Fredholm theory

23.6 FREDHOLM THEORY
common ratio λ/3. Thus, provided |λ| < 3, this infinite series converges to the value
λ/(3 − λ), and the solution to (23.39) is
3x
λx
=
.
(23.40)
3−λ
3−λ
Finally, we note that the requirement that |λ| < 3 may also be derived very easily from
the condition (23.38). y(x) = x +
23.6 Fredholm theory
In the previous section, we found that a solution to the integral equation (23.34)
can be obtained as a Neumann series of the form (23.36), where the resolvent
kernel R(x, z; λ) is written as an infinite power series in λ. This solution is valid
provided the infinite series converges.
A related, but more elegant, approach to the solution of integral equations
using infinite series was found by Fredholm. We will not reproduce Fredholm’s
analysis here, but merely state the results we need. Essentially, Fredholm theory
provides a formula for the resolvent kernel R(x, z; λ) in (23.36) in terms of the
ratio of two infinite series:
D(x, z; λ)
.
(23.41)
R(x, z; λ) =
d(λ)
The numerator and denominator in (23.41) are given by
D(x, z; λ) =
d(λ) =
∞
(−1)n
n=0
∞
n=0
n!
Dn (x, z)λn ,
(−1)n
dn λn ,
n!
(23.42)
(23.43)
where the functions Dn (x, z) and the constants dn are found from recurrence
relations as follows. We start with
D0 (x, z) = K(x, z)
and
d0 = 1,
(23.44)
where K(x, z) is the kernel of the original integral equation (23.34). The higherorder coefficients of λ in (23.43) and (23.42) are then obtained from the two
recurrence relations
b
Dn−1 (x, x) dx,
(23.45)
dn =
a
Dn (x, z) = K(x, z)dn − n
b
K(x, z1 )Dn−1 (z1 , z) dz1 .
(23.46)
a
Although the formulae for the resolvent kernel appear complicated, they are
often simple to apply. Moreover, for the Fredholm solution the power series
(23.42) and (23.43) are both guaranteed to converge for all values of λ, unlike
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