Questions about partial differential equations of hyperbolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

learn more… | top users | synonyms

3
votes
1answer
48 views

Advice on numerical solution for 2D hyperbolic PDE with zero flux boundary conditions

I would like to numerically solve a hyperbolic PDE of the form $\frac{\partial\theta_t}{\partial t}(x,y)+\frac{\partial\left[\theta_t \gamma_t^x\right]}{\partial x}(x,y)+\frac{\partial\left[\theta_t ...
0
votes
1answer
121 views

When is separation of variables an acceptable assumption to solve a PDE?

We know that one of the classical methods for solving some PDEs is the method of separation of variables. It works for known types of PDEs and many examples of physical phenomena are successfully ...
4
votes
2answers
130 views

Analytic solution of a system of linear, hyperbolic, first order, partial differential equations

In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form ...
3
votes
0answers
56 views

Local energy decay for variable-speed, divergence-form wave equation in non-trapping medium without obstacles

I'm looking for a reference in the literature describing local energy decay for solutions of a smooth-coefficient, variable-speed wave equation, in divergence form, with compactly-supported initial ...
0
votes
0answers
83 views

Wave operator for focusing NLS

Consider the NLS equation \begin{equation} \left\{ \begin{array}{rl} iu_t + \Delta u+u|u|^{\alpha}=0\\ u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\ \end{array}\right. \end{equation} where ...
5
votes
1answer
131 views

On a conjecture of Lions for the wave equation

In Control of Distributed Singular Systems p 236, JL Lions makes the conjecture : Let $\Omega$ be a domain in $\mathbb{R}^n$, $Q = \Omega \times ]0,T[$ and consider $\phi'' - \triangle \phi = F$ ...
3
votes
0answers
90 views

Rarefaction Shock Wave Interaction

I am interested in explicit solutions in 1D for the interaction of a rarefaction wave with a shock wave of arbitrary strength. The book Supersonic Flow and Shock Waves by Courant and Friedricks ...
1
vote
1answer
144 views

Double series solution of wave equation

Let $u(x,y,t)$ be the solution of wave equation $u_{tt}=u_{xx}+u_{yy}, 0\lt x\lt 1, 0\lt y\lt 1, t\ge 0,$ $u(x,y,0)=(x-x^2)(y-y^2), u_t(x,y,0)=0$ and $ u(x,y,t)=0$ on the boundary of the square. Then ...
3
votes
0answers
214 views

well-posedness of the transport equation

I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am ...