Exact Integration of a Nonlinear Model of Steady Heat Conduction/Radiation in a Wire with Internal Power

Abstract

The paper treats in one dimensional mixed heat transfer problem of steady conduction and radiation in a wire with internal source. We are led to a Cauchy problem consisting of a second order nonlinear ordinary differential equation. A special integrable case with two non independent left boundary conditions requires a hyperelliptic integral, for which a representation theorem has been established through the Gauss hypergeometric function ${_2\!F_1}$. The relevant steady solution is then found to grow monotonically with the distance from boundary, up to a certain limiting position where it suddenly jumps unbounded.