Invariant submodels of model of nonlinear diffusion in inhomogeneous medium with nonstationary absorption or source

Abstract

We carried out a complete group classification of the nonlinear differential equation of the model, describing the nonlinear diffusion process (or the process of heat propagation) in an inhomogeneous medium with nonstationary absorption or source. We found all models whose equations admit a nontrivial continuous Lie group of the transformations. We researched invariant submodels of the obtained models. Some invariant solutions, describing these submodels were found either explicitly, or their search was reduced to solving of the nonlinear integral equations. In particular, we obtained an invariant solution that describes the nonlinear diffusion process (or heat distribution process) with two fixed “black holes”, and invariant solution that describes the nonlinear diffusion process (or heat distribution process) with a fixed “black hole”, and with a moving “black hole”. The solutions with “black holes” can be used to describe the process of destruction of glaciers under the influence of the external heat source, for example a solar energy. Using other invariant solution, we studied a diffusion process (or heat distribution process) for which at the initial moment of the time at a fixed point a concentration (a temperature) and its gradient are specified. The solution of the boundary value problem describing this process reduces to the solution of nonlinear integral equations. We have established the existence and uniqueness of solutions of this boundary value problem under certain conditions. The mechanical significance of the obtained solutions is as follows: 1) they describe specific nonlinear diffusion processes (or heat distribution processes), 2) they can be used as tests in numerical calculations in studies of nonlinear diffusion (or heat distribution) in an inhomogeneous medium with nonstationary absorption or source, 3) they allow to assess the degree of adequacy of a given mathematical model to real physical processes, after carrying out experiments corresponding to these decisions, and estimating the arising deviations.