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Classification of PDEs

  1. Aug 18, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm doing a course on numerical solutions of PDE and I am waaaay out of my depth, having not covered differential equations previously. I spoke to my lecturer about this and he said I would be fine as the course is on FDM/FEM and not analytic solutions but this is week 4 and I am utterly lost. Before I throw in the towel, I would like to bang my head against it for a while longer and see if I can figure a few things out. This is my first hurdle - classification of PDEs. I have a set of exercises to work on for a test this week, and here is one of the questions ...

    2. Relevant equations

    Classify the PDE:

    [itex]tu_{xx} - (t-x)u_{xt} = xu_{tt} + u^{2}_{t}[/itex]

    and if possible find the equations of the characteristic curves.

    3. The attempt at a solution

    For a start, I find the notation confusing, but I think [itex]u_{xx}[/itex] is equivalent to [itex]\frac{du^{2}}{d^{2}x}[/itex]

    With that out of the way, I have a reference text in Numerical Methods for PDE by William Ames which I understand to be canonical, but I'm finding it really hard to follow, so I'm hoping somebody could explain it more simply for me.

    I need to find "conditions under which a knowledge of [itex]u, u_{x}[/itex] and [itex]u_{t}[/itex] serve to determine [itex]u_{xx}, u_{xt}[/itex] and [itex]u_{tt}[/itex] uniquely so the equation is satisfied" - to paraphrase the Ames text, and then put them in matrix form so I can find the determinant, and if it's not equal to zero then I can use the the discriminant of the quadratic formula to classify the PDE. I think if I can get this thing into matrix form I will be ok from there, but this first part is killing me! I understand there is the notion of a directional derivative involved in forming the equations that are then put into matrix form but I don't grasp it.

    Sorry to be vague! I really want to understand this but my brain just doesn't want to!
     
  2. jcsd
  3. Aug 19, 2012 #2
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