Resistor temperature dynamics
To begin, most resistors these days are rectangular surface mount resistors, not cylindrical.

Second, the resistive element is usually not a cylindrical conductor, but a resistive surface layer which is rather thin, both for SMT and through-hole cylindrical components.

Third, for a given dissipated power (P=R*i^2), the body temperature of a resistor will be mostly determined by heat fluxes to the soldered ends/pads, and each pad/mounting hole will have certain thermal impedance in each particular case of board layout (and associated thermal mass/path).

More, solving equations for cylindrical conductor would be not enough, you will need to "solve" for heat transfer by free convection into ambient air, which would depend on component orientation relative to the gravity field and air density. Plus you would need to add an effect of thermal radiation. 

In short, temperature of resistive element is defined by a balance between a given Joule dissipation and the rate of heat loss, which is usually unknown. So there is no "simple equation" to determine its temperature.

[A paradoxical example: if a resistor (heating element) is placed in a fully insulated enclosure, its temperature will rise to infinity]