing th e th ird tim e derivative w ith m ultiple characteristics. Uniqueness theorem s are proved using th e m ethod of energy integrals. T h e existence of a solution is shown w ith th e help of conditions im posed on given functions con structed as a series by th e Fourier m ethod and a regular solution of this series.
Acknowledgm ents
T h e au th o rs declare th a t th ey have no known com peting financial interests or personal relationships th a t could have appeared to influence th e work reported in this paper.
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2 Н ам анган и,нженерл'1к цуры лы с и нст ит ут ы , Н аманган, взб ек ст а н
Уак,ыт бойы нш а ушшш1 туы нды сы бар тертшш1 ретт1 т ец д еу уш ш ш екаралы к есеп ж ай ы н да
М а к а л а д а уакы т бойы нш а ушшш1 туы нды сы бар б1ртект1 тертшш1 ретт1 дербес туы нд ы лы д и ф ф е рен ц и алды к тецдеу уш ш тж б уры ш ты облы стагы ш екар ал ы к есеп карасты ры лды . К ойы лган есептщ шеш1мшщ ж а л гы зд ы гы энергия и н тегралдары эд1с1мен дэлелденген. А йны м алы ларды белу эд1сш крлданы п, есептщ шеш1м1 X (x) ж эн е Y(у) ею ф ункцияньщ кебейтшд1с1 турш де 1здест1ршед1. X (x) аньщ тау уш ш [0,p] кесш д!сш щ ш екарасы нда т е р т ш ек ар ал ы к ш арты бар тертшш1 ретт1 карапайы м д и ф ф ер ен ц и ал д ы к тендеуш , ал Y (у) аны ктау уш ш [0,q] кесш д!сш щ ш екарасы нда уш ш екаралы к ш арты бар ушшш1 ретт1 карапайы м д и ф ф ер ен ц и ал д ы к тецдеуд1 алам ы з. Б ерш ген ф у н к ц и ял ар га ш арттард ы кою аркы лы есептщ т у р а к т ы ш еш !мш щ бар е к е н д т ту р ал ы теорем а дэлелденедь Кой- ы лган есептщ шеш1м1 ш е к а з катар турш де куры лы п, катарды ц б ар л ы к ай н ы м алы ларга каты сты
Buketov
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к,атарды мушелеп диф ф еренциалдау м у м ю н д т аньщ талган. Б1рк,алыпты ж инак,талуды табу кез1нде
«юш1 б елпш » нелге тен; емес екен1 аньщ талды .
К ш т свздер: бастапк,ы-шетт1к есеп, Ф урье эд1с1, шеш1мнщ ж ал гы зд ы гы , бар болуы, меншжта мэн, менш1кт1 ф ун кц и я, функционалды к, к атар, аб солю ти ж эн е б1рк,алыпты жинак,талу.
Ю .П . А п а к о в 1,2, Д .М . М е л и к у з и е в а2
1 И н с т и т у т м а т ем а т и ки им ен и В.И . Романовского А кадем ии наук Р еспублики Узбекистан, Таш кент, Узбекистан;
2 Н ам анганский инж енерно-ст роит ельны й и н с т и т у т , Н ам анган, Узбекист ан
О граничной за д а ч е д л я уравнения четвёртого порядка, содер ж ащ его третью производную по времени
В статье рассм отрена к р ае вая за д а ч а в прям оугольной области д л я однородного диф ф еренциального уравнения в частны х производны х четвёртого порядка, содерж ащ его третью производную по време
ни. Единственность реш ения поставленной задачи док азан а методом интегралов энергии. И спользуя м етод разделения переменных, реш ение задачи ищ ется в виде произведения двух ф ункций X (x) и Y (у). Д л я определения X (x) получаем обыкновенное диф ф еренциальное уравнение четвёртого по
ряд к а с четы рьм я граничны м и условиями на границе сегмента [0,p], а д л я Y (у) - обыкновенное диф ф еренциальное уравнение третьего поряд ка с трем я граничны м и условиями н а границе сегмента [0,q]. Н ал агая определенные условия на заданны е функции, доказан а теорем а сущ ествования регу
лярного реш ения задачи. Реш ение поставленной задачи построено в виде бесконечного ряда, обос
нована возм ож ность почленного диф ф ерен ц и рован и я р я д а по всем переменным. П ри доказательстве равномерной сходимости установлена отличность от нуля «малого знаменателя».
К лю чевы е слова: начальн о-краевая задача, м етод Ф урье, единственность, сущ ествование, собственное значение, собственная ф ун кц и я, ф ункциональны й ряд, абсолю тная и равном ерная сходимость.
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