On construction of heat wave for nonlinear heat equation in symmetrical case

A. L. Kazakov ; P. A. Kuznetsov ; A. A. Lempert

The Bulletin of Irkutsk State University. Series Mathematics, Tome 11 (2015), pp. 39-53

Résumé

The nonlinear second-order parabolic equation with two variables is considered in the article. Under the additional conditions, this equation can be interpreted as the nonlinear heat equation (the porous medium equation) in case of dependence of the unknown function on two variables (time and origin distance). The equation has many applications in continuum mechanics, in particular, it is used for mathematical modeling of filtration of ideal polytropic gas in porous media. The authors research a special class of solutions which are usually called a “heat wave” in literature. The special feature of these solutions is that they are “sewn” together of two continuously butt-joined solutions (trivial and nonnegative). The solution of heat wave's type can has derivative discontinuity on the line of joint which is called as the heat wave's front (the front of filtration), i.e. smoothness of the solution, generally speaking, is broken. The most natural problem which has the solutions of this kind is so-called “the Sakharov problem of the initiation of a heat wave”. New solutions of this problem in kind of multiple power series in physical variables were constructed in the article. The coefficients of the series are determined from tridiagonal systems of linear algebraic equations. Herewith, the elements of matrixes of systems depend on the order of the matrixes and the condition of the diagonal dominance is not executed. The recurrent formulas of the coefficients were obtained.

Détail

Keywords: partial differential equations, porous medium equation, heat wave, power series.