An implicit series solution for a boundary value problem modelling a thermal explosion

An implicit series solution admitted by a boundary value problem modelled by a Lane–Emden equation of the second kind is obtained. The boundary value problem was derived by Frank-Kamenetskii to model the steady temperature in a vessel in which a thermal explosion is taking place. The Lane–Emden equation is reduced to an autonomous second-order ordinary differential equation by means of a coordinate transformation. The autonomous second-order ordinary differential equation is reduced to a first-order Abel equation. A power series solution of the first-order Abel equation is obtained. The power series solution of the Abel equation is transformed into an implicit series solution of the original Lane–Emden equation satisfying the boundary conditions of the original problem. We show that the implicit power series solution is valid for values of the dimensionless Frank-Kamenetskii parameter δ < 0.02 .