On a three-dimensional heat wave generated by boundary condition specified on a time-dependent manifold

A. L. Kazakov ; P. A. Kuznetsov ; L. F. Spevak

The Bulletin of Irkutsk State University. Series Mathematics, Tome 26 (2018), pp. 16-34

Résumé

The article is devoted to study the nonlinear heat equation (the porous medium equation) in the case of power nonlinearity. Three–dimensional problem of the initiation of a heat wave by boundary condition specified on a time–dependent manifold is considered. The wave has a finite velocity of propagation on the cold (zero) background. A new theorem of existence and uniqueness of the analytical solution (the main theorem) is proved. The solution is constructed in the form of a multiple power series with respect to independent variables. The coefficients of the series are computed recurrently by induction on the total order of differentiation: a system of algebraic equations of increasing dimension is solved at each step. The local convergence of the series is proved by majorant method using Cauchy–Kovalevskaya theorem. Thus, previously obtained results are generalize and reinforced which concern the solution of the problem of heat wave motion on the cold background. Besides, some particular cases are considered when the solution procedure can be reduced to the solution of a second order nonlinear ordinary differential equation unsolved with respect to the highest derivative. As the obtained ordinary differential equation can not be solved in quadratures, qualitative research is performed as well as the numerical experiments with the use of the boundary element method. The obtained results are interpreted with respect to the original problem of the heat wave motion.

Détail

Keywords: nonlinear heat equation; existence theorem; invariant solution; boundary element method; numerical experiment.