In this paper we investigate the linear stability and properties of the planar travelling non-adiabatic combustion front for the cases of zero and non-zero ambient temperature. The speed of the front is estimated numerically using the shooting and relaxation methods. It is shown that for given parameter values the solution either does not exist or there are two solutions with different values of the front speed, which are referred to as 'fast' and 'slow'. The Evans function approach extended by the compound-matrix method is employed to numerically solve the linear-stability problem for the travelling-wave solution. We demonstrate that the 'slow' branch of the solutions is unstable, whereas the 'fast' branch can be stable or exhibits Hopf or Bogdanov-Takens instability, depending on the parameter values.

We consider travelling wave solutions of a reaction-diffusion system corresponding to a single-step homogeneous premixed combustion scheme competitively coupled with an endothermic reaction. Properties of the travelling combustion fronts, such as the wave speed and the burnt temperature are derived numerically over a range of different parameter values, such as those describing the relative enthalpies, rates and activation energies of the endothermic and exothermic reactions. Unique combustion wave solutions are shown to exist for each distinct combination of the parameter values. These solutions are linearly stable if the heat release from the exothermic reaction is sufficiently large, otherwise the combustion waves develop pulsation. In particular, using a finite element package to numerically integrate the governing partial differential equations, period-1 and period-2 type oscillatory behaviour was observed prior to wave extinction.

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