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Classification of Second-Order PDE with Constant Coefficients

  1. Oct 3, 2013 #1
    1. The problem statement, all variables and given/known data

    I have 3 equations:

    [itex]\frac{\partial^2 u}{\partial t^2}+\frac{\partial^2 u}{\partial x \partial t}+\frac{\partial^2 u}{\partial x^2}[/itex]

    [itex]\frac{\partial^2 u}{\partial t^2}+4\frac{\partial^2 u}{\partial x \partial t}+4\frac{\partial^2 u}{\partial x^2}[/itex]

    [itex]\frac{\partial^2 u}{\partial t^2}-4\frac{\partial^2 u}{\partial x \partial t}+\frac{\partial^2 u}{\partial x^2}[/itex]

    I know their classification (first one is elliptical, second one is parabolic, third one is hyperbolic).
    I need to find their characteristics. How do I do that?
    What is a characteristic?
    I think that the one who is elliptical has no characteristic, the parabolic has only one and the hyperbolic has two.
     
  2. jcsd
  3. Oct 4, 2013 #2

    HallsofIvy

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    Staff Emeritus
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    Yes, that is true. But how could you know that (or what it means) if you don't know what a "characteristic" is?
    A characteristic is a curve upon which the partial differential equation separates into two ordinary differential equations. For example, if I associate the last formula (These are NOT equations because there is no "=". Did you mean "= 0"?) with "[itex]T^2- 4TX+ X^2[/itex] then by "completing the square" I get [itex]T^2- 4TX+ 4X^2- 3X^2= (T- 2X)^2- (\sqrt{3}X)^2= (T- 2X+\sqrt{3}X)(T- 2X- \sqrt{3}X)[/itex]

    So the "characteristics" are the curves [itex]t- (2-\sqrt{3})x= C[/itex] and [itex]t- (2+ \sqrt{3})x= C[/itex] for C any constant.

    Similarly, for the parabolic equation, we can write [itex]T^2+ 4TX+ 4X^2=N (T+ 2X)^2[/itex] and so have the single characteristic [itex]t+ 2x= C[/itex].
     
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