Electronic Journal of Differential Equations, Vol. 2001(2001), No. 76, pp. 1–8.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu (login: ftp)
EXISTENCE OF SOLUTIONS FOR ONE-DIMENSIONAL WAVE
EQUATIONS WITH NONLOCAL CONDITIONS
SERGEI A. BEILIN
Abstract. In this article we study an initial and boundary-value problem
with a nonlocal integral condition for a one-dimensional wave equation. We
prove existence and uniqueness of classical solution and find its Fourier representation. The basis used consists of a system of eigenfunctions and adjoint
functions.
1. Introduction
Certain problems of modern physics and technology can be effectively described
in terms of nonlocal problems for partial differential equations. These nonlocal
conditions arise mainly when the data on the boundary cannot be measured directly.
The first paper, devoted to second-order partial differential equations with nonlocal integral conditions goes back to Cannon [4]. Later, the problems with nonlocal integral conditions for parabolic equations were investigated by Kamynin [10],
Ionkin [9], Yurchuk [18], Bouziani [2]; problems for elliptic equations with operator
nonlocal conditions were considered by Mikhailov and Guschin [7], Scubachevski
[17], Paneiah [13].
Then, Gordeziani and Avalishvili [5], Bouziani [3] devoted a few papers to nonlocal problems for hyperbolic equations. Pulkina [14, 15] studied the nonlocal
analogue to classical Goursat problem.
In this paper we investigate the nonlocal analogue to classical mixed problem,
which involves initial, boundary and nonlocal integral conditions. In the rectangular
domain D = {(x, t) : 0 < x < l, 0 < t < T }, we consider the equation
LU ≡ Utt − Uxx = F (x, t)
(1.1)
with initial data
U (x, 0) = Φ(x), Ut (x, 0) = Ψ(x),
Dirichlet boundary condition
U (0, t) = 0
and the nonlocal condition
Z
(1.2)
(1.3)
l
U (x, t) dx = 0,
0
2000 Mathematics Subject Classification. 35L99, 35L05, 35L20.
Key words and phrases. Mixed problem, non-local conditions, wave equation.
c
2001
Southwest Texas State University.
Submitted August 21, 2001. Published December 10, 2001.
1
(1.4)
2
SERGEI A. BEILIN
EJDE–2001/76
where Φ(x), Ψ(x) are given, Φ(x) ∈ C[0, l] ∩ C 2 (0, l), Ψ(x) ∈ C[0, l] ∩ C 1 (0, l) and
satisfy the compatibility conditions
Z l
Z l
Φ(0) = 0, Ψ(0) = 0,
Φ(x) dx =
Ψ(x) dx = 0.
0
0
Note that we do not lose generality by assuming that (1.3) and (1.4) are homoRl
geneous. Indeed, if U (0, t) = m(t) and 0 U (x, t) dx = n(t), we introduce a new
unknown function v(x, t) = U (x, t) − W (x, t), where
W (x, t) = (1 −
2x
2x
)m(t) + 2 n(t).
l
l
Then (1.1) is converted into the similar equation
vtt − vxx = g(x, t),
g(x, t) = F (x, t) − LW,
while the Dirichlet and integral conditions are now homogeneous.
The presence of integral conditions complicates the application of standard techniques. Therefore, we first reduce (1.1)-(1.4) to an equivalent problem.
Lemma 1.1. Problem (1.1)-(1.4) is equivalent to (1.1)-(1.3) and
Z l
Ux (0, t) − Ux (l, t) =
F (x, t) dx.
(1.5)
0
Proof. Let U (x, t) is a solution of (1.1)-(1.4). Integrating (1.1) with respect to x
over (0, l), and taking in account (1.4), we obtain
Z l
Ux (0, t) − Ux (l, t) =
F (x, t) dx.
0
Let now U (x, t) be a solution of (1.1)-(1.3), (1.5). We need only to show that
Rl
U (x, t) dx = 0. For this end we integrate again (1.1) and obtain
0
d2
dt2
l
Z
U (x, t) dx = 0.
