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Questions tagged [hyperbolic-equations]

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.

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Solving a First Order Hyperbolic system of PDEs with an ill-conditioned matrix.

I have the system of non-homogeneous First Order One-Way Wave Equations $$\frac{\partial {\bf \overrightarrow u}}{\partial t}+{\bf \underline A}\frac{\partial {\bf \overrightarrow u}}{\partial x}={\bf\...
Sharat V Chandrasekhar's user avatar
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Hyperbola from JEE Mains [closed]

A point P moves on the hyperbola given by the equation x^2 - 4y^2 = 16 . Find the locus of the midpoint of the line segment joining the fixed point (6, -2) and the point P on the hyperbola. Determine ...
Aman Kumar Singh's user avatar
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Search for a general solution for a hyperbolic equation with a boundary condition on a sphere

I need help solving an interesting problem on partial differential equations. I got stuck with the problem of finding a general solution to the equation $$u_{xy} + u_{xz} + u_{yz} = 0$$ with a ...
Tom Sawyer's user avatar
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integral inequality in energy inequality

I encountered these two inequalities in reading Sogge's lectures on nonlinear wave equations (page 17). It seems natural and straightforward such that the author didn't give any hint but I cannot work ...
Bowen Zhao's user avatar
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First Order linear PDE with complex variable coefficients.

Consider the following first order linear equations: $$\partial_t u(t,x) = a(t,x)\nabla u(t,x)+b(t,x)u(t,x),u(0,x)=u_0(x).$$ If $a,b$ are real functions, this can be solved by characteristic method. ...
xinggu's user avatar
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2 votes
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Let $P$ be a point in the first quadrant that lies on the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$.

Let $a$ and $b$ be positive numbers such that $a>1$ and $b<a$. Let $P$ be a point in the first quadrant that lies on the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$. Suppose the tangent to ...
mathophile's user avatar
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Solution to Inviscid Burger's equation given piecewise initial condition

I'm taking Non-Linear PDEs course this semester. I'm stuck in this HW problem. Solve the IVP: $ u_t+uu_x =0, \ \ x \in \mathbb{R}, \ t\ge 0 $ $$ u(x,0) = \left\lbrace \begin{aligned} & 1 &&...
sai saandeep's user avatar
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1 vote
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Use the method of reflections to solve the Dirichlet problem on the half-line

The problem is: $$u_{xx}-u_{xy}-2u_{yy} = 0, 0<x<\infty, 0<y<\infty$$ $$u(x,0)=\phi(x),0<x<\infty$$ $$u_y(x,0)=\psi(x)$$ $$u(0,y)=0$$ I've worked out a general solution is $$ u(x,y) =...
izzitrin's user avatar
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solving the PDE $u_{xx}-u_{xy}- 2u_{yy} = 0$ with initial conditions $u(x,0) = \phi(x)$, $u_y(x,0) = \psi(x)$

I am trying to solve this PDE and not sure if I'm on the right track. I first found (using a change of variables) the general solution in the form $$u(x,y)=f(x-y)+g(x+\frac{y}{2})$$ Then, the initial ...
izzitrin's user avatar
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Existence and unique solution to a linear PDE

I'm doing an exercise with no solution, the question says $u_x+xu_y=0$ for $x,y\in \mathbb{R}$, with initial value $u(x,0)=f(x)$, where $f$ is a real function. Now the question ask me to impose some ...
kkk's user avatar
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2 answers
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Mixed PDE problem

Can someone help me with the following problem of PDE, namely $$u_{tt}+2u_{xt}-2u_{t}=0,u=u(x,t),u(x,0)=u_{t}(x,0)=e^{x}$$ Classify the given equation and write it in the canonical form. We were ...
Vuk Stojiljkovic's user avatar
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1 answer
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Inhomogeneous second order PDE solution

We have the following equation: $$u_{yy}-2u_{xy}+2u_x-u_y=4e^x$$ and we want to find its general solution. We have brought it to the canonical form: $$-2u_{\xi\eta}+u_\xi=2e^\xi$$ where $\xi=x, \eta=x+...
Britanica's user avatar
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Existence and Uniqueness Results for Second Order Linear Hyperbolic Equations on Unbounded Domains

I am studying second-order linear hyperbolic equations and their existence and uniqueness results. Most literature I've encountered discusses these results primarily in the context of bounded domains. ...
user477157's user avatar
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Seeking Modern References on Non-Homogeneous Boundary Value Problems in Hyperbolic PDEs

I am currently working on hyperbolic partial differential equations (PDEs) and specifically interested in non-homogeneous boundary value problems. I've heard about the books by Lions and Magnes, but I ...
user477157's user avatar
3 votes
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What is the difference between mathematical entropy and thermodynamic entropy?

