Partial Differential Equations: Symmetry Solutions

In the previous chapter, the infinitesimals and symmetry generators are calculated. In this chapter, the solutions using these symmetries will be discussed. For ODEs, it was shown in Chapter 5 that one-parameter Lie group symmetry reduces the order of the equation by one. In partial differential equations, the effect of symmetries is somewhat different. One-parameter symmetry reduces the number of independent variables by one. This is done by the so called similarity variables (group invariants) and the solutions are called similarity (group invariant) solutions. If the PDE has two independent variables, then employing one symmetry transforms the equation into an ODE which is easier to handle either analytically or numerically. It is always possible to achieve multiple reductions in the number of independent variables by employing more than one symmetry. The other option is to map a solution from a known solution using the symmetries. The requirement for this to happen is that the known solution should not be a group invariant solution, otherwise, the solution will map onto itself if the generator producing that solution is employed. More details will be given in the worked examples on the issue.   
The algorithm to calculate similarity solutions is as follows. Given a scalar PDE