The non-autonomous wave equation with general Wentzell boundary conditions

2005 ◽  
Vol 135 (2) ◽  
pp. 317-329 ◽  
Author(s):  
Angelo Favini ◽  
Ciprian G. Gal ◽  
Gisèle Ruiz Goldstein ◽  
Jerome A. Goldstein ◽  
Silvia Romanelli
Keyword(s):  
Hilbert Space ◽  
Wave Equation ◽  
Boundary Conditions ◽  
Abstract Cauchy Problem ◽  
Regularity Conditions ◽  
Well Posedness ◽  
One Dimensional ◽  
Wentzell Boundary Conditions ◽  
Image Position ◽  
Dimensional Wave

We study the problem of the well-posedness for the abstract Cauchy problem associated to the non-autonomous one-dimensional wave equation utt = A(t)u with general Wentzell boundary conditions Here A(t)u := (a(x, t)ux)x, a(x, t) ≥ ε > 0 in [0, 1] × [0, + ∞) and βj(t) > 0, γj(t) ≥ 0, (γ0(t), γ1(t)) ≠ (0,0). Under suitable regularity conditions on a, βj, γj we prove the well-posedness in a suitable (energy) Hilbert space

Time-periodic solutions to the one-dimensional wave equation with periodic or anti-periodic boundary conditions

2007 ◽  
Vol 137 (2) ◽  
pp. 349-371 ◽  
Author(s):  
Shuguan Ji ◽  
Yong Li
Keyword(s):  
Wave Equation ◽  
Weak Solution ◽  
Boundary Conditions ◽  
Periodic Solutions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
One Dimensional ◽  
Time Periodic Solutions ◽  
Time Periodic ◽  
Dimensional Wave

This paper is devoted to the study of time-periodic solutions to the nonlinear one-dimensional wave equation with x-dependent coefficients u(x)ytt – (u(x)yx)x + g(x,t,y) = f(x,t) on (0,π) × ℝ under the periodic boundary conditions y(0,t) = y(π,t), yx(0,t) = yx(π,t) or anti-periodic boundary conditions y(0, t) = –y(π,t), yx[0,t) = – yx(π,t). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. Our main concept is that of the ‘weak solution’. For T, the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.

scholarly journals Classical solution of the mixed problem for a one-dimensional wave equation with second-order derivatives at boundary conditions

2020 ◽  
Vol 55 (4) ◽  
pp. 406-412
Author(s):  
V. I. Korzyuk ◽  
S. N. Naumavets ◽  
V. A. Sevastyuk
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Classical Solution ◽  
Mixed Problem ◽  
Method Of Characteristics ◽  
Analytical Form ◽  
Second Order ◽  
Compatibility Conditions ◽  
One Dimensional ◽  
Dimensional Wave

This paper considers the mixed problem for a one-dimensional wave equation with second-order derivatives at boundary conditions. Using the method of characteristics, a classical solution to this problem is found in analytical form. Its uniqueness is proved under the relevant compatibility conditions.

On the integrals of a partial differential equation

1947 ◽  
Vol 43 (3) ◽  
pp. 348-359 ◽  
Author(s):  
F. G. Friedlander
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Partial Differential ◽  
One Dimensional ◽  
First Order ◽  
Progressive Waves ◽  
Image Position ◽  
Straight Lines ◽  
Dimensional Wave

The ordinary one-dimensional wave equationhas special integrals of the formwhich satisfy the first-order equationsrespectively, and are often called progressive waves, or progressive integrals, of (1·1). The straight linesin an xt-plane are the characteristics of (1·1). It follows from (1·2) that progressive integrals of (1·1) are constant on some particular characteristic, and are characterized by this property.

scholarly journals Mixed problem for a one-dimensional wave equation with conjugation conditions and second-order derivatives in boundary conditions

2020 ◽  
Vol 56 (3) ◽  
pp. 287-297
Author(s):  
V. I. Korzyuk ◽  
S. N. Naumavets ◽  
V. P. Serikov
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Mixed Problem ◽  
Analytical Form ◽  
Second Order ◽  
Conjugation Conditions ◽  
One Dimensional ◽  
Characteristics Method ◽  
Half Strip ◽  
Dimensional Wave

In this paper, we consider the boundary problem for the half-strip on the plane for the case of two independent variables. This mixed problem is solved for a one-dimensional wave equation with Cauchy conditions on the basis of the half-strip and boundary conditions for lateral parts of the area border containing second-order derivatives. Moreover, the conjugation conditions are specified for the required function and its derivatives for the case when the homogeneous matching conditions are not satisfied. A classical solution to this problem is found in an analytical form by the characteristics method. This solution is approved to be unique if the relevant conditions are fulfilled.

Thermally corrected solutions of the one-dimensional wave equation for the laser-induced ultrasound

2021 ◽  
Vol 130 (2) ◽  
pp. 025104
Author(s):  
Misael Ruiz-Veloz ◽  
Geminiano Martínez-Ponce ◽  
Rafael I. Fernández-Ayala ◽  
Rigoberto Castro-Beltrán ◽  
Luis Polo-Parada ◽  
...  
Keyword(s):  
Wave Equation ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave
Sign in / Sign up

Export Citation Format

Share Document