Page 612 - Handbook Of Integral Equations

If condition (3) is satisfied, then the operator L given by formula (9) is simultaneously a right
regularizer of the operator K:

∞

KL[y(x)] ≡ y(x)+ K ∗ (x, t)y(t) dt, (10)
–∞
where the function K ∗ (x, t) satisfies the condition

∞ ∞
2
|K ∗ (x, t)| dx dt < ∞. (11)
–∞ –∞
11.20-3. The Regularization Method
Consider the equation of the form

K[y(x)] = f(x), –∞ < x < ∞, (12)

where the operator K is defined by (6).
There are several ways of regularizing this equation, i.e., of its reduction to a Fredholm equation.
First, this equation can be reduced to an equation with a Cauchy kernel. On regularizing the last
equation by a method presented in Section 13.4, we can achieve our aim. This approach can be
applied if we can find, for given functions K 1 (x), K 2 (x), M(x, t), and f(x), simple expressions
for their Fourier integrals. Otherwise it is natural to perform the regularization of Eq. (12) directly,
without passing to the inverse transforms.
A left regularization of Eq. (12) involves the application of the regularizer L constructed in the
previous subsection to both its sides:
LK[y(x)] = L[f(x)]. (13)

It follows from (8) that Eq. (13) is a Fredholm equation

∞
y(x)+ K(x, t)y(t) dt = L[f(x)]. (14)
–∞
Thus, Eq. (12) can be transformed by left regularization to a Fredholm equation with the same
unknown function y(x) and the known right-hand side L[f(x)]. Left regularization is known to
imply no loss of solutions: all solutions of the original equation (12) are solutions of the regularized
equation. However, in the general case, a solution of the regularized equation need not be a solution
of the original equation.
The right regularization consists in the substitution of the expression

y(x)= L[ω(x)] (15)
for the desired function into Eq. (12), where ω(x) is a new unknown function. We finally arrive at
the following integral equation:
KL[ω(x)] = f(x), (16)
which is a Fredholm equation as well by virtue of (10):

∞
KL[ω(x)] ≡ ω(x)+ K ∗ (x, t)ω(t) dt = f(x), –∞ < x < ∞. (17)
–∞

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 595