Page 464 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 30: Neumann s Series

This series for the solution of the Fredholm integral equation of the second kind is known as Notes
Neumann Series.

30.2 Successive Approximation for the Resolvent Kernel

Writing in full the expression for the function y(x), we have
x
x
y ( ) f ( ) 1 ( ) 2 2 ( ) 3 3 ( ) ... ...(10)
x
x
x
Making use of (4) (6a) and (8) for , , ,...... into (10) we get
1 2 3
b b b
x
t
x
x
f
t
t
x
x
t
f
t
t
f
y ( ) f ( ) K ( , ) ( )dt 2 K 2 ( , ) ( )dt 3 K 3 ( , ) ( )dt ...
a a a
x
x
x
t
t
t
f ( ) K 1 ( , ) 2 K 2 ( , ) 3 K 3 ( , ) ... f ( )dt
t
x
b
f
t
x
t
x
y ( ) f ( ) H ( , , ) ( ) dt ..(11)
x
a
where the resolvent Kernel H(x, t, ) is given by the series
t
x
H ( , , ) K 1 ( , ) K 2 ( , ) 2 K 3 ( , ) .... ...(12)
t
t
t
x
x
x
Equation (12) is now the power series known again as Neumann Series.
As discussed in unit 29, we see that the resolvent Kernel is still analytic function of but is no
longer an entire function of . Also the resolvent Kernel satisfies the integral equation
x
u
K
H ( , , ) K ( , ) H ( , u , ) (u , )du ...(13)
u
x
x
u
1 1 1
Now the solution (11) is the unique L -solution of the equation (2), as f(x) and K(x, t) are L -class
2 2
and it exists in the whole domain of C(a, b). We now show that if the homogeneous equation (1)
for = has a certain non-trivial solution then with the help of equation (13) we obtain
0
( ) K ( , ) ( )dt
t
t
x
x
0 0 0
t
z
x
K
t
z
H ( , , ) ( )dt 2 ( )dt H ( , , ) ( , )dz
t
t
x
0 0 0 0 0 0
t
t
z
z
x
x
t
t
H ( , , ) ( )dt 2 H ( , , )dz K ( , ) ( )dt
0 0 0 0 0 0
z
H ( , , ) ( )dt H ( , , )dz ( )
t
z
t
x
x
0 0 0 0 0 0
0
This shows that if equation (2) has a unique non-trivial solution of the form (12) then the non-
trivial solution of the homogeneous equation (1) is (x), vanishes almost everywhere.
0
The above analysis process the following theorem to each quadratically integrable Kernel
K(x, t) there corresponds a resolvent Kernel H(x, t, ) which is analytic function of , regular at
least inside the circle K 1 and represented these by the power series (12). Let the domain
of existence of the resolvent Kernel in the complex plane be H. Then if f(x) also belongs to the
class L , the unique quadratically integrable solution of Fredholm s equation (2) valid in H is
2
given by (11).
For the proof of this theorem please refer to the treatment in the unit 29.
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