# Help with a 2nd order PDE involving mixed derivatives

1. Apr 2, 2012

### aurban

I have a PDE in two variables, $u$ and $v$, which takes the form

$\frac{\partial\psi}{\partial u\hspace{1pt}\partial v} + \frac{1}{r}\left(\frac{\partial r}{\partial u} \frac{\partial \psi}{\partial v} + \frac{\partial r}{\partial v}\frac{\partial\psi}{\partial u}\right)$

for an auxiliary field $r=r(u,v)$. 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 $r$ in both directions. My first idea was to define some intermediary fields $\rho = r\partial_v\psi$ and $\tilde{\rho} = r\partial_u\psi$, then write down the wave equation as

$\partial_u(r\rho)+\partial_v(r\tilde{\rho}) = \rho\partial_u r + r\partial_u\rho + \tilde{\rho}\partial_vr+r\partial_v\tilde{\rho} = 0.$

Then, making the ansatz

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

one arrives at a system of four first-order equations along characteristics: two for $\psi$ and one each for $\rho$ and $\tilde{\rho}$. (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. Apr 3, 2012

### hunt_mat

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.