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Chapter 1

Introduction to PDE

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.
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 first-order and 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.
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. The methods most commonly used in practice for obtaining
solutions to PDEs subject to given boundary conditions will be considered.
Most of the important PDEs of physics are second-order and linear. In order to gain
familiarity with their general form, some of the more important ones will now be briefly
discussed. These equations apply to a wide variety of different physical systems. Since,
in general, the PDEs listed below describe three-dimensional situations, the independent
variables are r and t, where r is the position vector and t is time. The actual variables
used to specify the position vector r are dictated by the coordinate system in use. For
example, in Cartesian coordinates the independent variables of position are x, y and z,
whereas in spherical polar coordinates they are r, θ and ϕ. The equations may be written
in a coordinate-independent manner, however, by the use of the Laplacian operator ∆ or
2
∇ .
Recall that an ordinary differential equation (ODE) contains an independent variable x
and a dependent variable y, which is the unknown in the equation. The defining property
of an ODE is that derivatives of the unknown function enter the equation. Thus, an

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