Doklady Physics, Vol. 45, No. 11, 2000, pp. 610–612. Translated from Doklady Akademii Nauk, Vol. 375, No. 2, 2000, pp. 188–190.
Original Russian Text Copyright © 2000 by Bykov, Shmidt.
MECHANICS
Exact Time-Dependent Solutions in the Simplest Model
of Chain-Flame Propagation
V. I. Bykov and A. V. Shmidt
Presented by Academician A. G. Merzhanov December 17, 1999
Received February 8, 2000
In the theory of combustion, the model of chain- found using namely this technique. The scheme of con-
flame propagation was proposed by Ya.B. Zel’dovich structing solution (3) to model (1) is as follows.
and D.A. Frank-Kamenetskiœ [1] (see also [2]). In suc-
ceeding years, this line of research was actively devel-
oped by many investigators (see, for example, [3–6]).
In modern notation, the following nonlinear equation
corresponds to the simplest model of chain-flame prop-
agation:
According to [7, 8], manifold
h(t, x, ux, uxx, …) = 0
(4)
is invariant with respect to equation (1), if
(Dt(h)
)
= 0,
{4}
(5)
{1}
ut = uxx + 2u2(1 – u),
(1) where Dt is the operator of total differentiation with
respect to time t. Relationship (5) is assumed to be met
which is referred to as the Zel’dovich model [5, p. 198]
by experts in mathematical modeling. At the same time,
in mathematical studies on the dynamics of “reaction +
diffusion” systems, problem (1) is treated as a part of
the so-called KPP (Kholmogorov–Petrovskiœ–
Piskunov) problem concerning the existence of travel-
ing-wave type solutions.
due to (1), (4), and their differential consequences with
respect to variable x.
Invariant manifold (4) for equation (1) is sought in
the form
uxx = α (u)ux + β(u).
(6)
Substituting (6) in (1), we obtain
There is a well known classical exact solution to
Eq. (1) [4]:
ut = α (u)ux + β(u) – 2u3 + 2u2.
(7)
1
Invariance condition (5) for manifold (6) with respect
to equation (7) has the form
----------------------------------
u1(t, x) =
,
(2)
1 + exp(x – 1)
utxx = Dt(α (u)ux + β(u)).
(8)
which has the form of a traveling wave with unity
velocity. In this study, we first find the exact time-
dependent solution to model (1):
Condition (8) must be met due to Eqs. (6), (7), and to
their differential consequences:
exp(t – x) – 1
2t + x + exp(t – x)
α''ux3 + (α ux + β)(2α'ux + α2 + β' – 6u2 + 4u)
--------------------------------------------
u 2 (t, x) =
,
(3)
+ ux(2α'α ux + α'β + α β' + β''ux – 12uxu + 4ux)
which, in contrast to classical self-similar solution (2),
makes it possible to also study the transient processes
leading to the steady-state wave. Furthermore, we gen-
eralize our consideration to the case of many spatial
variables.
(9)
= (α'ux + β')(α ux + β – 2u3 + 2u2)
+ α (α'u2x + α (α ux + β) + β'ux – 6uxu2 + 4uxu).
Considering (9) as the condition for which the third-
power polynomial in terms of ux becomes zero, we
obtain the following set of equations:
In recent years, the technique of constructing exact
solutions to the sets of equations of the reaction + dif-
fusion type was developed in the cycle of studies of
O.V. Kaptsov [7–10] and in paper [11]. Solution (3) is
α'' = 0, 2α'α + β'' – 12u + 4 = 0,
β + u3 – u2 = 0, β'(u2 – u3) – β(2u – 3u2) = 0.
Institute of Computer Modeling, Siberian Division,
Russian Academy of Sciences,
Akademgorodok, Krasnoyarsk, 660036 Russia
The solution to this set has the following form:
α = 3u – 1, β = –u3 + u2.
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