Page 689 - Handbook Of Integral Equations

14.3-4. The Successive Approximation Method

Consider the nonlinear Urysohn integral equation in the canonical form:

b

y(x)= K x, t, y(t) dt, a ≤ x ≤ b. (31)
a
The iteration process for this equation is constructed by the formula

b

y k (x)= K x, t, y k–1 (t) dt, k =1, 2, ... . (32)
a

If the function K(x, t, y) is jointly continuous together with the derivative K (x, t, y) (with respect
y
to the variables x, t, and ρ, a ≤ x ≤ b, a ≤ t ≤ b, and |y|≤ ρ) and if
b b

sup |K(x, t, y)| dt ≤ ρ, sup |K (x, t, y)| dt ≤ β < 1, (33)

y
a y a y
then for any continuous function y 0 (x) of the initial approximation from the domain {|y|≤ρ, a≤x≤b},
∗
the successive approximations (32) converge to a continuous solution y (x), which lies in the same
domain and is unique in this domain. The rate of convergence is defined by the inequality
|y (x) – y k (x)|≤ β k sup |y 1 (x) – y 0 (x)|, a ≤ x ≤ b, (34)
∗
1 – β x

which gives an a priori estimate for the error of the kth approximation. The a posteriori estimate
(which is, in general, more precise) has the form

β
|y (x) – y k (x)|≤ sup |y k (x) – y k–1 (x)|, a ≤ x ≤ b. (35)
∗
1 – β x
A solution of an equation of the form (31) with an additional term f(x) on the right-hand side
can be constructed in a similar manner.
Example 2. Let us apply the successive approximation method to solve the equation
1
2
5
y(x)= xty (t) dt – 12 x +1.
0
The recurrent formula has the form
1

y k (x)= xty 2 k–1 (t) dt – 12 5 x +1, k =1, 2, ...
0
For the initial approximation we take y 0 (x) = 1. The calculation yields
y 1 (x) = 1 + 0.083 x, y 2 (x) = 1 + 0.14 x, y 3 (x) = 1 + 0.18 x, ...
y 8 (x) = 1 + 0.27 x, y 9 (x) = 1 + 0.26 x, y 10 (x) = 1 + 0.29 x, ...
y 16 (x) = 1 + 0.318 x, y 17 (x) = 1 + 0.321 x, y 18 (x) = 1 + 0.323 x, ...

Thus, the approximations tend to the exact solution y(x)=1 + 1 x. We see that the rate of convergence of the iteration
3
process is fairly small.
Note that in Subsection 14.3-5, the equation in question is solved by a more efficient method.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 672