and
–
take the form
where
are
all functions of
andand
University Maths Notes: Calculus – Classifying Second Order Linear Partial Differential Equations in Two Variables
Second order partial differential equations intwo variables –
sayand–
take the form
whereare
all functions of
andand
Ifare
constant then the equation is constant coefficient.
If G=0 then the equation is homogeneous and if G neq 0 the equation is non – constant coefficient.
If non ofare
functions of u or any partial derivative of u, then the equation is
linear.
Equations of the form (1) may also be classified as parabolic, hyperbolic or elliptic.
Parabolic equations describe heat flow and diffusion processes and
satisfy
e.g
Hyperbolic equations describe vibrating systems and wave phenomena
and satisfy
e.g.
Elliptic equations describe steady state phenomena and
satisfy
e.g.
A function may be parabolic, hyperbolic or elliptic in different
parts of the
plane.
For example
has
so
is elliptic for
parabolic
for
and
parabolic for
On
the other hand
is
hyperbolic everywhere since