Page 353 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Elliptic Equations: ac > b 2

In this case we can make neither A nor C zero, since no real characteristic curves exist. Instead we
can simplify by making A = C and B = 0, so that the second derivative form the Laplacian 2 and
the canonical form is
+ + b ( , ) + b ( , ) + b = g( , ) ...(14)
1 2 3
Clearly Laplace s equation is in canonical form
In order to proceed, we must solve

b
A C = (a  2 x  2 y ) 2 (    ) c ( 2 y  2 y ) 0
x y
x y
B = a  ( b     ) c  0.
x x
y y
x y
x y
We can do this by defining x = + i , and noting that these two equations form the real and
imaginary parts of
2 2
a x 2b  c x 0
x y
and hence
 x b ac b 2
 y a ...(15)

Now is constant on curves given by dy + dx = 0, and hence from (15) on
y x
dy b ac b 2
...(16)
dx a
By solving (16) we can deduce , . For example consider elliptic equation
4
+ sech x = 0 ...(17)
xx yy
In this case = + i is constant on the curves given by
dy 2
i sech x ,
dx
and hence y i tanb x = constant. We can therefore take = y + i tanb x, and hence = y, = tanb
x. On writing (17) in terms of these variables, with (x, y) = ( , ), we find that the canonical
form is
2n
      , ...(18)
(1  2 )
in the domain | | < 1.

21.3 Classification of Second order Partial Differential Equations

Let us consider a function z of two independent variables x and y. Writing various partial
derivatives as

z z 2 z 2 z 2 z
p , q , r 2 , s , t 2 ...(1)
x y x xdy y

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