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29Bulletin KRASEC. Phys. & Math. Sci, 2015, V. 10, №. 1, pp. 4-10. ISSN 2313-0156
MATHEMATICS
MSC 47F52+47F05
ON THE STABILITY OF THE BOUNDARY VALUE PROBLEM FOR EVEN 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 we consider ill-posed problem for one even-order equation. The stability of the problem is proved with the additional assumption.
Key words: partial differential equations, ill-posed problem, boundary value problem, algebraic numbers, the simple continued fraction
Introduction
We consider following problem for even order equation:
= 0, k, p e N, 0 < x < n, 0 < t < an,
dx2k dt2P d2mu ,„ , d2mu
(0,t) = —^~ (n,t) = 0, m = 0,1,...,k- 1, 0 < t < an, d x
(x,0) = pj (x), j = 0,1,...,p- 1, 0 < x < n,
dx2m ' d x2m d ju
d tj d j u
—r (x, an) = Yj (x), j = 0,1,...,p- 1, 0 < x < n,
where a is a positive constant.
If k = p = 1 we get The Dirichlet problem for the vibrating string equation, which is a classical ill-posed problem due to its irregular behavior. Its solution may neither exists, nor be uniquely determined, nor depend continuously on the data (see [1]-[3]).
In [2], the Dirichlet problem for the wave equation was studied with the additional assumption of an "a priori" bound for the gradient of the solution. Case when p = 1, k e N was studied in [6].
The present research leads to some problems of Diophantine approximation. Let us note that formulate problem is ill-posed problem if k — p is even number. Therefore, the above problem cannot be suitably dealt with if a and pj (x), Yj (x), j = 0, 1, . . . , p - 1 are known within a certain approximation.
Yuldasheva Asal Victorovna - Ph.D. (Phys. Math.), Lecturer of the Dep. Differential Equations and Mathematical Physics, of the National University of Uzbekistan by M. Ulugbek, Tashkent.
©Yuldasheva A.V., 2015.
Main results and comments
Let çj (x), yj (x), ( j = 0,1,..., p - 1) be functions in C2k [0, n] such that ç )2l> (0) =
,(20
çj2,) (n) = Yj2i)(0) = Yj2,)(n) = 0, , = 0,1,...,k- 1, j = 0,1,...,p - 1 .
Let t, E, a, 5 be positive constants. We consider solutions u in C2/2p ([0, n] x (0, +^]) of the following problem:
№
r(2f)
d2ku Ä
dx2k dt2P
= 0, k, p G N, 0 < x < n, t > 0,
d 2m w d 2m w
^ (0,t) = ~dX2fn (n,t) = 0, m = 0 l,...,k-l, t > 0,
d ju
d tj
(x, 0) - çj (x)
d ju
d tj
tj k) - Yj (x)
L2[0,n ]
< 5nVE, j = 0,1,...,p- 1,
L2[0,n ]
< önVE, j = 0,1,..., p - 1, I Tj - a | < 5,
dxv +
d pu ~dfP
I dx < E, t > 0.
(1) (2)
(3)
(4)
(5)
for real numbers Tj, j = 0,1,...,p — 1 depending on u and satisfying \xj — a| < 5. The meaning | Tj — a | < 5 is that the final time an is known up to a given error.
We denote by r5 the set of all C^'2p ([0,n] x (0, +~]) solutions of (1-(5). We note that if 5 = 0, then the problem (1-(5) is reduced to the classical boundary value problem with additional assumption (5). It was studied in [6] that this problem may have no solutions.
Let DiamF5 = sup ||v — w||. Let v1, v2 G r5. Then there are Tj such that
v,wer5
d ju
d tj
and let
x, Tjn) - Yj (x)
L2[0,n ]
< 5n VE, i = 0,1, j = 0,1,..., p - 1, | Tij - a |< 5
u (x, t) = v1 (x, t) - v2 (x, t), (x, t) g [0, n] x [0, +<*>)
(6)
Then u g C2xfp ([0, n] x [0, +<^)). Moreover, u satisfies equation (1), conditions (2) and following
d ]U' ~ < 25n VË, j = 0,1,..., p - 1, (7)
L2[0,n ]
d tj
j (x, 0)
d ju
~dtJ
(x, an) - Yj (x)
L2[0,n ]
< 45nVE, j = 0,1,...,p - 1,
d^uY d xk)
+
d pu ~dfP
\dx < 4E, t > 0.
