1 Definitions

We begin by giving some definitions.

Suppose that $u=u\left(x,t\right)$ satisfies the second order partial differential equation

$A{u}_{xx}+B{u}_{xt}+C{u}_{tt}+D{u}_x+E{u}_t+Fu=G$

in which $A,\dots ,G$ are given functions. This equation is said to be

$\begin{array}{ccc}\hfill \text{ parabolic}& \hfill & \hfill \text{if}{B}^2-4AC=0\\ \hfill \\ \hfill \text{ hyperbolic}& \hfill & \hfill \text{if}{B}^2-4AC>0\\ \hfill \\ \hfill \text{ elliptic}& \hfill & \hfill \text{if}{B}^2-4AC<0\\ \hfill \\ \hfill \end{array}$

These may look like rather abstract definitions at this stage, but we will see that equations of different types give rise to mathematical models of different physical situations. In this Section we will consider equations only of the parabolic type. The hyperbolic type is dealt with later in this Workbook and the elliptic type is discussed in HELM booklet  33.