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Help with a 2nd order PDE involving mixed derivatives

  1. Apr 2, 2012 #1
    I have a PDE in two variables, [itex]u[/itex] and [itex]v[/itex], which takes the form

    \frac{\partial\psi}{\partial u\hspace{1pt}\partial v} + \frac{1}{r}\left(\frac{\partial r}{\partial u} \frac{\partial \psi}{\partial v} + \frac{\partial r}{\partial v}\frac{\partial\psi}{\partial u}\right)[/itex]

    for an auxiliary field [itex]r=r(u,v)[/itex]. It would be nice to have this equation in a form that is amenable to the method of characteristics, as there are independent evolution equations for [itex]r[/itex] in both directions. My first idea was to define some intermediary fields [itex] \rho = r\partial_v\psi [/itex] and [itex] \tilde{\rho} = r\partial_u\psi [/itex], then write down the wave equation as

    [itex] \partial_u(r\rho)+\partial_v(r\tilde{\rho}) = \rho\partial_u r + r\partial_u\rho + \tilde{\rho}\partial_vr+r\partial_v\tilde{\rho} = 0. [/itex]

    Then, making the ansatz

    [itex] -\left(\rho\partial_u r + r\partial_u\rho\right) = \tilde{\rho}\partial_vr+r\partial_v\tilde{\rho} = \gamma = \text{const.} [/itex]

    one arrives at a system of four first-order equations along characteristics: two for [itex]\psi[/itex] and one each for [itex]\rho[/itex] and [itex]\tilde{\rho}[/itex]. (Initial data is free for the last two fields along the characteristic for which it lacks an evolution equation.)

    My question is, does this seem reasonable or is there a better method for approaching this problem?
  2. jcsd
  3. Apr 3, 2012 #2


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    Homework Helper

    There are a number of things that you can do, the equation that you have written down is a hyperbolic equation, so characteristics are a possibility, the use of a Riemann function may also be of some use here as well.
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