0
By virtue of the compatibility conditions,
Z l
Z l
U (x, 0) dx = 0,
Ut (x, 0) dx = 0 .
0
0
Then
Rl
0
U (x, t) dx = 0 is a unique solution to homogeneous Cauchy problem.
Introduce a new unknown function u(x, t) = U (x, t) − w(x, t), where w(x, t) =
2 Rl
− x2l 0 F (x, t) dx. Then (1.1)-(1.3), (1.5) is transformed now into
utt − uxx = g(x, t),
u(x, 0) = ϕ(x),
ut (x, 0) = ψ(x),
(1.6)
(1.7)
u(0, t) = 0,
(1.8)
ux (0, t) = ux (l, t),
(1.9)
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WAVE EQUATION WITH A NONLOCAL CONDITION
3
where
Z
1 l
Ftt (x, t) dx −
F (x, t) dx,
l 0
0
Z
x2 l
ϕ(x) = Φ(x) +
F (x, 0) dx,
2l 0
Z
x2 l
ψ(x) = Ψ(x) +
Ft (x, 0) dx.
2l 0
x2
g(x, t) = F (x, t) +
2l
Z
l
2. Uniqueness
Theorem 2.1. There exists at most one solution to (1.6)-(1.9).
Proof. Let u1 (x, t), u2 (x, t) be two different solutions of (1.6)-(1.9). Then u(x, t) =
u1 (x, t) − u2 (x, t) is a nontrivial solution to the homogeneous problem
utt − uxx = 0,
u(x, 0) = 0,
u(0, t) = 0,
ut (x, 0) = 0,
ux (0, t) = ux (l, t).
As u ∈ C 1 (D̄) ∩ C 2 (D), then u(x, t) takes on certain value for x = l. Let u(l, t) =
µ(t). Consider mixed problem for the equation utt − uxx = 0 with homogeneous
initial data and the boundary conditions
u(0, t) = 0,
u(l, t) = µ(t).
Note, that µ(t) is required to satisfy the compatibility conditions µ(0) = 0 and
µ0 (0) = 0.
For all conditions to be homogeneous, we let ũ = u − xl µ(t). Then, taking in
account the compatibility conditions for µ(t), we obtain
x
ũtt − ũxx = µ00 (t),
l
ũ(x, 0) = 0, ũt (x, 0) = 0,
ũ(0, t) = 0, ũ(l, t) = 0.
It is well known that there exists unique solution ũ(x, t) to this problem [1], hence
u(x, t) assumes the form
Z t
∞
2l X (−1)k
kπ(t − τ )
kπx x
00
u(x, t) = 2
µ
(τ
)
sin
dτ
sin
+ µ(t).
π
k2
l
l
l
0
k=1
Now we find that
Z
∞
2 X (−1)k t 00
kπ(t − τ )
1
µ (τ ) sin
dτ + µ(t),
π
k
l
l
0
k=1
Z
∞
2 X 1 t 00
kπ(t − τ )
1
ux (l, t) =
µ (τ ) sin
dτ + µ(t)
π
k 0
l
l
ux (0, t) =
k=1
and consider
4
ux (l, t) − ux (0, t) =
π
Z
0
t
µ00 (τ )
∞
X
1
(2m − 1)π(t − τ )
sin
dτ.
2m − 1
l
m=1
4
SERGEI A. BEILIN
EJDE–2001/76
As in [6]
∞
X
π/4,
if 0 < x < π
sin (2m − 1)x
=
2m − 1
−π/4, if π < x < 2π .
m=1
Then by (1.9) we can write
0 = |ux (l, t) − ux (0, t)| =
Z
t
µ00 (τ ) dτ.
0
Taking into account the compatibility conditions µ(0) = µ0 (0) = 0, we easily obtain
µ(t) ≡ 0. Now from the uniqueness theorem [1], we obtain u(x, t) ≡ 0.
