Lectures on partial differential equations /

Undergraduate students of mathematics, physics, and engineering all study ordinary differential equations (ODEs), but few have a serious encounter with partial differential equations (PDEs) before graduate school, if then. Because PDEs model so many things, they display a vast diversity of behavior and admit no unified theory. So, with apologies to Tolstoy, every ODE book runs the same, but each PDE book frames the subject in its own way. Arnold (Steklov Mathematical Institute, Russia) has long held a reputation as one of the world's leaders in dynamics and geometry. His Lectures survey big ideas; accordingly, he largely suppresses both the functional analytic formalism and the delicate estimates so characteristic of the subject. He takes the viewpoint that the most important PDEs arise in physics and the most important mathematical ideas contributing to their solution derive from physical principles. Arnold concentrates on the simplest equations of a given type and shows how the key ideas play out. For example, he attacks the general theory of one first-order equation, first via wave-particle duality, then via Hamiltonian dynamics. The various lectures make dramatically different demands on readers; e.g., an accessible introduction to the vibrating string follows a very terse introduction to contact structures. The author's stature and the book's lucidity make this an essential acquisition for all college libraries. The volume by Ockendon and Ockendon (both, Oxford Centre for Industrial and Applied Mathematics, UK) updates the 1983 Inviscid Fluid Flows, by H. Ockendon and Alan B. Tayler. In this new book they view gas dynamics as a paradigm for more general wave propagation. Chapter 2, in which the authors derive the paradigmatic equations for inviscid compressible flow, will particularly benefit undergraduate readers. However, the assumption that readers have a background in basic fluid dynamics modeling will typically put portions of the book beyond their reach. Chapter 3 concentrates on the model for acoustics and the associated linear hyperbolic PDEs. The following chapter studies waves with a purely harmonic time-dependence by means of elliptic PDEs. The last chapters consider nonlinear effects and shock waves. Throughout, the authors develop serious applications in both the text and the exercises. Overall, this is a very inviting introduction to a notoriously difficult subject. ^BSumming Up: Arnold: Essential. Ockendon: Highly recommended. Both books: Upper-division undergraduates through faculty. D. V. Feldman University of New Hampshire