Simulation of homogeneous wave equation with non-homogeneous boundary condition and homogeneous initial condition

2013 ◽  
Author(s):  
Yu Tao ◽  
Li Dandan
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Homogeneous Boundary Condition ◽  
Initial Condition ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation

Compact attractor for a nonlinear wave equation with critical exponent

1990 ◽  
Vol 115 (1-2) ◽  
pp. 61-64 ◽  
Author(s):  
Orlando Lopes
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Phase Space ◽  
Critical Exponent ◽  
Growth Condition ◽  
Nonlinear Wave Equation ◽  
Compact Attractor ◽  
Homogeneous Wave ◽  
Coercive Condition ◽  
Homogeneous Wave Equation

SynopsisIn this paper we study the existence of a compact attractor for the solutions of the equation utt − Δu + cut + f(u) = h(t, x), x ∊ ℝ3. The phase space is H1 × L2 and periodicity in the x-variables is taken as a boundary condition. Besides the usual coercive condition, we assume f satisfies the growth condition |f′(u)|≦ a + bu2; this growth condition is critical because the embedding H1 → L6 is not compact. In the proof we use an Lp − H1.q estimate for the linear homogeneous wave equation.

Classical and generalized of a mixed problem– solutions for a non-homogeneous wave equation

2019 ◽  
Vol 484 (1) ◽  
pp. 18-20
Author(s):  
A. P. Khromov ◽  
V. V. Kornev
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Mixed Problem ◽  
Homogeneous Equation ◽  
Generalized Solution ◽  
Problem Solution ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation ◽  
Explicit Expressions

This study follows A.N. Krylov’s recommendations on accelerating the convergence of the Fourier series, to obtain explicit expressions of the classical mixed problem–solution for a non-homogeneous equation and explicit expressions of the generalized solution in the case of arbitrary summable functions q(x), ϕ(x), y(x), f(x, t).

scholarly journals Сlassical solution of the mixed problem for the one-dimensional wave equation with the nonsmooth second initial condition

2020 ◽  
Vol 64 (6) ◽  
pp. 657-662
Author(s):  
V. I. Korzyuk ◽  
J. V. Rudzko
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Mixed Problem ◽  
Method Of Characteristics ◽  
Smooth Boundary ◽  
Analytical Form ◽  
Initial Condition ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave

In this article, we study the classical solution of the mixed problem in a quarter of a plane and a half-plane for a one-dimensional wave equation. On the bottom of the boundary, Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. Smooth boundary condition is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. Uniqueness is proved and conditions are established under which a piecewise-smooth solution exists. The problem with linking conditions is considered.

scholarly journals A minimization approach to the wave equation on time-dependent domains

2020 ◽  
Vol 13 (4) ◽  
pp. 425-436 ◽  
Author(s):  
Gianni Dal Maso ◽  
Lucia De Luca
Keyword(s):  
Wave Equation ◽  
Weak Solutions ◽  
Space Time ◽  
Time Dependent ◽  
Existence Of Weak Solutions ◽  
Homogeneous Wave ◽  
Minimization Approach ◽  
Homogeneous Wave Equation

AbstractWe prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.

scholarly journals Mixed Problem for a Homogeneous Wave Equation with a Nonzero Initial Velocity and a Summable Potential

2020 ◽  
Vol 20 (4) ◽  
pp. 444-456
Author(s):  
Vitalii P. Kurdyumov ◽  
◽  
Avgust P. Khromov ◽  
Victoria A. Khalova ◽  
◽  
...  
Keyword(s):  
Wave Equation ◽  
Mixed Problem ◽  
Initial Velocity ◽  
Formal Solution ◽  
Fourier Method ◽  
Generalized Solution ◽  
Initial Position ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation ◽  
Summable Potential

For a mixed problem defined by a wave equation with a summable potential equal-order boundary conditions with a derivative and a zero initial position, the properties of the formal solution by the Fourier method are investigated depending on the smoothness of the initial velocity u′t(x, 0) = ψ(x). The research is based on the idea of A. N. Krylov on accelerating the convergence of Fourier series and on the method of contour integrating the resolvent of the operator of the corresponding spectral problem. The classical solution is obtained for ψ(x) ∈ W1p (1 < p ≤ 2), and it is also shown that if ψ(x) ∈ Lp[0, 1] (1 ≤ p ≤ 2), the formal solution is a generalized solution of the mixed problem.

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