Page 176 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 10: Green’s Function Method

Notes
d dz dy
x
x
= p ( ) y ( ) z ...(2)
dx dx dx
Integrating both sides of equation (2) we obtain
b b
dz dy
x
x
z
x
y
p ( ) y ( ) z ( ) [yL x ( ) ZL x ( )]dx a a b b ...(3)
,
dx dx
a a
Equation (3) is known as Green’s theorem in one dimension. If y(x) and z(x) both satisfy the
boundary conditions
p(a) y,(a) sin y(a) cos = 0
p(b) y,(b) sin y(b) cos = 0
p(a) z,(a) sin z(a) cos = 0
p(b) y,(b) sin z(b) cos = 0 ...(4)
Then for a, = a and b, = b, L.H.S. is zero and we get

b
x
x
x
L
y
[ ( )L x ( ) z ( ) ( )]dx 0 ...(5)
y
x
a
x
x
Suppose that two functions y 1 ( ) 0 and y 2 ( ) 0 satisfy
L (y ) = 0
x 1
p(a) y ,(a) sin y (a) cos = 0 ...(6)
1 1
and
L (y ) = 0
x 2
p(b) y , (b) sin y (b) cos = 0 ...(7)
2 2
respectively, and suppose that these two functions y (x) and y (x) are linearly independent.
1 2
Write
C = p( ) [y ( ) y, ( ) y, ( ) y ( )].
1 2 1 2
Differentiating C with respect to and making use of (2), we see, by virtue of (6) and (7), that C
must be constant. Moreover, the linear independence of y (x) and y (x) implies that C is not zero.
1 2
Now we define a function G(x, ) of two variables x and by
1
y
x
G(x, ) = y 1 ( ) ( ) (x )
2
C
1
y
= y 1 ( ) ( ) (x )
x
2
C
y
C = ( )p y 1 ( ) ( ) y 1 ( ) ( ) Constant
y
2
2
The function G(x, ) is called Green’s Function for the equation L (y) = 0 subject to the boundary
x
conditions (4). Obviously Green function G(x, ) has the following properties:
G(x, ) is continuous at any point (x, ) in the domain a , x b.
As a function of x, G(x, ) satisfies the given boundary conditions for every . ...(9)
If x , G(x, ) satisfies the equation L (G) = 0 as a function of x.
x
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