3. Existence
Obviously, the solution to the problem (1.6)-(1.9), if it exists, is a sum of solutions
to the following two problems:
Problem H
utt − uxx = 0,
u(x, 0) = ϕ(x),
u(0, t) = 0,
ut (x, 0) = ψ(x),
ux (0, t) = ux (l, t)
Problem N H
utt − uxx = g(x, t),
u(x, 0) = ut (x, 0) = 0,
u(0, t) = 0,
ux (0, t) = ux (l, t).
First consider problem H and use separation of variables. Let u(x, t) = X(x)T (t).
Substituting in the equation utt − uxx = 0 and taking into account (1.8), (1.9), we
obtain
X 00 (x) + λX(x) = 0, X(0) = 0, X 0 (0) = X 0 (l).
(3.1)
Note that problem (3.1) is not self-adjoint: The adjoint problem is
Y 00 (x) + λ̄Y (x) = 0,
Y 0 (l) = 0,
Y (l) = Y (0).
(3.2)
The eigenvalues and eigenfunctions of problem (3.1) are
λk = (
2πk 2
) ,
l
k = 1, 2, . . .
(3.3)
2πkx
(3.4)
l
respectively. Note, that for k > 0 the functions (3.4) are not orthonormal with X0 .
To construct a basis in L2 , we complete (3.4) by using adjoint functions.
Following M. Keldysh [11], we define an adjoint function X̃k , corresponding
eigenvalue λk from (3.3), as a solution to the boundary-valued problem
p
X̃k00 (x) + λk X̃k (x) = −2 λk Xk (x), X̃k (0) = 0, X̃k0 (0) = X̃k0 (l).
(3.5)
X0 = x,
Xk = sin
We obtain
2πkx
, k = 1, 2, . . .
l
Rewrite now a system of eigenvalue and adjoint functions of (3.1) as
X̃k (x) = x cos
X0 = x,
X2k−1 (x) = x cos
2πkx
,
l
X2k (x) = sin
2πkx
.
l
(3.6)
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WAVE EQUATION WITH A NONLOCAL CONDITION
5
In a similar way we find the system of eigenvalue and adjoint functions (3.2):
Y0 (x) =
2
,
l2
4
2πkx
cos
,
l2
l
Y2k−1 (x) =
Y2k (x) =
4(l − x)
2πkx
sin
,
l2
l
(3.7)
where for every λk with k > 0, X2k (x), Y2k (x) are eigenvalue functions, X2k−1 (x),
Y2k−1 (x) are adjoint functions of the problems (3.1) and (3.2) respectively. Direct
calculations show that (3.6) and (3.7) form a biorthogonal system for x ∈ (0, l):
Z l
(Xi , Yj ) =
Xi (x)Yj (x) dx = δij .
0
As it was shown in [8] the system (3.6) is complete and forms a basis in L2 (0, l).
Hence, an arbitrary function f (x) ∈ L2 (0, l) may be expanded as
f (x) = A0 X0 (x) +
∞
X
(A2k X2k (x) + A2k−1 X2k−1 (x)),
k=1
where
Ai =
Z
l
f (x)Yi (x) dx.
(3.8)
0
Returning to the separation variables technique, for T (t) we obtain
Tk (t) = ak sin
2πkt
2πkt
+ bk cos
.
l
l
We assume now that a solution to H is of the form
∞ X
lt
0
(A2k X2k + A2k−1 X2k−1 )Tk −
u(x, t) = A0 X0 +
A2k−1 X2k Tk .
2πk
(3.9)
k=1
Substitute Tk (t) and rewrite the coefficients. Then
u(x, t) =C0 X0 +
∞
X
(X2k (C2k sin
k=1
2πkt
2πkt
+ D2k cos
)
l
l
2πkt
2πkt
+ D2k−1 cos
)
l
l
2πkt
2πkt
− tX2k (C2k−1 cos
− D2k−1 sin
)).
l
l
+ X2k−1 (C2k−1 sin
(3.10)
The initial data (1.7) give us the following two equalities
ϕ(x) = C0 X0 +
∞
X
(D2k X2k + D2k−1 X2k−1 ),
k=1
∞ X
2πk
2πk
ψ(x) =
(
C2k − C2k−1 )X2k +
C2k−1 X2k−1 ,
l
l
k=1
and the coefficients can be found via formula (3.8).
