Second order partial differential equations in two variables – say
and
– take the form
where
are 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