Page 591 - Handbook Of Integral Equations

11.11-3. The Hopf–Fock Formula

Let us give a useful formula that allows one to express the solution of Eq. (1) with an arbitrary
right-hand side f(x) via the solution to a simpler auxiliary integral equation with an exponential
right-hand side.
Assume that in Eq. (1) we have

f(x)= e iζx , Im ζ >0, y(x)= y ζ (x), (32)

and moreover, conditions (15) and (16) hold. In this case

∞

y ζ (x)= e iζx + R(x, t)e iζt dt, (33)
0
where R(x, t) has the form (25). After some manipulations, we can see that

x
ˇ
y ζ (x)= M – (–ζ) 1+ R(t,0)e –iζt dt e iζx . (34)
0
On setting x = 0 in (34), we have
ˇ
y ζ (0) = M – (–ζ), (35)
and if the function K(x) describing the kernel of the integral equation is even, then

ˇ
y ζ (0) = M + (ζ). (36)
On the basis of formula (34), we can obtain the solution of Eq. (1) for a general f(x) as well
(see also Section 9.6):
1 ∞ ∞ iux
ˇ
ˇ
y(x)= F + (–ζ)y ζ (x) dζ, F + (u)= f(x)e dx. (37)
2π 0
–∞
Remark 1. All results obtained in Section 11.11 concerning Wiener–Hopf equations of the sec-
ond kind remain valid for continuous, square integrable, and some other classes of functions, which
are discussed in detail in the paper by M. G. Krein (1958) and in the book by C. Corduneanu (1973).
Remark 2. The solution of the Wiener–Hopf equation can be also obtained in other classes of
ˇ
functions for the exceptional case in which 1 – K(u) = 0 (see Subsections 11.9-1 and 11.10-5).
•
References for Section 11.11: V. A. Fock (1942), M. G. Krein (1958), C. Corduneanu (1973), V. I. Smirnov (1974),
P. P. Zabreyko, A. I. Koshelev, et al. (1975).

11.12. Methods for Solving Equations With Difference
Kernels on a Finite Interval

11.12-1. Krein’s Method
Consider a method for constructing exact analytic solutions of linear integral equations with an
arbitrary right-hand side. The method is based on the construction of two auxiliary solutions of
simpler equations with the right-hand side equal to 1. The auxiliary solutions are used to construct
a solution of the original equation for an arbitrary right-hand side.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 574