Invariant Solutions of Two Dimensional Heat Equation
Abstract
The symmetry group of a given differential equation is the group of transformations that translate the solutions of the equation into solutions. If the infinitesimal generators of symmetry groups of systems of partial differential equations are known, the symmetry group can be used to explicitly find particular types of solutions that are invariant with respect to the symmetry group of the system. The class of invariant solutions includes exact solutions that have direct mathematical or physical meaning. In this paper, using the well-known infinitesimal generators of some symmetry groups of the two-dimensional heat conduction equation, solutions are found that are invariant with respect to these groups. It is considered cases when conductivity coefficients of the two-dimensional heat conduction equation are power functions of temperature and conductivity coefficients are exponential functions of temperature. In first case invariant solutions contain well known self-similar solutions which are widely used in applications.