Solution of Boussinesq problem using Lie symmetries

Firstly the Lie point symmetries of cylindrically symmetric homogeneous Navier equations are obtained. Using the symmetries the general class of similarity solutions are found. The subclass that also satisfies the non-homogeneous system of the medium subject to a singular force is determined. Substituting the subclass into the non-homogeneous system, a system of ordinary differential equations is obtained. The solution of the system satisfying the boundary conditions of Bousssinesq problem gives the exact solution.