For thermodynamic entropy, the inequality based on 2nd law is $$dS \geq \frac{\delta Q}{T_{surr}}$$ However, the entropy inequality for hyperbolic PDE system is $$\partial_t \eta(U(x,t)) + \partial_x ...
newbie125's user avatar
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2 votes
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Calculate the weak derivative of a dual product

This question is about hyperbolic equations of divergence form. The result has been applied directly, without proof, in Evans' book. I do not think this question is trivial. Please help by giving a ...
tfatree's user avatar
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Numerical Partial Differential Equation

The solution of the difference diagram for some partial differential equations from the Fourier transform and Fourier analysis can be written in ${U_{n}}^{k}=q^{k}e^{in\xi}$ form. The condition for ...
Yasemin's user avatar
2 votes
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Solving linear second order hyperbolic PDE $\nabla \cdot \left( M \nabla u\right)=0$

Let $\Omega \subset \mathbb{R}^{2}$ be a bounded and connected domain with a smooth boundary $\Gamma$. Furthermore, let $M$ be a matrix-valued function, where the entries $m_{ij}$ are smooth and ...
SebastianP's user avatar
4 votes
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190 views

Solution to linear second order PDE

I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. Let $\Omega \subset \mathbb{R}^{2}$ be a bounded and connected domain ...
SebastianP's user avatar
1 vote
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Find the equation of the locus of a point the difference of whose distances from two fixed points is constant given their coordinates.

So the fixed points are $$F_1=(p_1,q_1)$$$$F_2=(p_2, q_2)$$ Mid-point of foci(centre) is $$\left(\cfrac{p_1+p_2}{2},\cfrac{q_1+q_2}{2}\right)=(c_x,c_y)$$ and the the point $P=(h,k)$ The equation is ...
Aurelius's user avatar
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Existence of solution to a advection-reaction equation with forcing term

Consider the advection-reaction equation in One-Dimension $\dfrac{\partial u}{\partial t} + \dfrac{\partial u}{\partial x} = u(1-u) + f(x,t); x\in\mathbb{R}, t>0$ with initial condition $u(x,0) = g(...
J.MD24's user avatar
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1 answer
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What are the initial conditions for this coupled advection/transport system?

Consider the coupled transport (i.e. advection) system $$ \begin{align} \partial_t u + b\partial_x \phi &= 0,\\ \partial_t \phi + b\partial_x u &= 0, \end{align} $$ where $u(x,t),\phi(x,t) \in ...
l'étudiant's user avatar
2 votes
0 answers
119 views

Classification of 1st order non linear system of PDEs

Consider the non linear, 1st order PDE system \begin{align} \xi_u^2+\eta_u^2=\xi_v^2+\eta_v^2=\left(1+\frac{\xi^2+\eta^2}{4} \right)^2, \end{align} for $\xi=\xi(u,v)$ and $\eta=\eta(u,v)$, with $u,v \...
DanielKatzner's user avatar
1 vote
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41 views

Integrability of modified diagonalizable Jacobian

I have a smooth function $f$ from $\mathcal{R}^N$ to $\mathcal{R}^N$. For each $x\in \mathcal{R}^N$ the Jacobian of $f$, $J_f$, is diagonalizable as $$ J_f(x)=S(x)\Lambda(x) {S(x)}^{-1}, $$ where the ...
Singularly perturbed's user avatar
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A question about norm of weighted Sobolev space $e^{\gamma t} H_\gamma^s$

This question really bothers me for a long time. The following space is used for handling initial boundary value problem for first order hyperbolic equation. I am reading Multidimensional Hyperbolic ...
vent de la paix's user avatar
2 votes
1 answer
111 views

Lax-Friedrichs Scheme for second-order hyperbolic PDE

For the second-order hyperbolic PDE on $\Omega = (0,1)$: $$ u_{tt}(x,t) = c^2 u_{xx}(x,t)\; , \quad \begin{cases} u(0,t) = L(t) \\ u(1,t) = R(t) \\ u(x,0) = f(x) \\ u_t(x,0) = g(x) \...
Mak's user avatar
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4 votes
3 answers
207 views

Is there an exact solution to $x \sinh\Big(\frac{1}{x}\Big) = a$?

