4.6 Classification and Limitations 281
whereA,B,C,and so forth are, in general, functions ofξandη.(WeuseGreek
letters for the independent variables to avoid implying any relations to space
or time.) Such an equation can be classified according to the sign ofB^2 − 4 AC:
B^2 − 4 AC<0: elliptic,
B^2 − 4 AC=0: parabolic,
B^2 − 4 AC>0: hyperbolic.BecauseA,B,andCare functions ofξandη(not ofu), the classification of an
equation may vary from point to point. It is easy to see that the heat equation
is parabolic, the wave equation is hyperbolic, and the potential equation is
elliptic. The classification of an equation determines important features of the
solution and also dictates the method of attack when numerical techniques are
used for solution.
The question naturally arises whether separation of variables works on all
equations. The answer is no. For instance, the equation
(
ξ+η^2)∂^2 u
∂ξ^2 +∂^2 u
∂η^2 =^0does not admit separation of variables. In general, it is difficult to say just
which equations can be solved by this method. However, it is necessary to have
B≡0.
The region in which the solution is to be found also limits the applicability
ofthemethodwehaveused.Theregionmustbeageneralized rectangle.By
this we mean a region bounded by coordinate curves of the coordinate system
of the partial differential equation. Put another way, the region is described
by inequalities on the coordinates, whose endpoints are fixed quantities. For
instance, we have worked in regions described by the following sets of inequal-
ities:
0 <x<a, 0 <t,
0 <x, 0 <t,
−∞<x<∞, 0 <t,
0 <x<a, 0 <y<b,
0 <r<c, −π<θ≤π.All of these are generalized rectangles, but only one is an ordinary rectangle.
An L-shaped region is not a generalized rectangle, and our methods would
break down if applied to, for instance, the potential equation there.
There are, as we know, restrictions on the kinds of boundary conditions
that can be handled. From the examples in this chapter it is clear that we need