Page 350 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 21: Classifications of Second Order Partial Differential Equations

It will be seen that the first three terms of equation (1) allow us to classify the equation into one Notes
of three distinct types: Elliptic, for example Laplace s equation, Parabolic, for example the diffusion
equation or Hyperbolic, for example the wave equation as follows:
2 2
V V
0 (Laplace equations for two variables x, y)
x  2 y  2
2 V V
K 2 (Diffusion equation)
x  t 
2 2
V 1 V
2 2 2 (Wave equation)
x  C t 
Each of these types of equation has distinctive properties. We would like to know about those
properties of equation (1) that are unchanged by any change of co-ordinates since these must be
of fundamental significance and not just a result of our choice of co-ordinate system. We can
write this change of co-ordinates as
(x, y) { (x, y), (x, y)}
with



( , )
0 ...(2)
y
x
 ( , )
If equation represents a model physical system, a change of co-ordinates should not affect its
qualitative behaviour. Writing (x, y)  ( , ) and using subscripts to denote partial derivatives,

we find that
2
2
     ,    2        
x x  x  xx x  x x  x x  xx  xx 
and similarly for the other derivatives. Substituting these into equation (1) gives us






Α  2B  +C  b 1 ( , )  b 2 ( , )  + ( , ) g ( , ) ...(3)

b

3
where
A a 2 x 2b  c 2 y ,
x x
B a  ( b     ) c 
x x
y y
y x
x y
....(4)
C a 2 2b  c 2 ,
x x y y
We do not need to consider other co-efficient functions b , ( , ), b ( , ), b ( , ).
1 2 3
We can express (4) in a concise matrix form as
A B  x  x a b  x  y
...(5)
B C  y  y b c  x  y
which shows that
2
A B a b ( , )


det det ...(6)
y
B C b c ( , )
x

( , )

In (6) = Jacobian of transformation.
( , )
x
y
2
This shows that the sign of a c b is independent of the choice of co-ordinate system which
allows us to classify the equation.
2
An Elliptic equation has ac < b , for example Laplace equation
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