Page 511 - Handbook Of Integral Equations

10.1-4. Dual Integral Equations of the First Kind
A dual integral equation of the first kind with difference kernels (of convolution type) has the form

∞
K 1 (x – t)y(t) dt = f(x), 0 < x < ∞,
–∞
∞
(8)
K 2 (x – t)y(t) dt = f(x), –∞ < x <0,
–∞
where the notation and the classes of functions and kernels coincide with those introduced above for
equations of convolution type.
In the general case, a dual integral equation of the first kind has the form
∞

K 1 (x, t)y(t) dt = f 1 (x), a < x < b,
a
∞

K 2 (x, t)y(t) dt = f 2 (x), b < x < ∞,
a
where f 1 (x) and f 2 (x) are the right-hand sides, K 1 (x, t) and K 2 (x, t) are the kernels of Eq. (9), and
y(x) is the unknown function. Various forms of this equation are considered in Subsections 10.6-2
and 10.6-3.
The integral equations obtained from (5)–(8) by replacing the kernel K(x – t) with K(t – x) are
called transposed equations.
Remark 3. Some equations whose kernels contain the product or the ratio of the variables x
and t can be reduced to equations of the form (5)–(8).
Remark 4. Equations (5)–(8) of the convolution type are sometimes written in the form in which
√
the integrals are multiplied by the coefficient 1/ 2π.
•
References for Section 10.1: I. Sneddon (1951), B. Noble (1958), S. G. Mikhlin (1960), I. C. Gohberg and M. G. Krein
(1967), L. Ya. Tslaf (1970), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), P. P. Zabreyko, A. I. Koshelev,
et al. (1975), Ya. S. Uflyand (1977), F. D. Gakhov and Yu. I. Cherskii (1978), A. J. Jerry (1985), A. F. Verlan’ and
V. S. Sizikov (1986), L. A. Sakhnovich (1996).

10.2. Krein’s Method

10.2-1. The Main Equation and the Auxiliary Equation
Here we describe a method for constructing exact closed-form solutions of linear integral equations
of the first kind with weak singularity and with arbitrary right-hand side. The method is based on the
construction of the auxiliary solution of the simpler equation whose right-hand side is equal to one.
The auxiliary solution is then used to construct the solution of the original equation for an arbitrary
right-hand side.
Consider the equation
a

K(x – t)y(t) dt = f(x), –a ≤ x ≤ a. (1)
–a
Suppose that the kernel of the integral equation (1) is polar or logarithmic and that K(x)isaneven
positive definite function that can be expressed in the form
K(x)= β|x| –µ + M(x), 0 < µ <1,

1
K(x)= β ln + M(x),
|x|
respectively, where β >0, –2a ≤ x ≤ 2a, and M(x)isasufficiently smooth function.
Along with (1), we consider the following auxiliary equation containing a parameter ξ (0 ≤ ξ ≤ a):
ξ
K(x – t)w(t, ξ) dt =1, –ξ ≤ x ≤ ξ. (2)
–ξ

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 493