(8) (9)
n
2
It is easy to verify (1), (2), (7)-(9). We can write the function satisfying (1) and (2) in the following form
, , ^ . I An sinnp (an — t) . k u(x,t) = £ sinnx <-k--+ Bnsinnpt }.
n>i I sin up an
Similarly, we can rewrite (7)-(9) as follows:
£ An < 852nE,
ni
£ BnsinV an < 3252E,
ni
d ku
d xk
(•, t )
+
L2 [0,n]
d pu
d tp
(•, t )
La[0,x ]
< 4E, t > 0.
(10) (11)
(12)
Defining:
we obtain from (12)
On = \ —
n ( An sin up (an — t)
+ Bn sin npt
sinnp an
dkuf x 2 (•, t )
d xk
< £ n2kOn2 < 4E,
L2[0,n ] n>1
whence
N
N
I u 0, t )\\2Li [0,n] = £ On2 + £ On2 < £ On2 + 4E
n=1 n=N+1
n=1
N 2k'
We now have following bound:
2 n l k n —2 N
\u (•, t )\\^ [0,n] < n max (sin up an
n=1N
£
n=1
AUsin2up (an — t) + BUsin2up an • sin2up t +
sin npt
sinup an
+2 |An||Bn| sinnp (an -1) Therefore, from (10) and (11) it follows that
+ 4E
+ N2k .
max 1 \\u (•, t)\L2[0,n] = \\u\\2 < 4052n2E ma
E[0,an] /L ' J n=1
t e[0,an ]
max I sinup an
,N
2 4E
+ NEk,N =1,2,....
Let
a=
1
a1 +
1
Û2 + ...
be the simple continued fraction for a, where the partial quotients an are integers such that, an > 1.
We consider the set of irrational numbers with bounded partial quotients, i.e. the numbers a, for which there exists a constant Aa satisfying an < Aa for all n . We note that if a is a quadratic irrational, then the expansion of a as a simple continued fraction is ultimately periodic, which implies that an has bounded partial quotients.
2
2
Then from theory of continued fractions (see [5] p.37) we easily obtain
max
n
ax fsin up an) <
1 ,N V )
-2
n
sin
(Aa + 2) Nr
, N = 1,2,
Since sinx > ^n3x for x e [0,%/3], we have for every N
2k
-8 2 n 2E (Aa + 2)2N
27
(13)
u\\2 < 52 n 2E (Aa + 2)2N p + , N = 1,2,...
N2k
Now let
2k
g (t) = 127052n2E(Aa + 2)2t P + 4Et
The minimum value of g for t > 0 is attained at
p
p
t =( ^ 2k ( p + 1)(Sn (Aa + 2))-k (p + 1)
Since g is an increasing function on the interval [t, we have
g([t + 1]) < g(t + 1).
We obtain
. Il2 160E/C ^..Jp.
|u\\2 <— (5n (Aa + 2)) p+1
1 +
27p\ (p+1)2k
p ■
(p+1)k
+ (5n (Aa + 2))
2k p
(14)
So we proved following
Theorem 1. Let a is an irrational number and has the simple continued fraction with bounded partial quotients. Then for (Diamr$) (14) is valid.
Now we use some results obtained in [4]. By corollary 6 of [7], since a has a type Q < there exist K = K (Q, a) > 0 and, for any 8 > 0, a number % e R\Q such that
li — a| < 5,
and
max (sin uni ) < I sin
n=1,N 1
n (3 — V5
-2
2N
(15)
(16)
for all N > K8 Q. From (15) it follows that \xj — a| < 8, for every Tj satisfying — Tj| < 28.
