A linear recursive scheme associated with the love equation
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A LINEAR RECURSIVE SCHEME ASSOCIATED WITH THE LOVE EQUATION Le Thi Phuong Ngoc · Nguyen Tuan Duy · Nguyen Thanh Long
Received: 4 June 2012 / Revised: 18 October 2012 / Accepted: 29 October 2012 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013
Abstract This paper shows the existence of a unique weak solution of the following Dirichlet problem for a nonlinear Love equation ⎧ utt − uxx − εuxxtt = f (x, t, u, ux , ut , uxt ), 0 < x < L, 0 < t < T , ⎪ ⎪ ⎨ u(0, t) = u(L, t) = 0, ⎪ ⎪ ⎩ ut (x, 0) = u˜ 1 (x), u(x, 0) = u˜ 0 (x), where ε > 0 is a constant and u˜ 0 , u˜ 1 , f are given functions. This is done by combining the linearization method for a nonlinear term, the Faedo–Galerkin method and the weak compactness method. Keywords Faedo–Galerkin method · Linear recurrent sequence · Existence of a unique weak solution Mathematics Subject Classification (2000) 35L20 · 35L70 · 35Q72 L.T.P. Ngoc Nha Trang Educational College, 01 Nguyen Chanh Str., Nha Trang City, Vietnam e-mail: [email protected] L.T.P. Ngoc e-mail: [email protected] N.T. Duy Department of Fundamental sciences, University of Finance and Marketing, 306 Nguyen Trong Tuyen Str., Dist. Tan Binh, HoChiMinh City, Vietnam e-mail: [email protected]
B
N.T. Long ( ) Department of Mathematics and Computer Science, University of Natural Science, Vietnam National University Ho Chi Minh City, 227 Nguyen Van Cu Str., Dist.5, Ho Chi Minh City, Vietnam e-mail: [email protected] N.T. Long e-mail: [email protected]
L.T.P. NGOC ET AL.
1 Introduction In this paper, we consider the following Dirichlet problem for a nonlinear Love equation: utt − uxx − εuxxtt = f (x, t, u, ux , ut , uxt ),
0 < x < L, 0 < t < T ,
u(0, t) = u(L, t) = 0,
(2) ut (x, 0) = u˜ 1 (x),
u(x, 0) = u˜ 0 (x),
(1)
(3)
where ε > 0 is a constant and u˜ 0 , u˜ 1 , f are given functions satisfying the conditions specified later. When f = 0, Eq. (1) is related to the Love equation utt −
E uxx − 2μ2 k 2 uxxtt = 0, ρ
(4)
presented by V. Radochová in 1978 (see [10]). This equation describes the vertical oscillations of a rod. It was derived from Euler’s variational equation of an energy function
L
T
dt 0
0
1 2 1 2 2 2 2 2 2 Fρ ut + μ k utx − F Eux + ρμ k ux uxtt dx, 2 2
(5)
where the parameters in (5) have the following meanings: u is the displacement, L is the length of the rod, F is the area of cross section, k is the radius of cross section, E is the Young modulus of the material and ρ is the mass density. By using the Fourier method, Radochová [10] obtained a classical solution of Problem (4) associated with the initial conditions (3) and the following boundary conditions: u(0, t) = u(L, t) = 0, or
u(0, t) = 0, εuxtt (L, t) + c2 ux (L, t) = 0,
(6)
(7)
where c2 = Eρ , ε = 2μ2 k 2 . On the other hand, the asymptotic behavior of solutions for Problems (3), (4), (6) or (7) as ε → 0+ was also established by the method of small parameters. Equations of Love waves or equations for waves of Love type
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