Assume we have a straight piece of wire with two end points $\displaystyle A$ and

$\displaystyle B$ and with length $\displaystyle L$ where $\displaystyle x_{A}=0$ and $\displaystyle x_{B}=L$. The wire

has non-ohmic resistance and hence the current is not proportional

to the potential difference, i.e. $\displaystyle \left(V_{A}-V_{B}\right)$. In

fact the current is a function of the voltage at $\displaystyle A$ and $\displaystyle B$, that

is $\displaystyle I=f\left(V_{A},V_{B}\right)$.

I know $\displaystyle f$ and hence I know the current. However, I do not know $\displaystyle V$

as a function of $\displaystyle x$ $\displaystyle \left(0<x<L\right)$. I tried several mathematical

tricks, mainly from the calculus of variation, trying to find $\displaystyle V\left(x\right)$

but I did not get a sensible result. Can any one suggest a method

(whether from the calculus of variation or other branches of mathematics)

to solve this problem and obtain $\displaystyle V\left(x\right)$.