## Exact integration of a nonlinear model of steady heat conduction/radiation in a wire with internal power.(English)Zbl 1141.35405

Summary: 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 $${_2F_1}$$. 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.

### MSC:

 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 80A20 Heat and mass transfer, heat flow (MSC2010) 34A34 Nonlinear ordinary differential equations and systems

### Keywords:

hyperelliptic integral; Gauss hypergeometric function