Introduction

Lie group of transformations and their employment in search of solutions of differential equations dates back to the seminal works of Sophus Lie, a Norwegian mathematician in the last quarter of 19th century. Detailed information about his life and work has been summarized by Fritzsche (1999). The theory is a unification of many ad hoc mathematical solution methods which outlines a deeper understanding of the properties of differential equations and their so called symmetries. In fact, a generalized and unified excellent method with a strong theoretical background has been presented by Sophus Lie (1893). The importance of the method was not fully understood and the work of Lie remained almost in the shadows until 1960s. From then, the importance of the generalized approach, especially being the only general method in attacking nonlinear differential equations in search of analytical solutions, was more realized and used as an efficient tool to handle differential equations. The mathematical models in vast areas such as heat transfer, fluid mechanics, vibrations, elasticity, plasticity, dynamics, quantum mechanics, optics, administrative sciences were successfully solved by employment of the symmetry methods. See the handbooks edited by Ibragimov (1993, 1995, 1996) for a partial but extensive review of the work up to 1996 on the topic as well as original work of Sophus Lie. It is common to collect analytical and numerical techniques for solving differential equations in the form of handbooks (Zwillinger, 1989) and the symmetry techniques find a place although very brief in such general handbooks. Compared to the power and utility of the techniques, symmetry methods still did not receive enough appreciation from the scientific community. The high and advanced theoretical background prevents many people to enter the subject. A simplified and more readable presentation of the topic might be useful to increase the number of researchers employing the methods. At least, some special transformations can be augmented into the differential equation courses at the undergraduate level. This book is intended to convey the techniques to a broader audience and may be considered as an initiate to attract more researchers to the field who are desperate in solving their physical problems.