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I am very beginner of PDEs. I want to study wave equation in 1D and 2D for numerical methods. Basic question is which type is a wave equation is, elliptic, parabolic, or hyperbolic?

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  • $\begingroup$ I think it is hyberbolic. Right? $\endgroup$ – user3519733 May 15 '14 at 0:29
  • $\begingroup$ Yes, it is hyperbolic. If you think of $\partial/\partial x=X$ and $\partial/\partial t=T$, the equation looks like $(X^2-T^2)u=0$, and this looks like the equation of a hyperbola. (The notion of the symbol of a differential operator makes this rigorous.) $\endgroup$ – Ted Shifrin May 15 '14 at 1:17
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Generally speaking, wave equations are hyperbolic. They have the similar form that $$\frac{\partial^2u}{\partial t^2}=a^2\Delta u,$$ where $\Delta$ is the Laplacian and $u$ is the displacement of the wave.

Typical examples are acoustic wave, elastic wave, electromagnetic, as well as Schrödinger equation.

In one dimensional, the equation is written as $$\frac{\partial^2u}{\partial t^2}=a^2\frac{\partial^2u}{\partial x^2}.$$

The general solution is $$u(x,t)=u(x-at)$$ or $$u(x,t)=u(x+at).$$

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    $\begingroup$ * grumble * Schroedinger equation is not a wave equation, and it is not hyperbolic. $\endgroup$ – Willie Wong May 15 '14 at 12:09

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