Is there an exact formula for solutions to the equation $x \sinh\Big(\frac{1}{x}\Big) = a$ where $a,x \in \mathbb{R}^+$? And if not, why? I tried to rearrange to apply Lambert W somewhere to no avail. ...
LeaG's user avatar
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0 answers
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Proving hyperbolic equality

While trying to prove a problem on catenoids intersecting orthogonally with $S^2$ I have to prove that if $a=\frac{1}{T\cosh(T)}$ where $T$ is the only real number such that $T\tanh(T)=1$ and $t$ ...
PunkZebra's user avatar
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0 votes
1 answer
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Conservation laws and compact support

Assume we have $F \in C^2(\mathbb{R})$ and $u_0 \in L^\infty(\mathbb{R})$ with sompact support. I am wondering whether the entropy solution to the Cauchy Problem on $(0,T) \times \mathbb{R}$: $$ \...
Hyperbolic PDE friend's user avatar
1 vote
0 answers
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Convergent Finite Difference Method for Variable Coefficient Hyperbolic Equation

Consider the hyperbolic equation $u_t+sin(x)u_x-yu_y=0$ on the domain $[-\frac{\pi}{2},\frac{\pi}{2}]\times[-\frac{\pi}{2},\frac{\pi}{2}]$ with some initial condition $u(x,y,0)=\phi(x,y)$. Set up a ...
Left Hand's user avatar
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0 answers
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How to define addition law in hyperboloid model(lorentz space) of hyperbolic space

I know mobius addition and Einstein addition are well defined in Poincaré ball model . But how to define addition in hyperboloid model(lorentz space) of hyperbolic space,and can we define the exact ...
Zoe.peace's user avatar
1 vote
1 answer
73 views

Demonstration of hyperbola in polar coordinates

I'm working on Matlab and I have the parameters a,b and c of a hyperbola. I'm working on polar coordinates because an Matlab example did so. I'm trying to get a matrix $2 \times N$ where each column ...
Alex Ferre's user avatar
3 votes
0 answers
111 views

PDEs with mixed time and space derivatives

New user here, please pardon my mistakes. During my research I was faced with the following type of variational equation $$ \int_0^T \int_{\Omega} \nabla u \cdot \nabla v -\partial_t u \partial_t v + \...
André Matos de Souza's user avatar
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0 answers
50 views

Why is $\text{Ric}(g)=0$ a quasi-linear PDE in harmonic coordinates?

While studying the dynamics of the Einstein vacuum equations $$ \text{Ric}(g)=0 $$ for $(M,g)$ unknwon, I've come across the statement that in harmonic coordinates $x^\lambda$ defined by $\Box_g x^\...
Gandalf The Gray's user avatar
4 votes
2 answers
185 views

A doubt on Theorem 2.6 from Pazy's book

I have been very confused about an argument on Theorem 2.6 from Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Here is the theorem and part of the proof: I ...
Lucas Linhares's user avatar
1 vote
1 answer
100 views

Trying to figure out WENO, and how to obtain the "+/-" values to use in the numerical fluxes.

I've been trying to read Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (Ed. by A. Quarteroni, Springer, 1998, doi:10.1007/BFb0096351) to get a general idea of WENO for the scalar/...
Vogtster's user avatar
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2 votes
0 answers
51 views

Is there a physical interpretation of the "test function" for weak solutions to differential equations?

Background Consider the inviscid Burgers' equation. \begin{align*} u_t + f(u)_x&=0 \end{align*} or, using $f(u)=\frac{1}{2}u^2$, \begin{align*} u_t + u u_x &=0 \end{align*} In the case of non-...
nwsteg's user avatar
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2 votes
1 answer
165 views

How to understand the changes of characteristics when we convert a high order PDE to first order PDEs?

