Symmetry reduction and invariant solutions for nonlinear fractional diffusion equation with a source term

Abstract

We consider a problem on constructing invariant solutions to a nonlinear fractional differential equations of anomalous diffusion with a source. On the base of an earlier made group classification of the considered equation, for each case in the classification we construct the optimal systems of one-dimensional subalgebras of Lie algebras of infinitesimal operators of the point transformations group admitted by the equation. For each one-dimensional subalgebra of each optimal system we find the corresponding form of the invariant solution and made the symmetry reduction to an ordinary differential equation. We prove that there are three different types of the reduction equations (factor equations): a second order ordinary differential equation integrated by quadratures and two ordinary nonlinear fractional differential equations. For particular cases of the latter we find exact solutions.