Page 504 - Handbook Of Integral Equations

9.9. The Successive Approximation Method

9.9-1. The General Scheme

1 . Consider a Volterra integral equation of the second kind
◦
x
y(x) – K(x, t)y(t) dt = f(x). (1)
a
Assume that f(x) is continuous on the interval [a, b] and the kernel K(x, t) is continuous for a ≤ x ≤ b
and a ≤ t ≤ x.
Let us seek the solution by the successive approximation method. To this end, we set

∞

y(x)= f(x)+ ϕ n (x), (2)
n=1

where the ϕ n (x) are determined by the formulas
x

ϕ 1 (x)= K(x, t)f(t) dt,
a
x x

ϕ 2 (x)= K(x, t)ϕ 1 (t) dt = K 2 (x, t)f(t) dt,
a a
x x
ϕ 3 (x)= K(x, t)ϕ 2 (t) dt = K 3 (x, t)f(t) dt, etc.
a a
Here
x
K n (x, t)= K(x, z)K n–1 (z, t) dz, (3)
a
where n =2, 3, ... , and we have the relations K 1 (x, t) ≡ K(x, t) and K n (x, t)=0 for t > x.
The functions K n (x, t) given by formulas (3) are called iterated kernels. These kernels satisfy the
relation
x

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.

2 . The successive approximations can be implemented in a more general scheme:
◦
x
y n (x)= f(x)+ K(x, t)y n–1 (t) dt, n =1, 2, ... , (5)
a
where the function y 0 (x) is continuous on the interval [a, b]. The functions y 1 (x), y 2 (x), ... which
are obtained from (5) are also continuous on [a, b].
Under the assumptions adopted in item 1 for f(x) and K(x, t), the sequence {y n (x)} converges,
◦
as n →∞, to the continuous solution y(x) of the integral equation. A successful choice of the
“zeroth” approximation y 0 (x) can result in a rapid convergence of the procedure.
Note that in the special case y 0 (x)= f(x), this method becomes that described in item 1 .
◦
Remark 1. If the kernel K(x, t) is square integrable on the square S = {a ≤ x ≤ b, a ≤ t ≤ b}
and f(x) ∈ L 2 (a, b), then the successive approximations are mean-square convergent to the solution
y(x) ∈ L 2 (a, b) of the integral equation (1) for any initial approximation y 0 (x) ∈ L 2 (a, b).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 486