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Even from casual observation it is apparent that most physical phenomena vary both in space and time. For example, the temperature of the atmosphere changes continuously at any given location and it varies significantly from point to point over the surface of the Earth. A consequence of this is that mathematical models of the real world almost inevitably involve both time and space derivatives. The objective of this book is to examine how to solve such problems using a computer; but to begin, we first consider more simplified situations. In this chapter we study problems involving only time derivatives and then in the next chapter we examine spatial problems. The remaining chapters then examine what happens when both time and space derivatives are present together in the problem.

In this chapter we investigate how to find the numerical solution of what are called two-point boundary value problems (BVPs). The most apparent difference between these problems and the IVPs studied in the previous chapter is that BVPs involve only spatial derivatives. What this means is that we consider how to solve a differential equation in an interval 0 < x < ℓ, where the solution is required to satisfy conditions at the two endpoints x = 0, ℓ. Examples of such problems are below.

To begin the study of finding numerical solutions of partial differential equations we begin with diffusion problems. In physical terms these are problems that involve motion or transport of particles (ions, molecules, etc.) from areas of higher concentration to areas of lower concentration. Simple examples are the spread of a drop of ink dropped into water and the melting of an ice cube. Diffusion is also a key component in the formation of dendrites when liquid metal cools, as well as in the chemical signals responsible for pattern formation (Figure 3.1). Other interesting applications of diffusion arise in the study of financial assets as expressed by the Black-Scholes theory for options pricing and in the spread of infectious diseases ([2002], [2001]).

We now begin the study of numerical wave propagation. Everyone has experience with traveling waves, whether it is waves on a lake, sound waves, or perhaps an earthquake or two. It is not particularly difficult to write down a reasonable-looking finite difference approximation to a wave equation. Most of the effort is invested in trying to determine whether the method actually works. There are unique complications for numerical wave propagation, and so to introduce the ideas we use one of the simplest mathematical equations that produces traveling waves. This is the advection equation, given as (4.1) $$ \frac{{\partial u}} {{\partial t}} + a\frac{{\partial u}} {{\partial x}} = 0,{\text{ }}for\left\{ \begin{gathered} - \infty < x < \infty , \hfill \\ 0 < t, \hfill \\ \end{gathered} \right. $$ where u ( x , 0) = g(x) . It is assumed that a is a positive constant.

In studying phenomena in such diverse areas as electrodynamics, fluid dynamics, and acoustics, it is almost inevitable to come across what is known as the wave equation. This ubiquitous equation is a prototype for many of the waves seen in nature, and it is the subject of this chapter. The specific problem we start with is the wave equation (5.1) $$ c^2 \frac{{\partial ^2 u}} {{\partial x^2 }} = \frac{{\partial ^2 u}} {{\partial t^2 }} = for\left\{ \begin{gathered} 0 < x < \ell , \hfill \\ 0 < t, \hfill \\ \end{gathered} \right. $$ where c is a positive constant. The boundary conditions are (5.2) $$ u(0,t) = u(\ell ,t) = 0, $$ and the initial conditions are (5.3) $$ u(x,0) = f(x),{\text{ }}u_t (x,0) = g(x). $$

One might title this chapter “The Challenges of Dimensionality,” or perhaps “Why One-Dimensional Models Aren’t So Bad After All.” The reason is that we address how to solve boundary value problems with more than one spatial variable, and this will require us to consider some unique challenges. To introduce the ideas we will limit the development to two dimensions and consider how to find the function u(x, y) that satisfies (6.1) $$ \nabla \cdot (a\nabla u) + b \cdot \nabla u + cu = f,{\text{ }}for{\text{ }}(x,y) \in D, $$ where (6.2) $$ \nabla \equiv (\frac{\partial } {{\partial x}},\frac{\partial } {{\partial y}}) $$ is the gradient and D is a bounded domain in the xy -plane, as indicated in Figure 6.1. Also, the functions a, b, c, f are smooth with a > 0 and c ≤ 0 on $$ \bar D = D \cup \partial D $$ , where ∂ D is the boundary of D . We will use a Dirichlet boundary condition, which means that the solution is specified around the boundary, and the general form is (6.3) $$ u = g(x,y),{\text{ }}for{\text{ }}(x,y) \in \partial D, $$ where g is given. The particular case of a = 1, b = 0 , c = 0 produces Poisson’s equation. If, in addition, f = 0, one obtains Laplace’s equation given as (6.4) $$ \nabla ^2 u = 0,{\text{ }}for{\text{ }}(x,y) \in D, $$ where (6.5) $$ \nabla ^2 \equiv \frac{{\partial ^2 }} {{\partial x^2 }} + \frac{{\partial ^2 }} {{\partial y^2 }} $$ is the Laplacian