  Analytical solutions to the Semenov thermal ignition problem for constant
volume burn governed by Arrhenius reaction kinetics are derived. Specifically,
an approximate analytical solution technique for the Arrhenius-Semenov
differential equation is derived for reaction orders $n \in \mathbb{R}_{>0}$
and exact solutions are also constructed for reaction orders $n \in \mathbb{N}:
n \leq 3$. The approximation technique relies on expansion of the respective
nondominant terms in the differential equation at the lower and upper bounds of
the reaction progress variable in order to create a pair of integrable series.
The two integrated series are then connected to create a single continuous
analytical solution. Excellent agreement is observed between the analytical
approximation and solutions obtained numerically. The presented approximation
constitutes a simple and robust strategy for solving the Arrhenius-Semenov
problem analytically.
