Asymptotic form of the solution of the heat conduction equation with a non-linear heat source in a thin rod

Abstract

The asymptotic form with respect to a small parameter of the boundary-layer solution of the initial and boundary-value problem for the heat conduction equation with a non-linear heat source in a thin flat rod is constructed. The thickness of the rod is taken as the small parameter and the thermal conductivity has the same order of smallness. The asymptotic form possesses a number of singularities. In particular, the principal term of its regular part is described by a non-degenerate equation and the boundary layers near the ends ofthe rod are of two scales ofmagnitude.