On the construction of a heat wave generated by a boundary condition on a moving border

Abstract

The paper deals with the construction of solutions to a nonlinear heat equation, which have the type of heat waves propagating over a cold (zero) background with a finite velocity. Such solutions are atypical for parabolic equations. They appear due to the degeneration of the parabolic type of equation on a manifold where the desired function becomes zero. Various kinds of boundary conditions provide the existence of solutions with the desired properties. The most complicated of them, specifying nonzero values of the desired function on a moving manifold, is considered in this paper. A new theorem of the existence and uniqueness of the solution to the heat wave initiation problem under the considered boundary condition is proved. A method for constructing an approximate solution based on expansion in radial basis functions and the collocation method is proposed. The solution is constructed in two steps. At the first step, we construct a solution in the domain situated between the specified moving manifold and the zero front, which is determined in the process of solving. A special variable change similar to hodograph transformation is used. At the second step, we complete the solution in the domain situated between the initial and actual position of the moving manifold. Calculations are made showing that the new approach gives good results and more stable convergence as compared with the boundary element method used by the authors earlier.