Page 687 - Handbook Of Integral Equations

where the parameters λ k are determined from the system of algebraic (or transcendental) equations:

"
λ k – H k (λ)=0, k =1, ... , n,
b (13)
"

H k (λ)= h k t, y(t) dt, "
λ = {λ 1 , ... , λ n }.
a
"
Into system (13), we must substitute the function y(x)= y(x, λ), which can be obtained by solving
Eq. (12).
The number of solutions of the integral equation is defined by the number of solutions obtained
from (12) and (13). It can occur that there is no solution.
Example 1. Let us solve the integral equation
1
3
y(x)= λ xty (t) dt (14)
0
with parameter λ. We write
1
3
A = ty (t) dt. (15)
0
In this case, it follows from (14) that
y(x)= λAx. (16)
On substituting y(x) in the form (16) into relation (15), we obtain
1
3
3 3
A = tλ A t dt.
0
Hence,
3
3
A = 1 λ A . (17)
5
For λ > 0, Eq. (17) has three solutions:
5 5
1/2 1/2
A 1 =0, A 2 = , A 3 = – .
λ 3 λ 3
Hence, the integral equation (14) also has three solutions for any λ >0:
5 1/2 5 1/2
y 1 (x) ≡ 0, y 2 (x)= x, y 3 (x)= – x.
λ 3 λ 3
For λ ≤ 0, Eq. (17) has only the trivial solution y(x) ≡ 0.
14.3-2. The Method of Integral Transforms
◦
1 . Consider the following nonlinear integral equation with quadratic nonlinearity on a semi-axis:
∞ 1
x

µy(x) – λ y y(t) dt = f(x). (18)
t t
0
To solve this equation, the Mellin transform can be applied, which, with regard to the convolution
theorem (see Section 7.3), leads to a quadratic equation for the transform ˆy(s)= M{y(x)}:
2
ˆ
µ ˆy(s) – λ ˆy (s)= f(s).
This implies

ˆ
2
µ ± µ – 4λf(s)
ˆ y(s)= . (19)
2λ
–1
The inverse transform y(x)= M { ˆy(s)} obtained by means of the Mellin inversion formula (if it
exists) is a solution of Eq. (18). To different signs in the formula for the images (19), there are two
corresponding solutions of the original equation.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 670