Boundary Value Problems

In the previous chapters, it is observed that a given differential equation may admit a number of symmetries. Different solutions corresponding to different symmetries are always possible. The practical problem is to choose the right solution for the physical problem at hand. But physical problems in the form of differential equations appear with constraints which are named as initial or boundary conditions. Many of the mathematical solutions produced with different symmetry generators would then be useless as it comes to satisfy the initial/boundary conditions.  
For nonlinear partial differential equations, in addition for the equation to be invariant under the given symmetry, the boundaries and the boundary conditions should also be invariant under the same transformation (Bluman and Kumei, 1989). This restricts the symmetries and usually a few of the symmetries may survive after the application of the symmetry generators to the boundaries and boundary conditions. What happens when a symmetry is admitted by the equation but not by the accompanying conditions? The answer is simple. For nonlinear PDEs, transformation of the equation in the form of similarity variables is possible but since the transformation of the conditions is not achieved, the reduced boundary value problem (BVP) cannot be expressed in terms of the new variables. The ideas will be exploited by treating sample problems.  
BVP with infinite or semi-infinite domains are better in applying symmetry methods since such conditions impose no further restrictions on the symmetries. In the case of nonlinear PDEs, if the whole domain is finite, it may happen that all symmetries of the equations may become useless when the finite boundaries and conditions are imposed on the equations.