Introduction

Partial differential equations (PDE) describe physical systems, such as solid and fluid mechanics, the evolution of populations and disease, and mathematical physics. The many different kinds of PDE each can exhibit different properties. For example, the heat equation describes the spreading of heat in a conducting medium, smoothing the spatial distribution of temperature as it evolves in time; it also models the molecular diffusion of a solute in its solvent as the concentration varies in both space and time. The wave equation is at the heart of the description of time-dependent displacements in an elastic material, with wave solutions that propagate disturbances. It describes the propagation of p-waves and s-waves from the epicenter of an earthquake, the ripples on the surface of a pond from the drop of a stone, the vibrations of a guitar string, and the resulting sound waves. Laplace's equation lies at the heart of potential theory, with applications to electrostatics and fluid flow as well as other areas of mathematics, such as geometry and the theory of harmonic functions. The mathematics of PDE includes the formulation of techniques to find solutions, together with the development of theoretical tools and results that address the properties of solutions, such as existence and uniqueness.

This text provides an introduction to a fascinating, intricate, and useful branch of mathematics. In addition to covering specific solution techniques that provide an insight into how PDE work, the text is a gateway to theoretical studies of PDE, involving the full power of real, complex and functional analysis. Often we will refer to applications to provide further intuition into specific equations and their solutions, as well as to show the modeling of real problems by PDE.

The study of PDE takes many forms. Very broadly, we take two approaches in this book. One approach is to describe methods of constructing solutions, leading to formulas. The second approach is more theoretical, involving aspects of analysis, such as the theory of distributions and the theory of function spaces.

1.1. Linear PDE

To introduce PDE, we begin with four linear equations. These equations are basic to the study of PDE, and are prototypes of classes of equations, each with different properties. The primary elementary methods of solution are related to the techniques we develop for these four equations.

For each of the four equations, we consider an unknown (real-valued) function u on an open set U [subset] Rn. We refer to u as the dependent variable, and x = (x1, x2, ..., xn) [member of] U as the vector of independent variables. A partial differential equation is an equation that involves x, u, and partial derivatives of u. Quite often, x represents only spatial variables. However, many equations are evolutionary, meaning that u = u(x, t) depends also on time t and the PDE has time derivatives. The order of a PDE is defined as the order of the highest derivative that appears in the equation.

The Linear Transport Equation:

ut + cux = 0. (1.1)

This simple first-order linear PDE describes the motion at constant speed c of a quantity u depending on a single spatial variable x and time t. Each solution is a traveling wave that moves with the speed c. If c > 0, the wave moves to the right; if c< 0, the wave moves left. The solutions are all given by a formula u(x, t) = f (x - ct). The function f = f ([xi]), depending on a single variable [xi] = x - ct, is determined from side conditions, such as boundary or initial conditions.

The next three equations are prototypical second-order linear PDE.

The Heat Equation:

ut = k Δ u. (1.2)

In this equation, u(x, t) is the temperature in a homogeneous heat-conducting material, the parameter k > 0 is constant, and the Laplacian Δ is defined by

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in Cartesian coordinates. The heat equation, also known as the diffusion equation, models diffusion in other contexts, such as the diffusion of a dye in a clear liquid. In such cases, u represents the concentration of the diffusing quantity.

The Wave Equation:

utt = c2 Δ u. (1.3)

As the name suggests, the wave equation models wave propagation. The parameter c is the wave speed. The dependent variable u = u(x, t) is a displacement, such as the displacement at each point of a guitar string as the string vibrates, if x [member of] R, or of a drum membrane, in which case x [member of] R2. The acceleration utt, being a second time derivative, gives the wave equation quite different properties from those of the heat equation.

Laplace's Equation:

Δ u = 0. (1.4)

Laplace's equation models equilibria or steady states in diffusion processes, in which u(x, t) is independent of time t, and appears in many other contexts, such as the motion of fluids, and the equilibrium distribution of heat.

These three second-order equations arise often in applications, so it is very useful to understand their properties. Moreover, their study turns out to be useful theoretically as well, since the three equations are prototypes of second-order linear equations, namely, elliptic, parabolic, and hyperbolic PDE.

1.2. Solutions; Initial and Boundary Conditions

A solution of a PDE such as any of (1.1)–(1.4) is a real-valued function u satisfying the equation. Often this means that u is as differentiable as the PDE requires, and the PDE is satisfied at each point of the domain of u. However, it can be appropriate or even necessary to consider a more general notion of solution, in which u is not required to have all the derivatives appearing in the equation, at least not in the usual sense of calculus. We will consider this kind of weak solution later (see Chapter 11).

As with ordinary differential equations (ODE), solutions of PDE are not unique; identifying a unique solution relies on side conditions, such as initial and boundary conditions. For...