# 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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### 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 ...
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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 ...
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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 ...
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### 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 ...
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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 ...
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