Page 26 - 6099

Chapter 3

Classification of second order

linear PDEs

Linear, second-order PDEs, as the examples shown above, are commonly encountered
in science and engineering applications. For that reason special attention is paid in this
lecture to these equations. We will consider the general second order linear PDE and will
reduce it to one of three distinct types of equations that have the wave, heat and Laplace’s
equations as their canonical forms. Knowing the type of the equation allows one to use
relevant methods for studying it, which are quite different depending on the type of the
equation. One should compare this to the conic sections, which arise as different types of
second order algebraic equations (quadrics). Since the hyperbola, given by the equation
2
2
2
x − y = 1, has very different properties from the parabola x − y = 0, it is expected
that the same holds true for the wave and heat equations as well. For conic sections, one
uses change of variables to reduce the general second order equation to a simpler form,
which are then classified according to the form of the reduced equation. We will see that
a similar procedure works for second order PDEs as well.
The general second order linear PDE has the following form

Au xx + 2Bu xy + Cu yy = F(x, y, u, u x , u y ), (3.1)

where the coefficients A, B and C are in general functions of the independent variables
x, y, but do not depend on the unknown function u. The classification of second order
equations depends on the form of the leading part of the equations consisting of the
second order terms.
Basically, the type of the above equation depends on the sign of the quantity
2
∆(x, y) = B (x, y) − A(x, y)C(x, y), (3.2)

which is called the discriminant for (3.1). The classification of second order linear PDEs
is given by the following.

Definition 3.0.1 At the point (x ; y ) the second order linear PDE
0
0
(3.1) is called

19