On the intermediate asymptotic solutions in some models of the combustion theory

We consider the travelling wave solutions of a nonlinear parabolic equation of the second order, namely the equation of the Kolmogorov–Petrovsky–Piskunov type with the heat release function on the right–hand side being analytical. We found a new analytic representation for such a solution or, more accurately, for its inverse function which is represented as a sum of an explicitly calculated summand and an auxiliary function defined on the unit interval. An algorithm for calculating the Taylor coefficients of that function at the right endpoint and at the interior points of the interval is constructed.

We establish a sufficient condition for for the mentioned auxiliary function to be analytical on the entire unit interval including its both endpoints. The obtained criterion for the analyticity allowed us to distinguish a countable dense set of values among the spectrum of the permissible values for the traveling wave velocity (the spectrum being a numerical ray defined by A.Kolmogorov, I.Petrovskii and N.Piskunov) for which the auxiliary function is analytic and consequently the inverse of the traveling wave solution is approximately representable by an explicit formula up to a term uniformly bounded on the unit interval.

There is a result of the analytical theory of the Abel defferential equation. In the proof of the criterion of analyticity we use a kind of Painleve test (or Fuchs–Kovalevskaya–Painleve test) applied to an accessorial equation namely to the Abel equation of the second kind. It became apparent that this equation satisfies the Painleve test when some additional parameter (defined in the text) takes the prescribed values. Moreover the family of solutions passed through the corresponding singular point of the equation consist of analytical functions when the conditions of test gets satisfied.

In the second part of the paper an analytic-numerical method is developed based on the representation described above. The method is applied to the problem of intermediate asymptotic regimes of the thermal combustion of a gas mixture reacting at the initial temperature under the condition of similarity of concentration and temperature fields. Some numerical results of the constructed method are presented.