Page 550 - Handbook Of Integral Equations

11.3. Solution as a Power Series in the Parameter.
Method of Successive Approximations

11.3-1. Iterated Kernels
Consider the Fredholm integral equation of the second kind:

b
y(x) – λ K(x, t)y(t) dt = f(x), a ≤ x ≤ b. (1)
a
We seek the solution in the form of a series in powers of the parameter λ:
∞
n
y(x)= f(x)+ λ ψ n (x). (2)
n=1
Substitute series (2) into Eq. (1). On matching the coefficients of like powers of λ, we obtain a
recurrent system of equations for the functions ψ n (x). The solution of this system yields

b
ψ 1 (x)= K(x, t)f(t) dt,
a
b b
ψ 2 (x)= K(x, t)ψ 1 (t) dt = K 2 (x, t)f(t) dt,
a a
b b
ψ 3 (x)= K(x, t)ψ 2 (t) dt = K 3 (x, t)f(t) dt, etc.
a a
Here
b
K n (x, t)= K(x, z)K n–1 (z, t) dz, (3)
a
where n =2, 3, ... , and we have K 1 (x, t) ≡ K(x, t). The functions K n (x, t)defined by formulas (3)
are called iterated kernels. These kernels satisfy the relation

b
K n (x, t)= K m (x, s)K n–m (s, t) ds, (4)
a
where m is an arbitrary positive integer less than n.
The iterated kernels K n (x, t) can be directly expressed via K(x, t) by the formula

b b b
K n (x, t)= ··· K(x, s 1 )K(s 1 , s 2 ) ...K(s n–1 , t) ds 1 ds 2 ... ds n–1 .
a a a

n–1
All iterated kernels K n (x, t), beginning with K 2 (x, t), are continuous functions on the square
S = {a ≤ x ≤ b, a ≤ t ≤ b} if the original kernel K(x, t) is square integrable on S.
If K(x, t) is symmetric, then all iterated kernels K n (x, t) are also symmetric.

11.3-2. Method of Successive Approximations
The results of Subsection 11.3-1 can also be obtained by means of the method of successive
approximations. To this end, one should use the recurrent formula
b
y n (x)= f(x)+ λ K(x, t)y n–1 (t) dt, n =1, 2, ... ,
a
with the zeroth approximation y 0 (x)= f(x).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 533