Kazakov A.   Orlov S.  

The method of heat wave construction for nonlinear heat (porous medium) equation

Reporter: Kazakov A.

The nonlinear second-order parabolic equation with two variables is considered. Under the additional conditions, this equation can be interpreted as the porous medium equation in case of dependence of the unknown function on two variables: time and distance from the origin. The equation has a wide variety of applications in continuum mechanics, for example, it is applicable for mathematical modeling of filtration of ideal polytropic gas in porous media or heat conduction. The authors deal with a special solutions which are usually called a heat waves. The special feature of such solution is that it consists of two continuously joined solutions. First of them is trivial and second one is nonnegative. The heat wave solution can have a discontinuity derivatives on the line of joint which is called the front of heat wave, i.e. smoothness of the solution, generally speaking, is broken. The most natural problem which has such solutions is so-called “the Sakharov problem of the initiation of a heat wave”. New solutions of considering problem in the form of multiple power series for physical variables are constructed. The coefficients of the series are obtained from tridiagonal systems of linear algebraic equations. Herewith, the elements of matrixes of this systems depend on the matrixes order and the condition of the diagonal dominance is not executed. The recurrent formulas for the coefficients are suggested.