Approximate Symmetry Theories

Although Lie group theory is a systematic and unified way of finding exact analytical solutions of differential equations, exact analytical solutions are rare for especially nonlinear and variable coefficient equations. If the equation does not admit an exact analytical solution, then no matter what method is employed, a solution cannot be expressed in the form of known functions. Lie group theory is not an exception, that is, if there is no analytical solution, then it cannot be deduced by Lie group theory also. If analytical solutions are not available, the next best choice is to search for approximate analytical solutions before resorting to purely numerical techniques. One widely used and well established approximate analytical solution technique is the perturbation technique applied successfully for over a century to differential equations (Nayfeh, 1973, 1981, 1985; Murdock, 1999; Hinch, 1991; Shivamoggi, 2003; Aziz and Na, 1984; Kevorkian and Cole, 1981; Van Dyke, 1975). The idea is then to unite the potentials of perturbation methods and symmetry methods to develop new methods for finding approximate solutions systematically.