A substance that reacts at initial temperature cannot sustain flame propagation with constant flame velocity, with temperature and composition being functions of a single variablx z = x -ut, u = const.
Nevertheless, a solution almost similar to uniform flame propagation but with time-dependent velocity is a good approximation to the exact solution. In typical cases it is the correct intermediate asymptotic between the initial period, when the specific form of igniting impulse is smoothed out, and the final period, when adiabatic self-ignition of the substance occurs independent of the approach of the flame. The differential equation is found governing the distribution of temperature and concentration in the flame front as well as the momentary value of the flame propagation velocity. This is the equation for flame propagation with a modified source function. This source function permits one to find the exact solution of the differential equation, contrary to the nonmodified source function that prohibits the existence of the solution. Therefore a natural answer is given to the old question of how to calculate flame propagation with a smooth source function, which is positive at initial temperature. Practical application to flame propagation in a preheated substance is also of interest.