  Filtration combustion is described by Laplacian growth without surface
tension. These equations have elegant analytical solutions that replace the
complex integro-differential motion equations by simple differential equations
of pole motion in a complex plane. The main problem with such a solution is the
existence of finite time singularities. To prevent such singularities, nonzero
surface tension is usually used.However, nonzero surface tension does not exist
in filtration combustion, and this destroys the analytical solutions. However,
a more elegant approach exists for solving the problem. First, we can introduce
a small amount of pole noise to the system. Second, for regularisation of the
problem, we throw out all new poles that can produce a finite time singularity.
It can be strictly proved that the asymptotic solution for such a system is a
single finger. Moreover, the qualitative consideration demonstrates that a
finger with 1/2 of the channel width is statistically stable. Therefore, all
properties of such a solution are exactly the same as those of the solution
with nonzero surface tension under numerical noise. The solution of the ST
problem without surface tension is similar to the solution for the equation of
cellular flames in the case of the combustion of gas mixtures.