Consider a second order PDE, \begin{eqnarray*} au_{tt}+bu_{tx}+cu_{xx} & = & 0, \end{eqnarray*} which is equivalent to the following first order PDEs (introducing new variables $p$ and $q$) \...
Knuseat's user avatar
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3 votes
1 answer
415 views

Finite Speed of Propagation for $u_{tt} - \Delta u + qu = 0$

I am looking to show that there is some type of finite speed of propagation property for PDEs of the form $$ u_{tt} -\Delta u + q u = 0 $$ Where we can assume that $q$ is as smooth or integrable ...
Andrew Shedlock's user avatar
1 vote
0 answers
38 views

Deriving Finite Difference Scheme for Goursat PDE on Triangular Domain

I have the following PDE: $k_{xx}(x, y) - k_{yy}(x, y) = \lambda(y)k(x, y)$ $k(x, 0) = 0$ $k(x, x) = -\frac{1}{2} \int_0^x \lambda(y) dy$ defined on the triangle $0 < y \leq x \leq 1$. ($k_{xx}$ is ...
Luke Bhan's user avatar
2 votes
1 answer
61 views

Nonlinear scalar conservation law with convex function

Suppose I have the scalar conservation law $u_{t}+(e^u)_{x}=0$. I want to determine the exact solution with the following initial data: $$ \mathring{u}(x) = \left\lbrace\begin{aligned} &2 & &...
Analyst_311419's user avatar
1 vote
1 answer
129 views

Inhomogeneous linear transport equation Cauchy problem

I am working through the first chapter of "Finite Difference Schemes and Partial Differential Equations" by Strikwerda and I am confused by this inhomogeneous problem (1.1.2): Show that $$ ...
Eilif's user avatar
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1 vote
0 answers
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Does the classification of second-order PDEs into hyperbolic, elliptic, and parabolic work for complex coefficients?

Suppose you have the following differential equation: $au_{xx}+ 2bu_{xy} + cu_{yy} = 0$. For real $a,b,c$ the classification into elliptical, hyperbolic, or parabolic is based on calculating the sign ...
Idieh's user avatar
  • 90
5 votes
1 answer
303 views

Anyone able to derive the following Riemann invariant relations?

The following was taken from Toro's Riemann Solvers and Numerical Methods for Fluid Dynamics, Equation (2.123) in Chapter 2. It states that for a general quasi-linear hyperbolic system $U_t + A(U)U_x =...
Winston Ong's user avatar
0 votes
0 answers
79 views

Weak solutions for 2D shallow water equations with source termes

My question is similar to the one asked in this post. I am trying to find the weak form solution for the following 2D shallow water equation (SWE) with source terms: $$ \begin{cases} \partial_t h + \...
Basilic's user avatar
1 vote
1 answer
73 views

Admissible solution to Riemann problem with unit jump

Question Consider the equation $$u_t + (u^4 /4)_x =0. $$ Find the admissible solution for each of the following initial data: $$u(x,0) = \begin{cases} 1 \quad & x<0 \\ 0 & x>0\end{cases} ...
Philomath's user avatar
1 vote
0 answers
158 views

Find Riemann function for hyperbolic PDE $u_{xy} + xyu_x = f(x,y)$

Find the Riemann function for the PDE $u_{xy} + xyu_x = f(x,y)$ and use it to show the solution to the problem $u_{xy} + xyu_x = 0$ in $x+y>0$ $u = x, u_y = 0$ on $x+y=0$ is $u(\xi, \eta) = -\...
user2080's user avatar
1 vote
0 answers
107 views

Change of coordinates to canonical form for general 2nd order hyperbolic PDE

I'm studying the way one changes a general 2nd-order, 2-dimensional, linear hyperbolic PDE $$a(x,y)u_{xx} + 2b(x,y) u_{xy} + c(x,y) u_{yy} + LOT = 0$$ into the canonical form $$A(\xi,\eta)u_{\xi\xi} + ...
Chris Wang's user avatar
2 votes
1 answer
120 views

Burgers PDE with piecewise constant initial condition

Let's deal with this Burgers PDE: $$\left\lbrace \begin{aligned} &u_t + uu_x = 0, \quad x\in \Bbb{R}, t>0\\ &u(x,0) = \varphi(x) \end{aligned} \right. $$ where $$\varphi(x) = \left\lbrace \...
Kώστας Κούδας's user avatar
2 votes
1 answer
135 views

Convergence of solution of non linear problem using heat equation

I am considering the following non linear problem: $u_{t}(t,x)-u_{xx}(t,x)+(u_{x}(t,x))^2=f(t,x)$ for $t>0, x \in (0,1)$ $u(0,x)=u_{0}(x)$ for $x \in [0,1]$ $u(t,0)=u(t,1)=0$ for $t>0$ where f ...
Aron's user avatar
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