Cable parameters calculation for a long 1000 V DC cable
The actual equations you want were derived by Lord Kelvin for use on the transatlantic cable system, and the math is known as CABLE THEORY.  Interestingly enough, a search on that will likely prove useless, as the most common hits will all be about neurons, for which the same math applies, and finding anything about real cables is difficult. 

Interestly enough, the Telegraphers Equations probably do the job (http://en.wikipedia.org/wiki/Telegrapher%27s_equations).  That provides all the differential equations you need to do the job.  If you need to solve this simultaneously in time and distance, you'll get hyperbolic partial differential equations, and the results will be catenaries involving hyperbolic trig functions (so THAT'S why they're on caculators) if you get to the right answer.

Sorry I can't walk you through the math.  It's not simple stuff.