Page 693 - Handbook Of Integral Equations

where f(x) ∈ L 2 (c, d) and y(t) ∈ L 2 (a, b), the Tikhonov regularization method leads to a regularized
nonlinear integral equation in the form

b

αy α (x)+ M t, x, y α (t), y α (x) dt = F x, y α (x) , a ≤ x ≤ b, (46)
a
d

M t, x, y(t), y(x) = K s, t, y(t) K s, x, y(x) ds, (47)

y
c
d

F x, y(x) = K t, x, y(x) f(t) dt, (48)

y
c
where α is a regularization parameter.
For instance, by applying the quadrature method on the basis of the trapezoidal rule, we can
reduce Eq. (46) to a system of nonlinear algebraic equations. An approximate solution of (45) is
constructed by the principle described above for linear equations (see Section 10.8).
•
References for Section 14.3: N. S. Smirnov (1951), P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. G. Tricomi (1985),
A. F. Verlan’ and V. S. Sizikov (1986).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 676