Exact Linearization of a Combustion Front: The F lambda Class Method

The Frank-Kamenetskii equation ∂_t U = ∂_xx U + U eλ U describes thermal explosion in a reactive medium. We show that this nonlinear PDE is exactly linearized by the transformation W = E_1(λ U), where E_1 is the exponential integral. The linearized equation ∂_t W = ∂_xx W + 1 admits an explicit solution W(x,t) = (etΔ W_0)(x) + t, where W_0(x) = E_1(λ U_0(x)). This yields a numerical method that is exact (no approximation of the nonlinearity), robust (handles blowup without instability), and fast (10–50× faster than standard ODE integrators). We validate the method on Gaussian and Lorentzian initial profiles, recover the theoretical extinction rates e⁻³ᵗ and e⁻ᵗ/t, and construct a phase diagram for ignition vs. extinction.