Rothe time-discretization method applied to a quasilinear wave equation subject to integral conditions
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This paper presents a well-posedness result for an initial-boundary value problem with only integral conditions over the spatial domain for a one-dimensional quasilinear wave equation. The solution and some of its properties are obtained by means of a suitable application of the Rothe time-discretization method. 1. Introduction Recently, the study of initial-boundary value problems for hyperbolic equations with boundary integral conditions has received considerable attention. This kind of conditions has many important applications. For instance, they appear in the case where a direct measurement quantity is impossible; however, their mean values are known. In this paper, we deal with a class of quasilinear hyperbolic equations (T is a positive constant):
∂v ∂2 v ∂2 v − = f x,t,v, , ∂t 2 ∂x2 ∂t
(x,t) ∈ (0,1) × [0,T],
(1.1)
subject to the initial conditions v(x,0) = v0 (x),
∂v (x,0) = v1 (x), ∂t
0 x 1,
(1.2)
and the boundary integral conditions 1 0
1 0
v(x,t)dx = E(t),
0 t T,
xv(x,t)dx = G(t),
0 t T,
(1.3)
where f, v0 , v1 , E, and G are sufficiently regular given functions. Problems of this type were first introduced in [3], in which the first author proved the well-posedness of certain linear hyperbolic equations with integral condition(s). Later, Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:3 (2004) 211–235 2000 Mathematics Subject Classification: 35L05, 35D05, 35B45, 35B30 URL: http://dx.doi.org/10.1155/S1687183904401071
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On a quasilinear wave equation with integral conditions
similar problems have been studied in [1, 4, 5, 6, 7, 8, 16, 24, 25] by using the energetic method, the Schauder fixed point theorem, Galerkin method, and the theory of characteristics. We refer the reader to [2, 9, 10, 11, 12, 13, 14, 15, 17, 21, 22, 23, 26] for other types of equations with integral conditions. Differently to these works, in the present paper, we employ the Rothe time-discretization method to construct the solution. This method is a convenient tool for both the theoretical and numerical analyses of the stated problem. Indeed, in addition to giving the first step towards a fully discrete approximation scheme, it provides a constructive proof of the existence of a unique solution. We remark that the application of Rothe method to this nonlocal problem is made possible thanks to the use of the so-called Bouziani space, first introduced by the first author, see, for instance, [4, 6, 20]. Introducing a new unknown function u(x,t) = v(x,t) − r(x,t), where
r(x,t) = 6 2G(t) − E(t) x − 2 3G(t) − 2E(t) ,
(1.4)
problem (1.1)–(1.3) with inhomogeneous integral conditions (1.3) can be equivalently reduced to the problem of finding a function u satisfying
∂2 u ∂2 u ∂u − 2 = f x,t,u, , (x,t) ∈ (0,1) × I, 2 ∂t ∂x ∂t ∂u u(x,0) = U0 (x), (x,0) = U1 (x), 0 x 1, ∂t 1 0
1 0
(1.5) (1.6)
u(x,t)dx = 0,
t ∈ I,
(1.7)
xu(x,t)dx = 0,
t ∈ I,
(1.8)
where
f x,t,u,
I := [0,T],
∂u ∂u ∂r ∂2 r := f x,t,u + r, + − 2, ∂t ∂t ∂t ∂t U0 (x
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