Page 184 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 10: Green’s Function Method

Here w function is called Green’s function for the problem. Since also Lz = f, we find that Notes
w z
[z w] = wz (ady bdx ) z dy w dx w f dxdy ...(9)
y x
AB AB

dz
Equation (7) enables us to find the value of z at the point P when is prescribed along the
dx
dz
curve C. When is prescribed, we make use of the following calculation
dx

(zw ) (zw )
[z w] [z w] = dx dy
B A x y
AB
to show that we can write equation (7) in the form

w
z
( ) ( )
z
[ ] P [zw ] B wz (a dy b dx ) z dx w dy (wf )dx dy
x y ...(10)
AB AB
Finally adding (9) and (10), we obtain the symmetrical results
1 1 z z
[z] = [zw ] [zw ] wz (a dy b dx ) w dy dx
P A B
2 2 y x
AB
1 w w
z dx dy ( wf dxdy
)
2 x y …(11)
AB
z z
So we can find z at any point in terms of prescribed values of ,z , , along a given curve.
x y

Self Assessment

3. If L denotes the operator
2 2 2
R S T P Q Z
x 2 x y y 2 x y
and M is the adjoint operator defined by

2 (Rw ) 2 (Sw ) 2 (Tw ) (Pw ) (Qw )
Mw 2 2 zw
x x y y x y
show that

n
(wLZ ZMw )dx dy U cos( , ) V cos( , ) ds
y
n
x
where is a closed curve enclosing an area and
z (Rw ) (Sw )
U Rw z z Pzw
x x y
z z (Tw )
V Sw Tw z Qzw .
x y y

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