# Multidimensional first order linear PDE

1. Dec 13, 2014

### Heimdall

Hello,

I have the following PDE :

$$v_y \frac{\partial f}{\partial y} + \Omega_x(y)\left(v_z\frac{\partial f}{\partial v_y} - v_y\frac{\partial f}{\partial v_z}\right) + \Omega_z(y)\left(v_y\frac{\partial f}{\partial v_x} - v_x\frac{\partial f}{\partial v_y}\right) = 0$$

(which is the steady state Vlasov equation in one spatial dimension and three velocity dimensions).

I know the analytic expression of the functions $$\Omega_{x}(y)$$ and $$\Omega_{z}(y)$$.

I also know the expression of the function f for y=0 and y=ymax, and know that f tends to 0 when v tends to (plus or minus) infinity.

is that enough to find a (numerical) solution to the PDE ? is there a method that should preferentially be used to solve it numerically? I've been looking a bit a the method of characteristics but I'm not sure I understand how that would help.

It does not look like an overly complicated equation (linear, first order) so I feel it should be possible to solve it.

I'd appreciate help like references, method names, explanations etc.

Thanks!

2. Dec 18, 2014

### Staff: Admin

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

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