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An implicit series solution for a boundary value problem modelling a thermal explosion. (English) Zbl 1211.65101
Summary: 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 $\delta <0·02$.
##### MSC:
 65L10 Boundary value problems for ODE (numerical methods)
##### Keywords:
Lane-Emden equation; series solution; Abel equation