Exact first integrals for a Lane-Emden equation of the second kind modeling a thermal explosion in a rectangular slab

The primary focus of the present paper will be the study of the exact first integral of a Lane-Emden equation of the second kind modeling a thermal explosion in a rectangular slab. Such results generalize those of Harley and Momoniat [Harley, C., Momoniat, E., 2008. J. Math. Anal. Appl. 344, 757-764], in which first integrals up to order ϵ were considered for the model. In particular, our results both generalize their results in the small ϵ regime and are valid in the large ϵ regime, for the k=0 case. As in Harley and Momoniat, we find that there is a critical value of δ beyond which solutions do not exist. Interestingly, we find that this critical value of δ is quite different than the one derived in Harley and Momoniat, thanks to the fact that we obtain exact, and not approximate, relations. Furthermore, we show that while multiple solutions exist in the case of thermal explosion in a rectangular slab, only one such solution is physically meaningful (positive over the domain). Hence, the physically meaningful solution is unique. Our exact analytical results are shown to be in agreement with numerical simulations.