Page 646 - Handbook Of Integral Equations

Remark 2. Equation (21) is a special case of the characteristic singular integral equation with
Hilbert kernel (see Subsections 13.1-2 and 13.2-5).
◦
2 . Consider the general singular integral equation of the first kind with Hilbert kernel
1 2π ξ – x
N(x, ξ) cot ϕ(ξ) dξ = f(x). (30)
2π 0 2
Let us represent its kernel in the form

ξ – x ξ – x ξ – x
N(x, ξ) cot = N(x, ξ) – N(x, x) cot + N(x, x) cot .
2 2 2
We introduce the notation
ξ – x
1
N(x, x)= –b(x), N(x, ξ) – N(x, x) cot = K(x, ξ), (31)
2π 2
and rewrite Eq. (30) as follows:

b(x) 2π ξ – x 2π
– cot ϕ(ξ) dξ + K(x, ξ)ϕ(ξ) dξ = f(x), (32)
2π 0 2 0
It follows from formulas (31) that the function b(x) satisfies the H¨ older condition, whereas the
kernel K(x, ξ) satisfies the H¨ older condition everywhere except possibly for the points x = ξ,at
which the following estimate holds:

A
|K(x, ξ)| < , A = const < ∞, 0 ≤ λ <1.
|ξ – x| λ
The general singular integral equation of the first kind with Hilbert kernel is frequently written in
the form (32). It is a special case of the complete singular integral equation with Hilbert kernel,
which is treated in Subsections 13.1-2 and 13.4-8.
•
References for Section 12.4: F. D. Gakhov (1977), F. D. Gakhov and Yu. I. Cherskii (1978), S. G. Mikhlin and
S. Pr¨ ossdorf (1986), N. I. Muskhelishvili (1992), I. K. Lifanov (1996).

12.5. Multhopp–Kalandiya Method

Consider a general singular integral equation of the first kind with Cauchy kernel on the finite interval
[–1, 1] of the form
1 1 ϕ(t) dt 1 1
+ K(x, t)ϕ(t) dt = f(x). (1)
π –1 t – x π –1
This equation frequently occurs in applications, especially in aerodynamics and 2D elasticity.
We present here a method of approximate solution of Eq. (1) under the assumption that this
equation has a solution in the classes indicated below.

12.5-1. A Solution That is Unbounded at the Endpoints of the Interval
According to the general theory of singular integral equations (e.g., see N. I. Muskhelishvili (1992)),
such a solution can be represented in the form

ψ(x)
ϕ(x)= √ , (2)
1 – x 2

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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