In Chapters 11–15, we saw how initial/boundary value problems for linear partial differential equations could be solved by first identifying an orthogonal basis of eigenfunctions for the relevant differential operator (usually the Laplacian), and then representing the desired initial conditions or boundary conditions as an infinite summation of these eigenfunctions. For each bounded domain, each boundary condition, and each coordinate system we considered, we found a system of eigenfunctions that was ‘adapted’ to that domain, boundary conditions, and coordinate system.

This method is extremely powerful, but it raises several questions as follows.

1. (1) What if you are confronted with a new domain or coordinate system, where none of the known eigenfunction bases is applicable? Theorem 15E.12, p. 353, says that a suitable eigenfunction basis for this domain always exists, in principle. But how do you go about discovering such a basis in practice? For that matter, how were eigenfunctions bases like the Fourier–Bessel functions discovered in the first place? Where did Bessel's equation come from?

2. (2) What if you are dealing with an unbounded domain, such as diffusion in ℝ3? In this case, Theorem 15E.12 is not applicable, and in general it may not be possible (or at least, not feasible) to represent initial/boundary conditions in terms of eigenfunctions. What alternative methods are available?

3. […]

This is a textbook for an introductory course on linear partial differential equations (PDEs) and initial/boundary value problems (I/BVPs). It also provides a mathematically rigorous introduction to Fourier analysis (Chapters 7, 8, 9, 10, and 19), which is the main tool used to solve linear PDEs in Cartesian coordinates. Finally, it introduces basic functional analysis (Chapter 6) and complex analysis (Chapter 18). The first is necessary to characterize rigorously the convergence of Fourier series, and also to discuss eigenfunctions for linear differential operators. The second provides powerful techniques to transform domains and compute integrals, and also offers additional insight into Fourier series.

This book is not intended to be comprehensive or encyclopaedic. It is designed for a one-semester course (i.e. 36–40 hours of lectures), and it is therefore strictly limited in scope. First, it deals mainly with linear PDEs with constant coefficients. Thus, there is no discussion of characteristics, conservation laws, shocks, variational techniques, or perturbation methods, which would be germane to other types of PDEs. Second, the book focuses mainly on concrete solution methods to specific PDEs (e.g. the Laplace, Poisson, heat, wave, and Schrödinger equations) on specific domains (e.g. line segments, boxes, disks, annuli, spheres), and spends rather little time on qualitative results about entire classes of PDEs (e.g. elliptic, parabolic, hyperbolic) on general domains.

Differential equations encode the underlying dynamical laws which govern a physical system. But a physical system is more than an abstract collection of laws; it is a specific configuration of matter and energy, which begins in a specific initial state (mathematically encoded as initial conditions) and which is embedded in a specific environment (encoded by certain boundary conditions). Thus, to model this physical system, we must find functions that satisfy the underlying differential equations, while simultaneously satisfying these initial conditions and boundary conditions; this is called an initial/boundary value problem (I/BVP).

Before we can develop solution methods for PDEs in general, and I/BVPs in particular, we must answer some qualitative questions. Under what conditions does a solution even exist? If it exists, is it unique? If the solution is not unique, then how can we parameterize the set of all possible solutions? Does this set have some kind of order or structure?

The beauty of linear differential equations is that linearity makes these questions much easier to answer. A linear PDE can be seen as a linear equation in an infinite-dimensional vector space, where the ‘vectors’ are functions. Most of the methods and concepts of linear algebra can be translated almost verbatim into this context. In particular, the set of solutions to a homogeneous linear PDE (or homogeneous linear I/BVP) forms a linear subspace.

This text has many advantages over most other introductions to partial differential equations.

Illustrations

PDEs are physically motivated and geometrical objects; they describe curves, surfaces, and scalar fields with special geometric properties, and the way these entities evolve over time under endogenous dynamics. To understand PDEs and their solutions, it is necessary to visualize them. Algebraic formulae are just a language used to communicate such visual ideas in lieu of pictures, and they generally make a poor substitute. This book has over 300 high-quality illustrations, many of which are rendered in three dimensions. In the online version of the book, most of these illustrations appear in full colour. Also, the website contains many animations which do not appear in the printed book.

Most importantly, on the book website, all illustrations are freely available under a Creative Commons Attribution Noncommercial Share-Alike License. This means that you are free to download, modify, and utilize the illustrations to prepare your own course materials (e.g. printed lecture notes or beamer presentations), as long as you attribute the original author. Please visit <http://xaravve.trentu.ca/pde>.

Physical motivation

Connecting the math to physical reality is critical: it keeps students motivated, and helps them interpret the mathematical formalism in terms of their physical intuitions about diffusion, vibration, electrostatics, etc. Chapter 1 of this book discusses the physics behind the heat, Laplace, and Poisson equations, and Chapter 2 discusses the wave equation.