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22Bulletin KRASEC. Phys. & Math. Sci, 2014, vol. , no. 2, pp. 18-22. ISSN 2313-0156
MSC 35C05
ON ONE PROBLE FOR HIGHER-ORDER EQUATION
A.V. Yuldasheva
National University of Uzbekistan by Mirzo Ulugbeka, 100174, Uzbekistan, Tashkent c., VUZ gorodok st. E-mail: [email protected]
In this paper not well posed problem for the even-order equation is studied. The stability of the problem is restored by additional conditions and conditions to domain.
Key words: partial differential equations of higher order, not well posed problem, method of separation of variables, simple continued fractions.
Problem definition
The present paper considers for the equation
d2ku d2u
= 0, k = 2n + 1, n G N, (1)
dx2k dt2
in the domain D = {(x,t) : 0 < x < n, 0 < t < 2n} a problem with the following conditions: d 2m u d 2m u
dx^(0,t) = dX2u(n,t) = 0 m = 0, l,...,k-l, 0 < t < 2n (2)
u(an,t)= f (t), 0 < t < 2n, (3)
where a is some constant from (0,1) and f(t)— is the given quite smooth function.
We shall show that if a is an irrational number, then the theorem of solution uniqueness
2k 2
of the problem (1), (2), (3) is valid in the class u e Cxt' (D).
Note that this problem is ill-posed, since a small change in the function f(t) under the norm Cs(s e N) may cause arbitrary large change of the solution u under the norm L2.
This problem may be regularized by a side condition, for example, by a priori estimate
n2n ' dku \2
dtdx < E2, 0 < t < 2n, (4)
d xk
0 0
where E is the defined constant.
Yuldasheva Asal Victorovna - Ph.D. (Phys. & Math.), Lecturer of the Dep. Differential Equations and Mathematical Physics, of the National University of Uzbekistan, Tashkent. ©Yuldasheva A.V., 2014.
The problem well-posedness
-,2k,2 ,
(2), (3), then u may be presented in the form of a series
^ i \ u(x,t) = ^ sinnx(ancosnkt + bnsinnkt\, (5)
n=1
and it follows from this representation that the function f (t) should have the form
^ / \ f (t) = ^ sin nan (an cos nkt + bn sin nkt\. (6)
n=1
Theorem 1. If a is a irrational number, the problem (1), (2), (3) does not have more
2k 2
than one solution of u e Cxy (D).
Proof. Indeed, if in (6) f = 0 , than an = bn = 0. Consequently, h u = 0. □ Remark. If a is a rational number, there is no uniqueness.
For example, let q be some natural number, then the function u (x,t) = sinqxcosqkt satisfies (1), (2) and
n
u(-, t)= 0, 0 < t < 2n. q
Defination. We shall indicate that the irrational number a have the order Q, if Q is the upper boundary of the numbers a, satisfying the inequality
P
a--
1
q
for any f g Q. It is known that almost all the numbers a have the order Q = 1 [3, ?].
H
The next statement is associated with the question on the solution stability depending on a and f. Here is an example.
Theorem 2. Let a be an irrational number. Then there is a sequence 2n of periodic
2k 2
functions fn g C™ (R), uniformly vanishing, and it is such un g Cxt' (D), satisfying (1), (2) and
un(an, t ) = fn(t ), (7)
the following relation holds
Proof. Let
\\m\\Un\\L2{D)=+c
(8)
fn(t ) =
Vnk
k
sin n t,
then
Unix, t ) =
nk sin nan)
i
k
sin n t sin nx.
It is known that [2] there is a sequence of such integers pn, qn that
\im qn = +<*>,
n—> œ
Pn
a - —
qn
1
< ~2 qn
q
and then the theorem statement appears from the following estimation
|sinqnan| = |sin - n| < —.
qn
Note, that for any given integer s there is such an irrational number a (for example, of the order s + 2) that the solution
un(x, t) = n-1-s(sin nan )-1 sin nkt ■ sin nx
of the problem (1), (2) and un(an,t) = n-1-ssinnkt satisfies the following estimation
lim \\un\\L2(D) = + ~ lim Ifn\\as(D) = 0,
n—y^ 2\ ) n—y^ ^ '
from which it is clear that the problem is ill-posed. □
Further we shall show that the problem is also unstable relative to a.
