Page 562 - Handbook Of Integral Equations

11.6-10. Skew-Symmetric Integral Equations

By a skew-symmetric integral equation we mean an equation whose kernel is skew-symmetric, i.e.,
an equation of the form
b

y(x) – λ K(x, t)y(t) dt = f(x) (31)
a
whose kernel K(x, t) has the property

K(t, x)= –K(x, t). (32)

Equation (31) with the skew-symmetric kernel (32) has at least one characteristic value, and all
its characteristic values are purely imaginary.

•
References for Section 11.6: E. Goursat (1923), R. Courant and D. Hilbert (1931), S. G. Mikhlin (1960), M. L. Krasnov,
A. I. Kiselev, and G. I. Makarenko (1971), J. A. Cochran (1972), V. I. Smirnov (1974), A. J. Jerry (1985), F. G. Tricomi
(1985), D. Porter and D. S. G. Stirling (1990), C. Corduneanu (1991), J. Kondo (1991), W. Hackbusch (1995), R. P. Kanwal
(1997).

11.7. An Operator Method for Solving Integral Equations
of the Second Kind

11.7-1. The Simplest Scheme
Consider a linear equation of the second kind of the special form

y(x) – λL [y]= f(x), (1)

2
where L is a linear (integral) operator such that L = k, k = const.
Let us apply the operator L to Eq. (1). We obtain

L [y] – kλy(x)= L [f(x)]. (2)

On eliminating the term L [y] from (1) and (2), we find the solution
1
y(x)= f(x)+ λL [f] . (3)
1 – kλ 2

Remark. In Section 9.4, various generalizations of the above method are described.

11.7-2. Solution of Equations of the Second Kind on the Semiaxis
1 . Consider the equation
◦

∞
y(x) – λ cos(xt)y(t) dt = f(x). (4)
0
In this case, the operator L coincides, up to a constant factor, with the Fourier cosine transform:

∞ π
L [y]= cos(xt)y(t) dt = F c [y] (5)
0 2
2
and acts by the rule L = k, where k = π (see Subsection 7.5-1).
2

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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