Page 680 - Handbook Of Integral Equations

◦
4 . Equations of the form
x

y(x)+ cosh λ(x – t) f t, y(t) dt = g(x), (9)
a
x

y(x)+ sinh λ(x – t) f t, y(t) dt = g(x), (10)
a
x

y(x)+ cos λ(x – t) f t, y(t) dt = g(x), (11)
a
x

y(x)+ sin λ(x – t) f t, y(t) dt = g(x) (12)
a
can also be reduced to second-order ordinary differential equations by double differentiation. For
these equations, see Section 5.8 in the first part of the book (Eqs. 22, 23, 24, and 25, respectively).
14.2-3. The Successive Approximation Method
◦
1 . In many cases, the successive approximation method can be successfully applied to solve various
types of integral equations. The principles of constructing the iteration process are the same as in
the case of linear equations. For Volterra equations of the second kind in the Urysohn form

x

y(x) – K x, t, y(t) dt = f(x), a ≤ x ≤ b, (13)
a
the corresponding recurrent expression has the form
x

y k+1 (x)= f(x)+ K x, t, y k (t) dt, k = 0,1,2, ... (14)
a
It is customary to take the initial approximation either in the form y 0 (x)≡0 or in the form y 0 (x)=f(x).
In contrast to the case of linear equations, the successive approximation method has a smaller
domain of convergence. Let us present the convergence conditions for the iteration process (14) that
are simultaneously the existence conditions for a solution of Eq. (13). To be definite, we assume
that y 0 (x)= f(x).
If for any z 1 and z 2 we have the relation
|K(x, t, z 1 ) – K(x, t, z 2 )|≤ ϕ(x, t)|z 1 – z 2 |

and the relation
x

K x, t, f(t) dt ≤ ψ(x)

a

holds, where
x b x

2
2
2
2
ψ (t) dt ≤ N , ϕ (x, t) dt dx ≤ M ,
a a a
for some constants N and M, then the successive approximations converge to a unique solution of
Eq. (13) almost everywhere absolutely and uniformly.
Example 2. Let us apply the successive approximation method to solve the equation
x 1+ y (t)
2

y(x)= dt.
0 1+ t 2
© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 663