On Analytic Solutions of the Problem of Heat Wave Front Movement for the Nonlinear Heat Equation with Source

Abstract

We continue our investigation of the special boundary-value problems for the nonlinear parabolic heat equation (the porous medium equation) in the article. In the case of a power-law dependence of the heat conductivity coefficient on temperature the equation is used for describing high-temperature processes, filtration of gas through porous media, migration of biological populations, etc. Moreover, the equation has specific nonlinear properties, which may be interesting from the point of view of physics as well as mathematics. For example, it is well known, that the described disturbances may have a finite velocity of propagation. The heat waves (waves of filtration) compose an important class of solutions to the equation under consideration. Geometrically, these solutions are constructed from two integral surfaces, which are continuously connected on a curve - heat wave front. We consider a boundary-value problem, which has such solutions. The research is carried out in the class of analytic functions by the characteristic series method. This method was suggested by R. Courant and then it was adapted for nonlinear parabolic equations in A.F. Sidorov’s scientific school. We have already researched similar problems in case of closed front without source. For each problem we constructed the solution in form of characteristic series and proved the exist theorem, which guaranteed the convergence. The paper deals with the flat-symmetrical problem with given front and source. The theorem of existence of the analytic solution(heat wave’s nonnegative part) was proved and the solution in form of the power series was constructed. Also we considered an interesting case in which the source is a power function (such cases are common in applications). It was shown that the original problem may be reduced to the Cauchy problem for nonlinear ordinary differential equation of the second order.

Keywords

• nonlinear heat equation with source
• porous medium equation
• heat wave
• characteristic series
• convergence
• exist theorem