I have a PDE in two variables, [itex]u[/itex] and [itex]v[/itex], which takes the form(adsbygoogle = window.adsbygoogle || []).push({});

[itex]

\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)[/itex]

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

[itex] \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. [/itex]

Then, making the ansatz

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

one arrives at a system of four first-order equations along characteristics: two for [itex]\psi[/itex] and one each for [itex]\rho[/itex] and [itex]\tilde{\rho}[/itex]. (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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Help with a 2nd order PDE involving mixed derivatives

**Physics Forums | Science Articles, Homework Help, Discussion**