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Hyperbolic equations

  1. Jan 27, 2007 #1
    (d2^u/dt^2) - (delta u) = 0 is called a hyperbolic equation.

    Why is this? What makes an equation a hyperbolic equation?
     
  2. jcsd
  3. Jan 30, 2007 #2

    HallsofIvy

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    The partial differential equation
    [tex]\frac{\partial^2u}{\partial t^2}= \frac{\partial^2 u}{\partial x^2}[/tex]
    and its extension to 2 or 3 space dimensions, is called "hyperbolic" in an obvious analogy with the hyperbolic equation
    [tex]x^2- y^2= 1[/tex]
    In addition, just as the hyperbola has two asymptotes, so the hyperbolic differential equation has two "characteristic" lines that can be used to solve the equation.

    The heat (or diffusion) equation,
    [tex]\frac{\partial u}{\partial t}= \frac{\partial u^2}{\partial x^2}[/tex]
    has only a single characteristic line and is a "parabolic" equation.

    Laplace's equation,
    [tex]\frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y^2}= 0[/tex]
    has no characteristic lines and is an "elliptic" equation.
     
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