I'm currently trying to solve the following nonlinear partial differential equation for $f(r,\theta)$:
$$ \frac{r^2f_{,rr}+rf_{,r}+f_{,\theta\theta}}{r\sqrt{1+f_{,r}^2+\frac{f_{,\theta}^2}{r^2}}}-\frac{\left[r^2f_{,r}\left(f_{,r}f_{,rr}+\frac{f_{,\theta}f_{,r\theta}}{r^2}-\frac{f_{,\theta}^2}{r^3}\right)+f_{,\theta}\left(f_{,r}f_{,r\theta}+\frac{f_{,\theta}f_{,\theta\theta}}{r^2} \right)\right]}{r\left(1+f_{,r}^2+\frac{f_{,\theta}^2}{r^2}\right)^{3/2}} = Ar $$
where $A$ is a constant. With the initial boundary conditions: $$ f(0,\theta)=0 $$
$$ f_{,r}(0,\theta)=constant. $$
$$ f_{,\theta}(r,0)=0 $$
Since it does not look to have a simple exact solution, I think that I have to approximate it numerically. Unfortunately, my experience with PDEs is very short so my question is on which methods should I look for to solve it? I have some experience with the $RK4$ method for nonlinear ODEs, but I'm not sure if this can be adapted for PDEs. Any help/references provided would be very appreciated. Thank you in advance!