Assume a solution to the problem N H is of the form
u(x, t) = V0 (t)X0 (x) +
∞
X
k=1
(V2k (t)X2k (x) + V2k−1 (t)X2k−1 (x)) ,
(3.11)
6
SERGEI A. BEILIN
EJDE–2001/76
where Vi (t) are unknown coefficients satisfying the initial conditions Vi (0) = Vi0 (0) =
0. Substitute (3.11) into the equation utt − uxx = g(x, t), where g(x, t) has been
expanded as a biorthogonal series:
g(x, t) = g0 (t)X0 (x) +
∞
X
(g2k (t)X2k (x) + g2k−1 (t)X2k−1 (x)),
k=1
with coefficients
gi (t) =
Z
l
g(x, t)Yi (x) dx,
i = 0, 1, . . .
0
We obtain
∞ X
4π 2 k 2
2πkx
+
+
V2k (t) sin
2
l
l
k=1
∞
X
4π 2 k 2
2πkx
00
+
V2k−1
(t) +
V2k−1 (t) x cos
l2
l
V000 (t)x
+
k=1
∞
X
V2k−1 (t)
k=1
00
V2k
(t)
2πkx
4πk
sin
l
l
= g0 (t)X0 (x) +
∞
X
(g2k (t)X2k (x) + g2k−1 (t)X2k−1 (x)).
k=1
Thus we have a Cauchy problem for the system of ordinary differential equations
V000 (t) = g0 (t)
4πk πk
( V2k (t) + V2k−1 (t)) = g2k (t)
l
l
4π 2 k 2
00
V2k−1
(t) +
V2k−1 (t) = g2k−1 (t)
l2
00
+
V2k
with initial data
V0 (0) = V00 (0) = 0,
0
V2k (0) = V2k
(0) = 0,
0
V2k−1 (0) = V2k−1
(0) = 0,
which has a unique solution
Z t
V0 (t) =
(t − τ )g0 (τ ) dτ,
0
Z t
1
kπ(t − τ )
V2k−1 (t) =
g2k−1 (τ ) sin
dτ,
kπ 0
l
Z t
1
kπ(t − τ )
V2k (t) =
(g2k (τ ) − 4πkV2k−1 (τ )) sin
dτ.
kπ 0
l
Theorem 3.1. Let:
(1) g(x, t) ∈ C 2 (D), gx (x, t) ∈ C[0, l] for all t ∈ (0, T ), |g(x, t)| ≤ P, (x, t) ∈ D
(2) ϕ ∈ C[0, l] ∩ C 2 (0, l), ψ ∈ C[0, l], ϕ(0) = 0, ϕ0 (0) = ϕ0 (l), ψ(0) = 0.
Then there exists the solution to (1.6)–(1.9),
u(x, t) ∈ C(D̄) ∩ C 1 (D̄ \ {t = T }) ∩ C 2 (D)
which has the form of a sum of (3.9) and (3.11).
EJDE–2001/76
WAVE EQUATION WITH A NONLOCAL CONDITION
7
Series Proof. It is sufficient to prove uniform convergence of the series (3.9)
and (3.11) and the series, obtained with formal differentiation. Let |ϕ0 (x)| ≤ M1 ,
|ϕ00 (x)| ≤ M2 , |ψ(x)| ≤ N , |ψ 0 (x)| ≤ N1 , |gx | ≤ P1 , |gxx | ≤ P2 .
Integrating Ci , Di , Vi by parts and taking in account the abovementioned assumptions, we obtain:
1 M2 l
1 l(lM2 + 2M1 )
, |D2k−1 | ≤ 2 2 ,
k2
π2
k π
1 l(N1 + 2N )
1 N1 l
|C2k | ≤ 2
, |C2k−1 | ≤ 2 2 ,
2
k
2π
k π
1 4T 2 (2p1 + P2 l)
1 2T P1
|V2k | ≤ 2
, |V2k−1 | ≤ 2
,
k
π2
k π2
|D2k | ≤
and hence the series (3.9) and (3.11) and the series, obtained with formal differentiation, converge uniformly.
