Analytical and numerical solutions to the problem on a heat wave initiating for the nonlinear heat equation with a source

Abstract

The article deals with the nonlinear heat (porous medium) equation with a source. This equation has a large number of applications, as well as non-standard non-linear properties. An important class of solutions to the equation is heat waves (waves of filtration), propagating on a zero background with a finite velocity. This paper continues the cycle of articles devoted to the construction and study of heat waves with a closed front. Here we consider a problem with a given boundary condition with a source in cases of plane, circular and spherical symmetry. The solution is constructed in the form of a double power series, the theorem of existence and uniqueness is proved. We also look for some exact solutions of an equation with a power source, the construction of which reduces to the integration of ordinary differential equations (ODE) with a singularity. To solve the obtained ODE, we propose a computational algorithm based on the boundary element method. The results of numerical calculations are presented and discussed.