20

Partial differential equations:

general and particular solutions

In this chapter and the next the solution of differential equations of types

typically encountered in the physical sciences and engineering is extended to

situations involving more than one independent variable. A partial differential

equation (PDE) is an equation relating an unknown function (the dependent

variable) of two or more variables to its partial derivatives with respect to

those variables. The most commonly occurring independent variables are those

describing position and time, and so we will couch our discussion and examples

in notation appropriate to them.

As in other chapters we will focus our attention on the equations that arise

most often in physical situations. We will restrict our discussion, therefore, to

linear PDEs, i.e. those of first degree in the dependent variable. Furthermore, we

will discuss primarily second-order equations. The solution of first-order PDEs

will necessarily be involved in treating these, and some of the methods discussed

can be extended without difficulty to third- and higher-order equations. We shall

also see that many ideas developed for ordinary differential equations (ODEs)

can be carried over directly into the study of PDEs.

In this chapter we will concentrate on general solutions of PDEs in terms

of arbitrary functions and the particular solutions that may be derived from

them in the presence of boundary conditions. We also discuss the existence and

uniqueness of the solutions to PDEs under given boundary conditions.

In the next chapter the methods most commonly used in practice for obtaining

solutions to PDEs subject to given boundary conditions will be considered. These

methods include the separation of variables, integral transforms and Green’s

functions. This division of material is rather arbitrary and has been made only

to emphasise the general usefulness of the latter methods. In particular, it will

be readily apparent that some of the results of the present chapter are in

fact solutions in the form of separated variables, but arrived at by a different

approach.