Page 682 - Handbook Of Integral Equations

14.2-4. The Newton–Kantorovich Method

A merit of the iteration methods when applied to Volterra linear equations of the second kind
is their unconditional convergence under weak restrictions on the kernel and the right-hand side.
When solving nonlinear equations, the applicability domain of the method of simple iterations is
smaller, and if the process is still convergent, then, in many cases, the rate of convergence can be
very low. An effective method that makes it possible to overcome the indicated complications is the
Newton–Kantorovich method. The main objective of this method is the solution of nonlinear integral
equations of the second kind with constant limits of integration. Nevertheless, this method is useful
in the solution of many problems for the Volterra equations and makes it possible to significantly
increase the rate of convergence compared with the successive approximation method.
Let us apply the Newton–Kantorovich method to solve a Volterra equation of the second kind
in the Urysohn form
x

y(x)= f(x)+ K x, t, y(t) dt. (15)
a
We obtain the following iteration process:

y k (x)= y k–1 (x)+ ϕ k–1 (x), k =1, 2, ... , (16)
x

ϕ k–1 (x)= ε k–1 (x)+ K x, t, y k–1 (t) ϕ k–1 (t) dt, (17)
y
a
x

ε k–1 (x)= f(x)+ K x, t, y k–1 (t) dt – y k–1 (x). (18)
a
The algorithm is based on the solution of the linear integral equation (17) for the correc-
tion ϕ k–1 (x) with the kernel and right-hand side that vary from step to step. This process has a high
rate of convergence, but it is rather complicated because we must solve a new equation at each step
of iteration. To simplify the problem, we can replace Eq. (17) by the equation
x

ϕ k–1 (x)= ε k–1 (x)+ K x, t, y 0 (t) ϕ k–1 (t) dt (19)
y
a
or by the equation
x

ϕ k–1 (x)= ε k–1 (x)+ K x, t, y m (t) ϕ k–1 (t) dt, (20)
y
a
whose kernels do not vary. In Eq. (20), m is fixed and satisfies the condition m < k – 1.
It is reasonable to apply Eq. (19) with an appropriately chosen initial approximation. Otherwise
we can stop at some mth approximation and, beginning with this approximation, apply the simplified
equation (20). The iteration process thus obtained is the modified Newton–Kantorovich method. In
principle, it converges somewhat slower than the original process (16)–(18); however, it is not so
cumbersome in the calculations.
Example 4. Let us apply the Newton–Kantorovich method to solve the equation
x
2
y(x)= [ty (t) – 1] dt.
0
The derivative of the integrand with respect to y has the form

K y t, y(t) =2ty(t).
For the zero approximation we take y 0 (x) ≡ 0. According to (17) and (18) we obtain ϕ 0 (x)= –x and y 1 (x)= –x. Furthermore,
y 2 (x)= y 1 (x)+ ϕ 1 (x). By (18) we have
x

4
2
ε 1 (x)= [t(–t) – 1] dt + x = 1 4 x .
0
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 665