Invariant solutions of the two-dimensional heat equation

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 are known, then we can find solutions that are invariant under this group. For systems of partial differential equations, the symmetry group can be used to explicitly find particular types of solutions that are themselves invariant under a certain subgroup of the full symmetry group of the system. For example, solutions of an equation with partial derivatives of two independent variables, invariant under a given one-parameter symmetry group, are found by solving a system of ordinary differential equations. The class of solutions that are invariant with respect to a group includes many exact solutions that have immediate 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. First we consider the two-dimensional heat conduction equation with a source that describes the process of heat propagation in a flat region. For this case, a family of exact solutions was found, depending on an arbitrary constant. Then invariant solutions of the two-dimensional heat conduction equation without source are found.