1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook