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Singular Perturbations Expert Brings the Theory to Life

By Ferdinand Verhulst

BOOK REVIEW: Historical Developments in Singular Perturbations. By Robert E. O’Malley, Springer, New York, 2014, 256 pages, $53.00.

There exist books on the history of mathematics that avoid mathematics and emphasize social interactions, and mathematics books that offer historical notes. The book under review is in a third category: Its extensive sketch of the development of the theory of singular perturbations is interspersed with historical considerations and anecdotes about the mathematics and the people involved. Such a book could be written only by an author who has devoted most of his working life to this topic, always with an eye on human interactions and the background of people in the field.

Bob O’Malley, a former president of SIAM (1991-92), has contacts in many different countries. He is the author of a large number of papers on singular perturbations and a number of valuable books. His book Thinking about Ordinary Differential Equations typifies his approach: Starting from a fascination with phenomena, he seeks deeper insight about them.

The theory of singular perturbations has a somewhat special place in mathematics. As we know, algebra has a profound and highly structured collection of theories, and work on famous basic conjectures helps to unify these theories. Analysis is different. Analysis has theorems and rules, though more exceptions than rules, a wealth of assorted theories, few basic conjectures, and many different points of view. The theory of singular perturbations, which is part of (asymptotic) analysis, represents a typical, and for some people somewhat distasteful, example of the problem with analysis. To algebraists (in the unlikely case that they have heard about singular perturbations), the theory looks like a bag of tricks—people who work on singular perturbation problems perform mental ballet and acrobatics, using mathematical tools that are sometimes concrete, sometimes abstract. 

The boundary value problem 
\varepsilon \frac{d^2 \phi}{dx^2} - \phi = -1-x^2,\,\,x \in (0, 1),\,\,\phi(0)=2,\,\phi(1)=0 

can serve as a relatively simple example. Here \(\varepsilon\) is a small positive parameter. In a crude approach we take \(\varepsilon =0\) to obtain the function 


Figure 1. The process of smoothly connecting interior solutions and local solutions near the boundary to obtain a global approximation is known as “matching.”
as a possible approximation of the solution of the boundary value problem. But how can \(\phi_0(x)\) be an approximation when it does not even satisfy the boundary conditions? The answer is that \(\phi_0(x)\) is a good approximation in the interior of [0, 1] but not near the boundary. Near the endpoints \(x = 0\) and \(x = 1\), we can construct good approximations from local solutions, and by smoothly connecting these “boundary layer solutions” to the interior approximation (a process called “matching”) we obtain an approximation over the entire domain; see Figure 1. The term “singular” is used because the unperturbed problem \((\varepsilon = 0)\) does not produce an acceptable approximation over the entire domain. 

Before 1900, a few people had noticed singular perturbation phenomena, but the theory really got its start in Heidelberg in 1904, with a lecture by Ludwig Prandtl at the International Congress of Mathematicians. Felix Klein, a leading mathematician of the time who arranged a chair for Prandtl in Göttingen, praised Prandtl for “his strong power of intuition and great originality of thought with the expertise of an engineer and the mastery of the mathematical apparatus.”

Prandtl’s breakthrough was in constructing approximate solutions of the nonlinear partial differential equations of fluid dynamics without and with friction. Although the motion of an airplane can be understood only when friction is considered, the corresponding equations are extremely difficult. Prandtl introduced the notion of a boundary layer, a thin layer near the aircraft wings where friction gives rise to solutions very different from those far from the wings. The idea of constructing different types of solutions in different parts of the spatial domain and using special matching rules turned out to be exceptionally fruitful. The insight that solutions of differential equations in different time and space domains can be qualitatively very different moved research a huge step forward. 

Many mathematicians and engineers subsequently started to use Prandtl’s ideas. A remarkable number of them were German–Jewish mathematicians who later had to leave Nazi Germany. They went to Great Britain and the U.S., where the field gained momentum, stimulated further by work of G.I. Taylor, S. Goldstein, and N. Levinson. After the Second World War, singular perturbation theory became an important area of research again in continental Europe, especially in France, the Soviet Union, and The Netherlands.

The book under review describes many aspects of the historical development and is a didactic introduction to singular perturbation theory. The book presupposes a working knowledge of ordinary differential equations; notwithstanding the many technical details, it provides more of an introduction and description of ideas than a mathematically rigorous treatment. For proofs and technical details, the author has compiled a useful list of 538 references.

The text should be very useful for introductory lectures on singular perturbations. It starts with a discussion of the asymptotics of divergent series, followed by the intriguing topic of matching solutions of linear and nonlinear equations. Partially unsolved problems are discussed, with attention to different points of view. This part, with its strong appeals to the reader’s intuition, is mathematics that is full of life. Subsequent chapters deal with turning points, canards, and multiscale methods, for oscillatory as well as boundary layer problems.

The book is a great gift to the world of mathematical analysis.

Ferdinand Verhulst is a professor at the Mathematisch Instituut, University of Utrecht, the Netherlands.

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