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Write down the ode satisfied by a characteristic curve

  1. Aug 21, 2011 #1
    1. The problem statement, all variables and given/known data

    i)write down the general form of a semi lenear first order pde in the unknown u(x,y)
    ii)write down the ode satisfied by a characteristic curve in the x-y plane for your pde
    ii)give a careful derivation of the ode satisfied by u(x,y) along such a charcteristic curve.

    2. Relevant equations



    3. The attempt at a solution

    i)[itex] a(x,y) \frac{\partial u}{\partial x} + b(x,y) \frac{\partial u}{\partial y} = g(x,y,u) [/itex]

    ii) the characteristic traces are given by [itex]\frac{dx}{dt}[/itex] = a(x,y) and [itex]\frac{dy}{dt}[/itex] = b(x,y) and [itex]\frac{du}{dt}[/itex] = g(x,y,u) so is one of these what i'm looking for?

    iii) since [itex]\frac{dx}{dt}[/itex] = a(x,y) along our characteristic we get t in term of x, provided a(x,y) [itex]\neq[/itex] 0. We can also express y in terms of x.

    the chainrule gives[itex]\frac{dy}{dt} = \frac{dy}{dx}\frac{dx}{dt}[/itex] and so [itex] \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{b(x,y)}{a(x,y)} [/itex] along the projected trace

    => the projected trace satisfies the ODE [itex]\frac{dy}{dx} = \frac{b(x,y)}{a(x,y)}[/itex]
     
  2. jcsd
  3. Aug 21, 2011 #2
    Re: P.D.E.s

    Anyone know where i can find information on region of influence of initial conditions?
    I cant get my head around it...
     
  4. Aug 22, 2011 #3
    Re: P.D.E.s

    Can anyone confirm, is my theory right here?
     
  5. Aug 22, 2011 #4

    hunt_mat

    User Avatar
    Homework Helper

    Re: P.D.E.s

    I would give the thumbs up with the above.
     
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