Theorem 3. Let p,q G N,p < q and {an} be a sequence of irrational numbers
converging to P. And let un g C^'2 (D) be the solution of the problem (1), (2) and
q '
un (an,t) = sinqkt, then
Hm\\un\\L2(D) =+<*>.
k
The solution is written in the form un(x,t) = S1"q *anqx , from which the theorem
^ 1 ' sin qann
statement is obvious.
Thus, a side condition is required.
d k \ ^
—r | dtdx < E2. dxk
0 0
Problem with a bounded solution
Let a,£,E be some positive constants, and a g (0,1).
2k 2
Let f g L2 (0,2n). The function class u g Cx t' (D) satisfying (1), (2) and
\\U (an, ■) - f 11L2(0,2n) < £,
(9)
d ku
d xk
< E.
(10)
¿2(D)
is indicated by r (e, E). The condition (9) substitutes the condition (3), and the priori estimate (10) is required for the problem to be well-posed. We introduce the following notations
2n
|u|| = sup / u2(x,t)dt,
(11)
xG[0,n ]
0
A(£, E )= sup ||v — w|
v,wGr(£ ,E )
Theorem 4. Let a be a rational number, a = P, (p, q) = 1 and
q
E
Then
If q =
q2 < 2- . £
E
A (£, E) < 3 e .
2E\k £
, then
A (e,E) <
2k 2
Proof. Let v,w e r(e,E), then u = v — w e Cx,t, (D) satisfies (1), (2) and
d ku
\u (an, 011^(0,2*) < 2£,
d xk
< 2E.
L2(D)
Since u is represented as (5), the conditions (15) are rearranged as
p 4e2
£ (al + b2) sin2 n-% < ,
n=1 q n
C 17 2
J2kf „2 , u2\ - 8E
£ n2k(a2n + b2n) <
n=1
n2
whence it follows
From (5) we have
p £2 8£2 £ (sin2 nqn + n2^ ) (an + bü < —.
n=l
= n max £ (a2 + b^) sin2 nx < % £ (al + ^.
xG[0,n ] n=1
n=1
It follows from (12)
A2 (e, E) < n supj £ (a2n + bl) : an, bn\ ,
\n=1
satisfying (18).
According to the Lagrange multiplier role we find
2 2 p 2k e2 1
A2 (£, E) < 8£2 min (sin2 n + r2k)-1, r G N
q
E2
From (13) we obtain
p £2 £2 sin2 r-n + r2k—> q2k, 1 < r < q.
q E2 E2
(13)
(14)
(15)
(16)
(17)
(18)
(19)
Substituting this estimation into (19) we obtain (14). The theorem has been proved. □
2
u
Assume that a is an irrational number expanded into a continued fraction
1
a = —
ai + 1
«2 + ...
Theorem 5. Let a g (0,1) be an irrational number and at < Ka, then
i
A (£, E) < 3 . (20)
Ka + 2 X2 2
Proof. To make sure that this estimation is valid, note that it follows from the theorem 4 that the estimation (19) is true for A (e,E) then from the condition at < Ka [3] we obtain
2. - 27
Then
sin ran >-^—, r> 1.
" 4 (Ka + 2)2r2 "
• , 27 2k £2 1 £ V27
min <-^--+ r2k—> >
rGNv| 4 (Ka + 2)2 r2 E2 J -E (Ka + 2)' Thus, the required estimation follows from the above
A2 (£,E) < 9ee[K0+2
□
References
1. Frosali G. Papi. On the stability of the Dirichlet problem for the vibrating string equation. Ann. Scuola Norm. Sup. Pisa, 1979, vol. 6, pp. 719-728.
2. Yuldasheva A.V. Ob odnoj zadache dlya uravneniya vysokogo poryadka [A problem for the equation of high order]. Doklady Akademii nauk Respubliki Uzbekistan - Reports of the Academy of Sciences of the Republic of Uzbekistan, 2012, no. 5, pp. 11-14.
3. Hinchin A.Ya. Cepnye drobi [Continued fractions]. Leningrad, 1961. 112 p.
Original article submitted: 23.10.2014
