Group Classification Problems

Generalization of differential equations to cover a wide range of problems by insertion of arbitrary functions or parameters other than the dependent variables of the system is a common task performed in mathematical modeling. The specific forms of these parameters/functions represent special solutions for the problem. Some special forms may inherit more exact analytical solutions than the others. To determine those special forms which lead to more analytical solutions then constitute a group classification problem in the context of symmetry methods.  
The specific value of parameters or the specific forms of functions may have direct consequences on the number of symmetry base generators admitted by the equations. If the parameters/functions are required to be arbitrary, that is in their most general forms, then, minimal numbers of symmetries are retrieved. The base symmetry generators corresponding to these minimum numbers of symmetries are named as the principle Lie algebra. When the parameters/functions attain special forms, extra symmetry generators are added to the Lie algebra which is called as the extensions of the principal Lie algebra. Only the case of arbitrary parameters/functions appearing in the equations will be treated in this chapter. Such arbitrary parameters/functions appearing in the boundary conditions will be delayed until next chapter on boundary value problems.