Our team of undergraduate physicists are familiar with finding numerical approximations to the following Poisson-like PDE central to our plasma research in a torus:
$\nabla^2 V = \frac{f(V)}{R^2}$
where $f(V)$ is an arbitrary function of the electric potential V, and $R$ is the radial cylindrical coordinate (assume $\phi$ symmetry, so we have two variables: $(R,z)$). The boundary condition is that $V=0$ on the torus boundary (the variable $R$ resides in the torus).
We know the following about $f$ and the physical solutions $V$:
$V$ is a smooth and always-negative function, which goes to zero at the toroidal boundary, and has an extremum near the geometric center of the torus' innards.
$f(V)$ is a specified smooth function of the potential such that $f(0)=0$, is always-positive, and goes to zero past some maximum potential $V_0$.
We've noticed that if $f(V)$ is expanded in a power series in $V$, then we'd might as well solve only:
$ \nabla^2 V = \frac{V^n}{R^2}$
where n is any positive (or negative, if that's convenient to the maths) integer.
Could you provide a name for either PDE, a method of approximating it algebraically, or some guidance in simplifying it?
Thanks for reading this!
