One method of solving Fredholm equations. Say you have a Fredholm integral equation of the second kind:

a
f(x) = g(x) + ∫ K(x,t) f(t) dt
b

To solve it, we take the limit of

n i
f (x) = ∑ λ u (x)
n i=0 i

as n approaches ∞, where u_{i}(x) is:

u (x) = g(x)
0
b
u (x) = ∫ K(x,t )g(t ) dt
1 a 1 1 1
b b
u (x) = ∫ ∫ K(x,t )K(t , t ) g(t ) dt dt
2 a a 1 1 2 2 2 1
b b b
u (x) = ∫ &int ... ∫ K(x,t )K(t , t ) ...
n a a a 1 1 2
K(t ,t ) g(t ) dt dt ... dt
n-1 n n n n-1 1