Page 458 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 29: Fredholm Equations Solution by the Method of Successive Approximation

This is called Neumann Series. Substituting (2) into (1) we obtain Notes
1
x
f ( ) 1 ( ) 2 2 ( ) .... K ( , ) f ( ) 1 ( ) 2 2 ( ) .... du f ( ) ...(3)
x
x
x
x
x
u
u
u
0
Comparing the powers of on both sides we have
1
x
( ) K ( , ) ( )du 0
x
f
u
u
1
0
x
f
u
u
x
( ) K ( , ) ( )du
1
1 1 1
x
( ) K ( , ) ( )du K ( , ) K ( , u ) (u )du
u
u
x
u
f
x
u
2 1 1 1 1
0 0 0
1
K 2 ( , y 1 ) (u 1 )du
x
f
0
1 1
u
u
u
u
x
x
K ( , ) ( )du K ( , ) ( )du
f
3 2 3
0 0

1 1
u
K ( , ) ( )du K ( , )du (for n 1, 2, ...)
x
x
u
u
n n 1 n
0 0
In the above we have
x
u
u
K 2 ( , ) K ( , u 1 ) (u 1 , )du 1
x
K
x
K ( , ) K ( , u K (u , )du . ...(4)
x
u
u
)
3 1 2 1 1
........................................................
and so on.
More generally
u
x
x
K ( , ) K ( , u )K (u , )du [n = 2, 3, 4, ...; r = 1, 2,..., n 1; K = K ...(5)
u
n
1
r
n
r
1
1
1
Thus the series for the resolvent Kernel H(x, u, ) is given by
u
x
u
x
u
x
x
u
H ( , , ) K ( , ) K 2 ( , ) 2 K 3 ( , ) ... n K n ( , ) ...(6)
x
u
The solution then is given by
f
x
x
u
y ( ) f ( ) H ( , u , ) ( )du ...(7)
x
The main difference from the Volterra case is that the series for the resolvent Kernel (6) now
converges only for sufficiently small values of | |. In other words, although H(x, u, ) is still
analytic function of it is no longer an entire function of .
29.2 Lower Bound for the Radius of Convergence
We shall now determine a lower bound for the radius of convergence of the power series (6).
We observe that if we preserve the basic hypothesis i.e. that the Kernel K(x, y) is an L Kernel,
2
u
x
u
x
i.e. K 2 K 2 ( , )dxdu A 2 ( )dx B 2 ( )du N 2 ...(8)
where
1 2 1 2
1 1
B
u
x
x
x
u
A ( ) K 2 ( , )du , ( ) K 2 ( , )dx ...(9)
u
0 0
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