References
[1] A. V. Bitzadze, Urawnenija matematicheskoj fiziki, M.,“Nauka”, 1976.
[2] A. Bouziani, On a class of parabolic equations with nonlocal boundary conditions, Bulletin
de la Classe des Sciences, Académie Royale de Belgique, T.X, 1999, p. 61-77.
[3] A. Bouziani, Solution forte d’un problème mixte avec conditions non locales pour une classe
d’équations hyperboliques, Bulletin de la Classe des Sciences, Académie Royale de Belgique,
T. VIII, 1997, p. 53-70.
[4] J. R. Cannon, The solution of the heat equation subject to specification of energy. Quart.
Appl. Math., 21, n2, 155-160, 1963.
[5] D. G. Gordeziani, G. A. Avalishvili, On the constructing of solutions of the nonlocal initial
boundary problems for one-dimentional medium oscillation equations, Matem. Modelirovanie,
12, N. 1, 2000, 94-103.
[6] I. S. Gradshtein, I. M. Ryzhik, Tables of integrals, sums, series and products, M., 1963.
[7] A. K. Gushin, V. P. Mikhailov, On solvability of nonlocal problems for second-odered elliptic
equation, Matem. Sbornik, V. 185, 1994, p. 121-160.
[8] V. A. Ilyin, Necessary and sufficient properties of being a basis of a subsystem of eigenfunctions and adjoint functions for Keldysh bundle for odrinary differential operators, Doklady
Acad. Nauk SSSR, 227, N. 4, 1976, p. 796-799.
[9] N. I. Ionkin, Solutions of boundary value problem in heat conductions theory with nonlocal
boundary conditions, Differents. Uravn., Vol. 13, N2, 1977, p. 294-304.
[10] L. I. Kamynin, A boundary value problem in the theory of the heat conduction with nonclassical boundary condition, Z. Vychisl. Mat. Fiz., 4, N6, 1964, p. 1006-1024.
[11] M. V. Keldysh, On eigenvalues and eigenfunctions of certain classes of not self-adjoint equations, Doklady Acad. Nauk SSSR, 87, 1951, p. 11-14.
[12] S. Mesloub, A. Bouziani, On a class of singular hyperbolic equation with a weighted integral
condition, Internat. J. Math. & Math. Sci., Vol. 22, No. 3, 1999, pp. 511-519.
[13] B. P. Paneiah, On certain nonlocal boundary problemsfor linear differential operators, Matem.
zametki., 1984, v. 35, N.3, p. 425-434.
[14] L. S. Pulkina, A nonlocal problem with integral conditions for hyperbolic equations, Electron.
J. Ddiff. Eqns., Vol. 1999(1999), No. 45, 1-6.
[15] L. S. Pulkina, On solvability in L2 of nonlocal problem with integral conditions for a hyperbolic equation, Differents. Uravn., V. N. 2, 2000.
[16] A. A. Samarskii, Some problems in differential equations theory, Differents. Uravn., Vol. 16,
n11, 1980, p. 1221-1228.
[17] A. L. Skubachevski, G. M. Steblov, On spectrum of differential operators with domain nondense in L2 , Dokladi AN USSR, 1991. v. 321, N.6, p. 1158-1163.
[18] N. I. Yurchuk, Mixed problem with an integral condition for certain parabolic equations,
Differents. Uravn., Vol. 22, N. 12, 1986, p. 2117-2126.
8
SERGEI A. BEILIN
EJDE–2001/76
Sergei A. Beilin
Department of Mathematics, Samara State University, 1, Ac.Pavlov st., 443011 Samara
Russia
E-mail address: sbeilin@narod.ru, awr@ssu.samara.ru