1. ## Non-linear problem

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 $\displaystyle V\left(x\right)$.

2. ## Re: Non-linear problem

Hi !
The general solution, only based on the first wording, is :
V(x) = Va +(f(x)-f(xa))(Vb-Va)/(f(xb)-f(xa)) or V(x) = Va +(f(x)-f(0))(Vb-Va)/(f(L)-f(0))
where f(x) is any continuous function.
You cannot determine what kind of function f(x) is without a descriptive physical model for the electrical behaviour from A to B.