A hyperbolic PDE is a PDE that has a well-posedness initial value problem for the first $n-1$ derivatives. The Cauchy problem can be solved locally for arbitrary initial date along any non-characteristic hypersurface.
Questions tagged [hyperbolic-equations]
497 questions
13
votes
2 answers
Method of characteristics for Burgers' equation with rectangular data
Using the method of characteristics, find a solution to Burgers' equation
\begin{cases}
u_t+\left(\frac{u^2}{2} \right)_x =0 & \text{in }\mathbb{R}\times(0,\infty) \\
\qquad \qquad \, \, u=g & \text{on } \mathbb{R} \times\{t=0\}
\end{cases}
…
Cookie
- 13,952
13
votes
2 answers
Unbounded entropy solution to Burgers' equation
I need to solve Problem 3.5 - 11 p. 164 of the book Partial Differential Equations by Lawrence C. Evans (2nd ed., AMS, 2010):
Show that
$$
u(x,t) = \begin{cases}
-\dfrac{2}{3}\left(t+\sqrt{3x+t^2}\right); & \text{if } 4x + t^2 >0\\
…
DaveNine
- 2,159
12
votes
1 answer
How can we prove that a non-linear equation of motion for a classical scalar field satisfies causality?
Let $\phi$ be a real-valued scalar field in $N$-dimensional spacetime with coordinates $(t,\vec x)$, and consder the equation of motion
$$
(\partial_t^2-\nabla^2)\phi(t,\vec x)+V'\big(\phi(t,\vec x)\big)=0
\tag{1}
$$
where $\nabla$ is the gradient…
Chiral Anomaly
- 992
10
votes
1 answer
Solving a simple Shallow Water model
I need to solve with basic methods this simple Shallow Water Model:
$$\begin{bmatrix}h\\ hv\end{bmatrix}_t+\begin{bmatrix}hv\\ hv^2+\frac{1}{2}gh^2\end{bmatrix}_x=\begin{bmatrix}0\\ 0\end{bmatrix}$$
where $h$ is the height of the water, $v$ is the…
Quiet_waters
- 1,475
10
votes
2 answers
The Rankine-Hugoniot jump conditions for conservation and balance laws
Consider two problems:
$$(1) \hspace{1cm}
u_t+f(u)_x = 0,
$$
$$(2) \hspace{1cm}
u_t+f(u)_x = g(u).
$$
Problem (1) represents system of conservation laws, and problem (2) represents system of balance laws (or conservation laws with source…
Mark
- 608
9
votes
1 answer
Entropy Solution of the Burgers' Equation
I am working on the following problem, which gives the Burgers' equation $$u_t + uu_x=0$$ with the initial data $$g(x)= \begin{cases}1, & x < 0, \\ 2, & 0 < x < 1,\\ 0, & x > 1.\end{cases}$$ It then asks to find the entropy solution of $u(x,t)$…
9
votes
3 answers
Prove an identity for the continuous integral solution of the conservation law
This is an exercise in Evans, Partial Differential Equations (1st edition), page 164, problem 13:
Assume $F(0) = 0, u$ is a continuous integral solution of the conservation law
$$
\left\{ \begin{array}{rl}
u_t + F(u)_x = 0 &\mbox{ in $\mathbb{R}…
user76189
- 91
9
votes
1 answer
Are characteristics of $u_t+f(u)_x=0$ always straight lines?
I am studying conservation laws and reviewing the papers I get a doubt. Consider
$$u_t+f(u)_x=0$$ with $f$ smooth a conservation law and take the characteristics
$$x(t)\,\, ; \,\, x'(t)=f'(u(x(t),t))\,\, ; x(0)=x_0$$
Are they always straight lines…
Quiet_waters
- 1,475
9
votes
1 answer
How to test the weak solution to hyperbolic conservation law?
Consider an inviscid Burgers equation:
$u_t + u u_x = 0.$
With the initial data:
$ u(x,0) = \left\{ \begin{array}{ll}
0 \quad & \text{if} \quad x < 0, \\[0.5em]
1 & \text{if} \quad x > 0.\end{array} \right. $
Possible…
nevermind
- 175
8
votes
2 answers
singular shock solutions for strictly hyperbolic system of conservation law
Shallow water equation
\begin{eqnarray}
\rho_t+q_x=0\\
q_t + \left(q^2 +\frac{\rho ^2}{2} \right)_x=0
\end{eqnarray}
being a strictly hyperbolic equation does not admit delta shock solutions.
Where as some strictly hyperbolic systems such as…
Rosy
- 1,087
8
votes
1 answer
Burgers' equation - Integrate discontinuity over rectangle
I am studying conservation laws and hyperbolic systems, particularly, Burgers' equation and shocks, and have a doubt at page 40 of the book Numerical Methods for Conservation Laws by R.J. LeVeque (Birkäuser, 1992).
I could not understand the…
Quiet_waters
- 1,475
8
votes
1 answer
Burgers' equation $u_t + uu_x =0$ with $u(x,0)=x$
Solve Burgers' equation $$u_t + uu_x =0,$$
with $u=u(x,t)$ and the side condition $u(x,0)=x$.
I am not sure on how to find the solution $u(x,t)$. I have learned the method of characteristics. I am neither sure on how to use the side condition…
Sapna Kumar
- 129
7
votes
1 answer
Integral form of the conservation law $u_t+f(u)_x=0$
Consider the conservation law given by
$$u_t+f(u)_x=0$$
We know that in general weak solutions are not smooth but are bounded in $L^{\infty}$ norm (they do not belong to Sobolev spaces).
However while deriving the numerical schemes most of the…
Rosy
- 1,087
7
votes
1 answer
Why does a classical solution of the wave equation have to be $C^2$?
A (classical) solution of the wave equation
$$
u_{tt}-c^2u_{xx}=0,\qquad (x,t)\in\mathbb{R}\times\mathbb{R}^*_+,
$$
is required to be of class $C^2$. Why?
I mean, why would one impose that all second partial derivatives, even $u_{xt}$ , which does…
Wang
- 324
7
votes
2 answers
Is it possible to solve a hyperbolic moving boundary problem?
J. L. Davies says in his book,
"The basic principle in PDEs is that boundary value problems are associated with elliptic equations while initial value problems, mixed
problems, and problems with radiation effects at boundaries are
associated…
Nanashi No Gombe
- 1,252