Solution of ordinary differential equations by continuous groups (Q2770554)

There are numerous techniques for finding exact solutions to differential equations which in most cases are applicable to particular classes of problems. It turns out that quite a few methods have a common feature exploiting symmetries of differential equations. More than a century ago, the Norwegian mathematician Sophus Lie used continuous groups to develop a systematic transformation theory for finding exact solutions to ordinary and partial differential equations. NEWLINENEWLINENEWLINEThis textbook is aimed on the presentation of a selfcontained exposition of the theory of continuous groups with an application to solving ordinary differential equations. It does not require a priori knowledge of group theory and algebraic concepts since the material necessary for understanding the text is provided. NEWLINENEWLINENEWLINELogically, the book is divided into three parts. The first part, entitled Background, is formed by Chapters 1 to 4, where continuous one-parameter groups are introduced and the method of characteristics is explained. The second part, entitled Ordinary Differential Equations, consists of Chapters 5 to 7, where the techniques for solving ODEs by group methods are explained. Finally, a massive array of tables, where the catalogues of the first- and second-order equations along with other useful information are collected, form the third part of the book. The core of the book is the second part where in three chapters first-order, higher-order and second-order ODEs are studied. Chapter 5 deals with the simplest case of first-order equations. The main idea is to use an inverse approach in which one starts with a known symbol. First, the general form for the ODE NEWLINE\[NEWLINE f(x,y,y')=0 \tag{1} NEWLINE\]NEWLINE is found which is invariant under the group represented by the symbol. Second, starting again with the known symbol, one finds a transformation of the coordinates that replaces equation (1) with a separable form. This relation is then integrated and the original variables are reintroduced. Therefore, an exact analytical solution to equation (1) is established in terms of quadratures, which means that any first-order ODE invariant under a known group can be analytically solved. In Chapter 6, the one-parameter theory of the previous chapter is extended to higher-order equations. It is shown that, if an \(n\)th-order ODE is invariant under a known group, its order can be reduced by one and further reduction may or may not be possible. A compendium of second-order equations invariant under a one-parameter group is provided in section 6.4. Finally, in Chapter 7, definition and classification of two-parameter groups, invariance, and canonical coordinates are discussed. It is shown that every second-order ODE invariant under a two-parameter group can be completely solved in quadratures. This is achieved by determining the classification, or type, for the group and introducing appropriate canonical coordinates that result in a separable form for the ODE which is twice integrable. The techniques explained in this chapter are especially important due to the fact that the one-parameter theory of Chapter 6 does not generally provide a full solution to a second-order ODE. NEWLINENEWLINENEWLINESince the book has been written by an engineer, the mathematical prerequisites and the level of difficulty are kept within the limits accessible by most engineering and science students. This text is a welcome contribution to the literature on group methods for solving ODEs.