If u is defined by (6), we obtain from (8) d ju
d tj
(x, )
< 45^V/E, j = 0,1,...,p - 1. (17)
L2[0,n ]
Therefore, u satisfies conditions (1), (2), (7), (9) and (17). The solutions of the problem (1), (2), (7), (9) and (17) u g C2xk,2p ([0, n] x [0, +<*>)) of the form
, , ^ . [ An sinw (Bn -1) . k
u(x,t) = £ sinnx <-k--+ Bnsinnpt
n>1 [ sinnp Bn
which satisfies (10), (12) and
£ Bnsin2npBn < 7252E. (18)
n>1
As in proof of theorem 1 we obtain
o / k \ -2 4E
lluM2 < 805 2n 2E max sin np Bn + , N = 1,2,....
n=1,NV / N2k
Using (16) and sinx > |x for all x g [0, f], we obtain
o 80 i i 2k 4E a
llull2 <-52n2EN2k + 4Ekr, N > K5-e. (19)
(3 -V5)2 N
Let
2k
g(t) = , 8V,2 52n2Et p + 42k, t > 0. (20)
(3 -v^)2 t
The minimum of g for t > 0 is attained at
^ —p
/3-A/5\ k0
t = ( p ) 2k( p + 1)f 3-V^ k(p + 1)
207 \ K5 J
We choose 5 as
k(p +1)
P / ^ \^TT] k0 (P + 1) - P 0 < 5 « k(|) 2k(p + ^ k(p + 1)[ . (21)
It follows from (21) that t < K5—e. Let N be the integer > K5—e for which the right side of (20) is minimum. Since g is increasing on the interval [t, , N satisfies K5—e < N < K5—e + 1. Hence
||u||2 < g (K5—e + A ,
and finally
, „о 80n2E
u2 <
(3 -v^)
■ k K 8 P-в + 8 p
2k
P 4E 8 2ke + —kvt
(22)
which proofs following:
Theorem 2. Let a be an irrational number and has a type Q < ^ Then for any
fixed в,
a
a +1
< в < 1, there is constant K = K(в, a) > 0 such that
, ll2 80n2E
u2 <
(3 -v^)
p
kk
—-в —
K 8 P + 8 P
2k
P 4E 8 2ke
+
for any 0 < 8 < < K (20) 2k(P + 1) ( k(p + 1)
k( P +1)
I_) ke (p + 1) - P
We conclude with the proof of the following: Theorem 3. The problem (1)-(5) is stable if and only if a is irrational. Moreover,
if a is irrational then lim (DiamFs) = 0 uniformly in pj (x), Yj (x), (j = 0,1,...,p — 1)
8 ^0
Proof. Let a e Q. By corollary 9 of [4], there exist a function f (8) such that
lim f (8) = lim 8 f (8) = 0,
(23)
and, for any sufficiently small 8, a number % e Q, satisfying (15) and (16) for all N > f (8) . The same argument given in the proof of theorem 2 shows that
N2 < g(f (8) + 1),
where g is defined by (20), i.e.
, ll2 80n2E U 2 <
(3 -V5)2
■ к t f (8 ) 8 p + 8 p
2k P
+
4E
f(8)
2k
By (23), this yields
□
References
lim (DiamF8 ) = 0.
80
1. Bourgin-R. Duffin. D.G. The Dirichlet problem for the vibrating string equation. Bull. Amer. Math. Soc., 45(1939), 851-858.
2. Fox-C. Pucci. D. The Dirichlet problem for the wave equation. Ann. Mat. Pura Appl. (IV), vol. XLVI (1958),pp. 155-182.
3. John F. The Dirichlet problem for a hyperbolic equation. Amer. J. Math., 63(1941), pp. 141-154.
4. Viola C. Diophantine approximation in short intervals, Ann. Scuola Norm. Sup. Pisa, 6(1979), pp. 703-717.
5. Khinchin A. Ya. Continued fractions. The Universiry of Chicago Press, 1964, 112 p.
6. Yuldasheva А.V. On one problem for high-order. Reports of the Academy of Sciences of Uzbekistan, Tashkent, 2012, №5, pp. 11-14.
Original article submitted: 23.03.2015
