Classification of PDEs: Understanding and Solving for Unique Solutions

[Adyssa]

Homework Statement

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 ...

Homework Equations

Classify the PDE:

$tu_{xx} - (t-x)u_{xt} = xu_{tt} + u^{2}_{t}$

and if possible find the equations of the characteristic curves.

The Attempt at a Solution

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

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 $u, u_{x}$ and $u_{t}$ serve to determine $u_{xx}, u_{xt}$ and $u_{tt}$ 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!

 

[christoff]

See http://en.wikipedia.org/wiki/Partial_differential_equation#Classification for classification.