# Chapter 1

# First Ideas

We will begin a study of partial differential equations by deriving equations modeling diffusion processes and wave motion. These are widely applicable in the physical and life sciences, engineering, economics, and other areas. Following this, we will lay the foundations for the Fourier method, which is used to write solutions for many kinds of problems, and then solve two eigenvalue/eigenfunction problems that occur frequently when this method is used.

The chapter concludes with a proof of a theorem on the convergence of Fourier series.

# 1.1 Two Partial Differential Equations

## 1.1.1 The Heat, or Diffusion, Equation

We will derive a partial differential equation modeling heat flow in a medium. Although we will speak in terms of heat flow because it is familiar to us, the heat equation applies to general diffusion processes, which might be a flow of energy, a dispersion of insect or bacterial populations in controlled environments, changes in the concentration of a chemical dissolving in a fluid, or many other phenomena of interest. For this reason the heat equation is also called the diffusion equation.

Consider a bar of material of constant density, ρ, having uniform cross sections with area *A.* The lateral surface of the bar is insulated, so there is no heat loss across this surface.

Place an *x*-axis along the length, *L*, of the bar and assume that at a given time, the temperature is the same along any cross section perpendicular to this axis, although it may ...