Page 352 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 21: Classifications of Second Order Partial Differential Equations

As a less trivial example, consider the hyperbolic equation Notes
4
sech x = 0 ...(9)
xx yy
Equation (7) shows that the characteristics are given by
2
2
2
4
dy 2 sech x dx = (dy + sech x dx) (dy sech x dx) = 0

and hence
dy 2
sech x
dx
The characteristics are therefore
y tanb x = constant,
and the characteristic co-ordinates are

= y + tanb x, = y tanb x. On writing (9) in terms of these variables with = (x, y) = ( , ), we
find that its canonical form is

(  )(    )
= 2 ...(10)
[4 (  ) ]
2
in the domain ( ) < 4.

Parabolic Equation ac = b 2

In this case
A = a 2 x 2b  c 2 y (p x q y ) 2
x y
2 2 2
C = a x 2b  c y (p x q y )
x y
so we can construct one set of characteristic curves. We therefore take to be constant on the
2
curves pdy qdx = 0. This gives us A = 0 and since AC + B , B = 0. For any set of curves where is
constant that is never parallel to the characteristics, C does not vanish, and the canonical form is
+ b ( , ) + b( , ) + b ( , ) = g( , ) ...(11)
1 3
We can now see that the diffusion equation is in canonical form.
As a further example, consider the parabolic equation

2
 + 2cosec y  + cosec y  = 0 ...(12)
xx xy yy
The characteristic curves satisfy
2
2
2
2
dy 2 cosec y dxdy + cosec y dx = (dy cosec dx) = 0,
and hence
dy
cosec y
dx
The characteristic curves are therefore given by x + cos y = constant, and we can take = x +
cos y as the characteristic. A suitable choice for the other co-ordinate is = y. On writing (12) in
terms of these variables, with (x, y) = ( , ), we find that its canonical form is

2
= sin cos ...(13)
in the whole ( , ) plane.

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