Page 681 - Handbook Of Integral Equations

If y 0 (x) ≡ 0, then
x dt
y 1 (x)= = arctan x,
1+ t 2
0
2
x 1 + arctan t

3
y 2 (x)= dt = arctan x + 1 3 arctan x,
0 1+ t 2
x 1 + arctan t + arctan t
1 3
2
3
7
1
5
y 3 (x)= 3 dt = arctan x + 1 3 arctan x + 3⋅5 arctan x + 7⋅9 arctan x.
0 1+ t 2
On continuing this process, we can observe that y k (x) → tan(arctan x)= x as k →∞, i.e., y(x)= x. The substitution
of this result into the original equation shows the validity of the result.
Example 3. For the nonlinear equation
x
2
y(x)= [ty (t) – 1] dt
0
we must obtain the first three approximations. If we set y 0 (x) = 0, then
x
y 1 (x)= (–1) dt = –x,
0
x
3
4
y 2 (x)= (t – 1) dt = –x + 1 4 x ,
0

4
8
10
1
1
7

t
t + t
y 3 (x)= x 16 t – 1 5 2 – 1 dt = –x + 1 4 x – 14 1 x + 160 x .
2
0
◦
2 . The successive approximation method can be applied to solve other forms of nonlinear equations,
for instance, equations of the form
x
y(x)= F x, K(x, t)y(t) dt
a
solved for y(x) in which the integral has x as the upper integration limit. This makes it possible to
obtain a numerical solution by applying small steps with respect to x and by linearization at each
step, which usually provides the uniqueness of the result of the iterations for an arbitrary initial
approximation.
3 . The initial approximation substantially influences the number of iterations necessary to obtain
◦
the result with prescribed accuracy, and therefore when choosing this approximation, some additional
arguments are usually applied. Namely, for the equation
x

Ay(x) – Q(x – t)Φ y(t) dt = f(x),
0
where A is a constant, a good initial approximation y 0 (x) can sometimes be found from the solution
of the following (in general, transcendental) equation for ˜y 0 (p):
˜
˜

A ˜y 0 (p) – Q(p)Φ ˜y 0 (p) = f(p),
˜
˜
where ˜y 0 (p), Q(p), and f(p) are the transforms of the corresponding functions obtained by means of
the Laplace transform. If ˜y 0 (p)isdefined, then the initial approximation can be found by applying
–1
the Laplace inversion formula: y 0 (x)= L { ˜y 0 (p)}.
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 664