Tectonic movements may be under way in the world of applied mathematics. The modelling complexities of many situations generate uncertainties that pose challenges for traditional approaches. Methods concerned with less prescriptive, more descriptive universalities may be in the ascendancy as management techniques replace mathematics in the engineering curriculum. Some engineering researchers even claim "all the calculations have already been done - the main problem is now data management".
The authors of An Introduction to Partial Differential Equations , Yehuda Pinchover and Jacob Rubenstein, draw on a wealth of teaching experience at Stanford University, University of California, Los Angeles, and the Technion. Their book spans both introductory and advanced material, making it suitable for use right through a UK undergraduate degree. It requires knowledge of multivariate calculus but builds through basics such as separation of variables up to variational methods and function spaces. It stresses the practical applications of rigour, introduced with a touch sufficiently light not to put off the average student.
Although covering similar topics, An Introduction to Partial Differential Equations with Matlab by Matthew Coleman feels quite different in approach.
Readers are assumed to be familiar with the package in question, but thereafter it is used mainly in the exercises. Pinning the book to a particular software package does not affect the logical progression of the text.
Both books cover separation of variables, characteristics, Sturm-Liouville theory, higher-dimensional equations and Green's functions. Both postpone numerical methods to the final chapter. Pinchover and Rubinstein's book contains brief sections on variational methods and function spaces.
Coleman's has separate appendices dealing with the rigour of uniform convergence, existence and uniqueness. Readers new to the subject will find Coleman's appendix cataloguing important partial differential equations in their natural surroundings quite useful.
More significant differences lie in the choice of emphasis. Pinchover and Rubinstein dedicate a 40-page chapter to first-order equations, including the analysis of shocks. Coleman dedicates three 50-page chapters to Fourier analysis, integral transform solutions and the properties of Bessel functions and orthogonal polynomials. Coleman's more explicit, extended style would probably allow its use as an advanced graduate or reference text for UK engineers or physicists.
Both texts contain eye-catching original features. For example, Pinchover and Rubinstein consider the optimum depth for a wine cellar to control temperature fluctuations. Coleman precedes each chapter with an historical prologue. In the prelude to chapter two, we learn that Fourier's work on the heat equation may have started during an enforced two-year sojourn in Egypt after the Battle of the Nile - which makes his fundamental work one indirect achievement of Nelson that was overlooked in the recent bicentennial celebrations.
A positive enhancement to both texts could be the inclusion of a (brief) discussion of applications of PDEs to areas such as biological models.
Biomathematics is an area of research that is comparatively young, but it is also very active. Many undergraduate degrees now include such courses.
If applied mathematics is to prove the nay-saying engineering data managers wrong, new texts must strive to reflect such exciting applications.
Christopher Howls is senior lecturer in applied mathematics, Southampton University.
An Introduction to Partial Differential Equations. First edition
Author - Yehuda Pinchover and Jacob Rubinstein
Publisher - Cambridge University Press
Pages - 371
Price - £65.00 and £26.00
ISBN - 0 521 84886 5 and 61323 X