Analytical solution of non-linear Heat conduction in long wire carrying current covered by an insulator

In this study analytical solution of conduction heat transfer in long circular wire of carrying current is investigated. When wire of carrying current is used in miniaturized electronic devices, heat recovery from wire plays a major role on amount of waste of energy and indeed needs to be estimated precisely specially in expensive environment where any underestimate dimensions cause costly damages. Different conditions would arise by choosing different approaches in which wire and its insulator cover considered to be lumped or distributed. Also in reality, the most insulator materials have low thermal conductivity such that varies with temperature. This feature results in non-linearity and complexity in governing equation of heat conduction and needs an innovative method to solve analytically. Using novel change of variable causes non-linear ODE to reduce to simpler form and straightforward solution.