Page 546 - Handbook Of Integral Equations

11.1-4. Dual Integral Equations of the Second Kind

A dual integral equation of the second kind with difference kernels (of convolution type) has the
form

∞
y(x)+ K 1 (x – t)y(t) dt = f(x), 0 < x < ∞,
–∞
∞
(5)
y(x)+ K 2 (x – t)y(t) dt = f(x), –∞ < x <0,
–∞
where the notation and the class of the functions and kernels coincide with those introduced for the
equations of convolution type in Subsection 10.1-3.
In a sufficiently general case, a dual integral equation of the second kind has the form

∞
y(x)+ K 1 (x, t)y(t) dt = f 1 (x), a < x < b,
a
(6)
∞
y(x)+ K 2 (x, t)y(t) dt = f 2 (x), b < x < ∞,
a
where f 1 (x) and f 2 (x) (and K 1 (x, t) and K 2 (x, t)) are the known right-hand sides (and the kernels)
of Eq. (6) and y(x) is the function to be found. These equations can be studied by the methods
of various integral transforms with reduction to boundary value problems of the theory of analytic
functions and also by other methods developed for dual integral equations of the first kind (e.g., see
I. Sneddon (1951) and Ya. S. Uflyand (1977)).
The integral equations obtained from (2)–(5) by replacing the kernel K(x – t)by K(t – x) are
said to be transposed to the original equations.
If the right-hand sides of Eqs. (1)–(6) are identically zero, then these equations are said to be
homogeneous. For the case in which the right-hand side of an equation of the type (1)–(6) does not
vanish on the entire domain, the corresponding equation is said to be nonhomogeneous.

Remark 3. Some equations whose kernel contains the product or the ratio of the variables x and
t can be reduced to Eqs. (2)–(5).
Remark 4. Sometimes equations of convolution type of the form (2)–(5) are written in the form
√
in which the integrals are multiplied by the coefficient 1/ 2π.
Remark 5. The cases in which the class of functions and kernels for equations of convolution
type (in particular, for Wiener–Hopf equations) differs from those introduced in Subsections 10.1-3
are always mentioned explicitly (see Sections 11.10 and 11.11).
•
References for Section 11.1: E. Goursat (1923), F. Riesz and B. Sz.-Nagy (1955), I. G. Petrovskii (1957), B. Noble
(1958), M. G. Krein (1958), S. G. Mikhlin (1960), L. V. Kantorovich and G. P. Akilov (1964), A. N. Kolmogorov and
S. V. Fomin (1970), L. Ya. Tslaf (1970), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), J. A. Cochran (1972),
V. I. Smirnov (1974), P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. D. Gakhov and Yu. I. Cherskii (1978), A. G. Butkovskii
(1979), L. M. Delves and J. L. Mohamed (1985), F. G. Tricomi (1985), A. J. Jerry (1985), A. F. Verlan’ and V. S. Sizikov (1986),
A. Golberg (1990), D. Porter and D. S. G. Stirling (1990), C. Corduneanu (1991), J. Kondo (1991), S. Pr¨ ossdorf and
B. Silbermann (1991), W. Hackbusch (1995), R. P. Kanwal (1997).

11.2. Fredholm Equations of the Second Kind With
Degenerate Kernel

11.2-1. The Simplest Degenerate Kernel
Consider Fredholm integral equations of the second kind with the simplest degenerate kernel:

b

y(x) – λ g(x)h(t)y(t) dt = f(x), a ≤ x ≤ b. (1)
a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 529