关键词: Lane-Emden equation, bifurcation, thermal explosion, analytical method

Abstract: We study a modified version of the Lane-Emden equation of the second kind modelling a thermal explosion in an infinite cylinder and a sphere. We first show that the solution to the relevant boundary value problem is bounded and that the solutions are monotone decreasing. The upper bound, the value of the solution at zero, can be approximated analytically in terms of the physical parameters. We obtain solutions to the boundary value problem, using both the Taylor series (which work well for weak nonlinearity) and the δ-expansion method (valid for strong nonlinearity). From here, we are able to deduce the qualitative behavior of the solution profiles with a change in any one of the physical parameters.

Key words: pipe conveying fluid, internal resonance, stability, bifurcation, Lane-Emden equation, bifurcation, thermal explosion, analytical method