In the linear PDE of second order in two variables,

if is formally replaced by , by , by , by and by , then associated with equation (7) is a polynomial of degree two in and

The mathematical nature of the solutions of equation (7) are
largely determined by the algebraic properties of the
polynomial . In turn, the computational strategy that
one selects to numerically solve (7) is strongly
influenced by the mathematical nature of the solution. Thus,
before embarking on a quantitative analysis of a partial
differential equation of the form (7), it is important
that a computational scientist have an idea of the
qualitative nature of the solution. Much of this qualitative
understanding of the solution can be obtained via the
following classification scheme. P(a,b) and along with it,
the PDE (7) is classified as hyperbolic, parabolic, or
elliptic according as its discriminant, , is
positive, zero, or negative. Note that the type of equation
(7) is determined solely by its principal part (the
terms involving the highest-order derivatives of **u**) and that
the type will generally change with position in the **xy**-plane
unless **a**,**b**, and **c** are constants.