Introduction

What is a partial differential equation?

It is tea-time, and you have decided to make some smoked salmon sandwiches. A slight inconvenience is that you have only just removed the loaf of bread from the freezer and need to let it defrost. How long is this going to take?

The way in which the temperature rises for any small region in the interior of the loaf depends on the rate at which heat is conducted into that region. This in turn depends on the temperature gradients within the loaf. Thus there is a relationship between the spatial and the time derivatives of the temperature T of the bread. This relationship is a partial differential equation in that it involves partial derivatives of T (with respect to x and with respect to t).

This is typical of partial differential equations. In contrast to ordinary differential equations, which have only one independent variable (see Chapter 5), here we consider differential equations which involve at least two independent variables. Because the dependent variable of our partial differential equation (T in the above example) is a function of these independent variables, the derivatives are necessarily partial ones. The solution of the equation then involves finding a specific functional dependence for, say, T in terms of position and time, which satisfies the particular requirements of the given problem.

Specific examples

Here we describe and derive some of the more common examples of partial differential equations.