Solutions by Special Group Transformations

The general Lie group theory requires extensive computations and the theory is much involved. In this chapter, as an introduction to the topic, special group transformations admitted by the differential equations will be considered which does not require extensive algebra. In this way, the general ideas behind the symmetry methods can be given in a simple and understandable way before discussing the general theory. Translational symmetries, scaling symmetries, spiral group symmetries are the most common symmetries that are inherited by the differential equations. It is an easy task to check whether such symmetries are admitted by the given equation. If the special transformation is determined, then solutions based on such symmetries can be constructed in a systematic way. Both ordinary and partial differential equations will be considered in this chapter. A unified approach including translational, scaling and spiral symmetries will also be discussed.  
In the direct analysis, the equation to be solved is given and the test yields the special transformations admitted by the equation which then produces the so called symmetry solutions. The reverse method is to find the general structural form of the equation for which a given special group of transformation is admitted. This reverse method will also be discussed in this chapter. Another important topic is the group classification problem which arises when the equation contains arbitrary parameters or arbitrary functions.  Such cases are investigated under the topic of general Lie group theory. However, in this section, group classification is done with respect to a given special group transformation as a preliminary introduction to the topic. Chapter 2 is self- contained and using the ideas, the applied oriented researchers may start finding solutions to their equations without a general grasp of the Lie Group theory which will follow in the subsequent chapters. For applications of special Lie group of transformations to physical problems, see Pakdemirli and Yürüsoy (1998), Pakdemirli and Şahin (2006a, 2006b), Hansen and Na (1968), Timol and Kalthia (1986), Pakdemirli (1992, 1993, 1994a, 1994b), Kılıç et al. (2004), Hayat et al. (2013), Abbasbandy et al. (2008).