The Frank-Kamenetskii equation ∂ₜ U = ∂ₓx U + U e^λ U describes thermal explosion in a reactive medium. We show that this nonlinear PDE is exactly linearized by the transformation W = E₁ (λ U), where E₁ is the exponential integral. The linearized equation ∂ₜ W = ∂ₓx W + 1 admits an explicit solution W (x, t) = (e^tΔ W₀) (x) + t, where W₀ (x) = E₁ (λ U₀ (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.
Judicael Brindel (Thu,) studied this question.