On the construction of heat wave in symmetric case

All-Russian conference on Nonlinear Waves: Theory and New Applications

Kazakov, A.L.^1 ; Lempert, A.A.^1

Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, Russia^1

关键词: Diagonal dominance;    Discontinuous derivatives;    Physical variables;    Porous medium equation;    Recurrent formulae;    Second-order parabolic equations;    Special solutions;    Tridiagonal systems;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/722/1/012016/pdf
DOI  :  10.1088/1742-6596/722/1/012016

来源: IOP

【 摘 要 】

A nonlinear second-order parabolic equation with two variables is considered. Under 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 heat waves. A special feature of such solution is that it consists of two continuously joined solutions. The first of them is trivial and the second one is nonnegative. The heat wave solution can have discontinuous 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 the so-called "the Sakharov problem of the initiation of a heat wave". New solutions of the 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 matrices of this systems depend on the matrix order and the condition of the diagonal dominance is not fulfilled. The recurrent formulas for the coefficients are suggested.