Similarity Methods for Differential Equations

Description

The aim of this book is to provide a systematic and practical account of methods of integration of ordinary and partial differential equations based on invariance under continuous (Lie) groups of trans­ formations. The goal of these methods is the expression of a solution in terms of quadrature in the case of ordinary differential equations of first order and a reduction in order for higher order equations. For partial differential equations at least a reduction in the number of independent variables is sought and in favorable cases a reduction to ordinary differential equations with special solutions or quadrature. In the last century, approximately one hundred years ago, Sophus Lie tried to construct a general integration theory, in the above sense, for ordinary differential equations. Following Abel's approach for algebraic equations he studied the invariance of ordinary differential equations under transformations. In particular, Lie introduced the study of continuous groups of transformations of ordinary differential equations, based on the infinitesimal properties of the group. In a sense the theory was completely successful. It was shown how for a first-order differential equation the knowledge of a group leads immediately to quadrature, and for a higher order equation (or system) to a reduction in order. In another sense this theory is somewhat disappointing in that for a first-order differ­ ential equation essentially no systematic way can be given for finding the groups or showing that they do not exist for a first-order differential equation.

MR0460846

From the authors' introduction: "The work presented here falls naturally into two parts; the aims and partial contents of these parts is now sketched for purposes of orientation.
"Part 1: Ordinary differential equations. For ordinary differential equations the aim is a theory of integration or reduction to quadratures. For first-order equations this means that special cases can be reduced to essentially the same case, one of quadrature. The canonical case of quadrature occurs when a variable is missing. If the general equation is (1) $F(x,y,dy/dx)=0$ the special case is (2) $F(x,dy/dx)=0$. For one of the (possibly many) branches of (2) we can write (3) $dy/dx=G(x)$, and the general solution is (4) $y=\int_{x_0}^xG(\zeta)d\zeta+\alpha$, $\alpha=\text{const}$. In the sense represented by (3) and (4) the problem is regarded as solved or reduced to quadrature. In this sense the theory accomplishes all that can reasonably be expected of it. For higher order equations or systems the aim is the reduction of the problem to lower order plus a suitable number of quadratures, and this can be carried out for a definite class of problems.
"Part 2: Partial differential equations. For first-order partial differential equations we take the (restricted) point of view that a sufficiently complete integration theory is given by the theory of characteristics. This connects the solution of partial differential equations with the integration of systems of ordinary differential equations and hence with results of Part 1. It may, however, be useful to look at some first-order equations directly from the point of view of transformations and invariance. For higher order equations or systems the aim is a reduction in the number of variables. A typical result is the statement that a solution $u(x,y,t)$ of a particular partial differential equation in three independent variables must be representable as (5) $u(x,y,t)=t^{-1}F(x/t,y/t)$. This procedure can possibly be repeated more than one time. The special case when a partial differential equation contains only two independent variables is particularly important since the problem is reduced to an ordinary differential equation. Further, the methods of Part 1 may be applied. In many physical problems of interest the resulting equation which needs to be studied (together with a suitable number of quadratures) is of first order. In this favorable case the structure of all possible solutions in the phase plane provides complete information on the structure of a class of solutions to the original partial differential equation. It also may provide the basis for a method of numerical integration. Another method, which can be used to obtain the same results in special cases, arises not directly from transformation theory but rather from dimensional analysis. The basic idea is that all physical problems must be expressible in dimensionless variables. This idea is applied to the variables entering a problem for a partial differential equation. For example, if $(x,y)$, some independent variables which are space coordinates with the physical dimensions of `length', enter the problem, then it can be concluded that only the combination $x/y$ (or equivalent) can enter the problem. Evidently, this represents a reduction in the number of variables. However, it should be remarked that the failure of dimensional analysis to predict similarity (i.e., a reduction in the number of essential variables) does not necessarily rule out similarity for the problem. The connection between dimensional analysis and similarity is discussed later. What has been outlined above is the main content of this book. However, various related topics which enter naturally are discussed as the opportunity arises. Among these are asymptotic and local behavior, superposition of similarity solutions for linear cases. Finally, it should be remarked that the methods used apply equally well to non-linear and linear cases. The ideas used represent one of the few systematic methods of attacking nonlinear problems, with an eye to obtaining exact solutions.''