# 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.
Keywords: travelling wave solutions, ﬂame propagation, intermediate asymptotics, Kolmogorov –
Petrovskii – Piskunov equation, Abel equation of the second kind, Painleve test.

UDC:
517.927.4