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43
Математика, механика, информатика
yflK 517.95
M. B. Frost
SOLVABILITY OF AN INITIAL-BOUNDARY PROBLEM FOR A LOADED WAVE EQUATION
Solvability of an initial-boundary problem for a loaded wave equation is proved. The proof of the solution uniqueness is based on the a priori estimate of the solution. A sequence of Galerkin approximations is constructed for the solution existence to be proved.
Keywords: loaded wave equation, uniqueness, solution existence.
A loaded equation is an equation with partial The following equality is valid: derivatives that contain the values of functionals of the t
sought function. Such equations are encountered in j(U(x),b(x,t)ux(x,t))dT = (£7(x),ux(x,t)b(x,t))-modeling certain physical processes [1] and also in solving inverse problems of mathematical physics [2; 3].
Let us consider the following initial-boundary problem for the function u (x,t), (x,t) e Q : [0,l] x [0,T] :
-(U (x ), ux (x,0 )b (x,0 ))-J(U (x ), ux (x, т) (x, т)) т.
(9)
J (сu, )dt <J|—||cf + ^IKf Idt
(10)
The following estimates are obtained:
T
u„ - uxx - b (x t) j (T ~T)uxx (x T) dT = c (^ t) , (1)
0
u (x,0) = p (x), ut (x,0) = n (x), (2)
u (0, t) = u (l, t) = 0, (3)
p (0) = n (0) = p (l ) = n (l ) = 0. (4)
Here, b (x, t), c (x, t), p (x), and n (x) are specified Here, e is an arbitrary positive number and B is the
functions. maximum value of |b (x,t)| in Q . Taking into account
To obtain an a priori estimate of the solution, we Eqs. (9, 10), and (8), we obtain multiply Eq. (1) by ut and integrate the result with
respect to x from 0 to l :
((x),Ux (x,t)b(x,t))|< Вi2T\U\f +1||u
Y/2
x)d т ,
V
2I -INI dt,
I|U| I2 + Til
d ( 1m ||2 1m ||2 I
dt I \u<\ + 2 I lux| I l = (Ux , but )+(C, u> )• (5)
|ut|P +1 |ux||2 < C2 + BT||ux112 +
+aT21 \\ut\2 dt + РГ21 \\u\2 dt.
(11)
Here, \[\\ and (•, •) are the norm and the scalar Her6, the positive constants a, P , and C2 depend °n the
values of |b (x, t)|, |bt (x, t), and |bx (x, t) in
product in L2 (0, l),
T
U = U(x) = j(T -x)ux (x,x)dt .
0
By virtue of conditions (3), we have
(Ux,but) = -(U,bxu, + bua).
maximum
(6)
Q . Integrating Eq. (11) with respect to t from 0 to T . we obtain
Integrating Eq. (5) with allowance for Eqs. (2) and (7) from 0 to t , we obtain
(1 -aT3) j||ut||2 dt + (1 -BT-PT3) j||ux||2 dt < C2T . (12)
0 0
(7)
Under the conditions
(1 -aT3 )>8j > 0, 1 - BT-PT3 >82 > 0, (13)
Here,
~l|ut||2 + ~l|ux||2 = C +J(c, ut )dt -
22 0
t t -J(U, bu )dt -J(U, bux )dt.
dp (x )
(8)
where 8j and 82 are arbitrarily small fixed positive numbers, we obtain the sought a priori estimate
TT
Ik 11 dt < const, jl lull dt > const. (14)
0
0
C1 = 21" (x )H2 + \
dx
This estimate means that u is bounded in the space W1 (Q) of functions that have generalized first-order derivatives integrated with a square.
Вестник Сибирского государственного аэрокосмического университета имени академика М. Ф. Решетнева
Estimate (14) immediately yields the theorem of the solution uniqueness, because the constants in Eq. (14) should be equal to zero for different solutions to exist.
The solution existence is proved with the Galerkin method. We seek for the Galerkin approximation in the form
m
um (x, t) = X Vmk (t)Vk (x),
k=1
where vk (x), k = 1, 2,... is the basis and qmk (0) = pk, q'mk (0) = nk, where mk, nk, k = 1,2,... are the coefficients of the expansion of the functions p ( x ) and n (x) in the basis vk (x). The basis is found by solving the problem
v"k(x) + !kvk (x) = 0 , Vk (0) = Vk (/) = 0,
where |k are eigenvalues. Obviously, this basis has the form
/ \ • nkx n2k2 , , „
vk (x) = sin—, Ik k = 1,2,...
From the condition of orthogonality, we obtain a system of ordinary differential equations for the functions qmk (t)
i T
m -um ■
tt "xx
\
= 0,
(15)
Thus, a theorem follows from here. Let
с (x, t)e L2 (Q) , b (x, t) e L°° (Q), bx (x, t)e LL (Q), b, (x, t)e LL (Q);
m (x) e W 2(Q) , n (x)<
L2 (Q)
there exists a
-J(T-TX (xT)dT-c,vk
v 0
which has to be solved under the conditions
qmk (0) = pk , q'mk (0) = nk. (16)
After the same considerations as those performed for obtaining the a priori estimate (14), we obtain
T T
JI\u^\dt < Cj, JI\u^\dt < C2,
00
where the constants C1 and C2 are independent of m . From this estimate, there follows that the system of ordinary equations (15) with conditions (16) is solvable and that it is possible to choose a subsequence from the Galerkin approximations, which converges to a certain function, which is a solution of the initial problem.
and condition (13) be satisfied. Then, unique solution of problem (1-4), and
u £ L2 (0,T), W2(0,l), ut e l2 (q).
To conclude, we should note that a similar theorem is also valid if we take for 0 < t < T u = u (x, y, t),
(x, y) e G; u = 0, (x, y) e dG, replace uxx in Eq. (1) by the Laplace operator Au, and replace the functions
b (x, t) and c (x, t) by B (x, y, t) and C (x, y, t). The
basis vk = vk (x,y) and eigenvalues k, k = 1,2,... are determined in this case by solving the following problem:
Avk +lvk = ^ (^ y)£ G, vk |dG = 0.
An initial-boundary problem for a one-dimensional (in terms of the spatial variable) loaded wave equation is considered. This equation contains a functional of the sought function. An a priori estimate of the solution is obtained, which is used to prove the solution uniqueness. A sequence of Galerkin approximations is constructed; a converging subsequence that is a solution of the initial problem is chosen from this sequence. The results of this paper can be used to justify the correctness of models of some physical processes and to solve inverse problems of mathematical physics.
References
1. Nakhushev A. M. Loaded differential equations and their applications // Dif. Uravn. 1983. Vol. 19. № 1. P. 86-94.
2. Frost M. B. Uniqueness of the solution of an inverse problem of determining the source of oscillations of an elastic body // Sib. Zh. Industr. Mat. 1998. Vol. 1. № 2. P. 199-2001.
3. Kozhanov A. I. One nonlinear loaded parabolic equation and related inverse problem // Mat. Zametki. 2004. Vol. 76. № 6. P. 840-853.
M. Б. Фрост
РАЗРЕШИМОСТЬ НАЧАЛЬНО-КРАЕВОЙ ЗАДАЧИ ДЛЯ УРАВНЕНИЯ, ОПИСЫВАЮЩЕГО ВОЛНУ НАГРУЖЕНИЯ
Доказана разрешимость начально-краевой задачи для уравнения, описывающего волну нагружения. Доказательство единственности базируется на априорных оценках решения. Построена последовательность Галер-кина, которая позволила доказать существование решения.
Ключевые слова: уравнение, описывающее волну нагружения, единственность и существование решения.
© Frost M. В., 2011
