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where ε > 0 is the regularization parameter. This equation is a Fredholm equation of the second
kind, so it can be solved by the methods presented in Chapter 11, whence the solution exists and is
unique.
On taking a sufficiently small ε in Eq. (2), we find a solution y ε (x) of the equation and substitute
this solution into Eq. (1), thus obtaining
b

K(x, t)y ε (t) dt = f ε (x), a ≤ x ≤ b. (3)
a
If the function f ε (x) thus obtained differs only slightly from f(x), that is,
f(x) – f ε (x) ≤ δ, (4)
where δ is a prescribed small positive number, then the solution y ε (x) is regarded as a sufficiently
good approximate solution of Eq. (1).
The parameter δ usually defines the error of the initial data provided that the right-hand side of
Eq. (1) is defined or determined by an experiment with some accuracy.
For the case in which, for a given ε, condition (4) fails, we must choose another value of the
regularization parameter and repeat the above procedure.
The next subsection describes the regularization method suitable for equations of the first kind
with arbitrary square-integrable kernels.

10.8-2. The Tikhonov Regularization Method
Consider the Fredholm integral equation of the first kind
b
K(x, t)y(t) dt = f(x), c ≤ x ≤ d. (5)
a
Assume that K(x, t) is any function square-integrable in the domain {a ≤ t ≤ b, c ≤ x ≤ d},
f(x) ∈ L 2 (c, d), and y(x) ∈ L 2 (a, b). The problem of finding the solution of Eq. (5) is also ill-posed
in the above sense.
Following the Tikhonov (zero-order) regularization method, along with (5) we consider the
following Fredholm integral equation of the second kind (see Chapter 11):
b
εy ε (x)+ K (x, t)y ε (t) dt = f (x), a ≤ x ≤ b, (6)
∗
∗
a
where
d d
∗
∗
∗
K (x, t)= K (t, x)= K(s, x)K(s, t) ds, f (x)= K(s, x)f(s) ds, (7)
c c
and the positive number ε is the regularization parameter. Equation (6) is said to be a regularized
integral equation, and its solution exists and is unique.
Taking a sufficiently small ε in Eq. (6), we find a solution y ε (x) of the equation and substitute
this solution into Eq. (5), thus obtaining
b

K(x, t)y ε (t) dt = f ε (x), c ≤ x ≤ d. (8)
a
By comparing the right-hand side with the given f(x) using formula (4), we either regard f ε (x)
as a satisfactory approximate solution obtained in accordance with the above simple algorithm, or
continue the procedure for a new value of the regularization parameter.
Presented above are the simplest principles of finding an approximate solution of the Fredholm
equation of the first kind. More perfect and complex algorithms can be found in the references cited
below.
•
References for Section 10.8: M. M. Lavrentiev (1967), A. N. Tikhonov and V. Ya. Arsenin (1979), M. M. Lavrentiev,
V. G. Romanov, and S. P. Shishatskii (1980), A. F. Verlan’ and V. S. Sizikov (1986).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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