Page 652 - Handbook Of Integral Equations

13.1-2. Integral Equations With Hilbert Kernel

A complete singular integral equation with Hilbert kernel has the form
2π

1 ξ – x
a(x)ϕ(x)+ N(x, ξ) cot ϕ(ξ) dξ = f(x), (11)
2π 0 2
where the real functions a(x), f(x), and N(x, ξ) and the unknown function ϕ(x) satisfy the H¨ older
condition (see Subsection 12.2-2), with the function N(x, ξ) satisfying the condition with respect to
both variables.
The integral equation (11) can also be written in the following equivalent form, which is
frequently used. We transform the kernel as follows:
ξ – x ξ – x ξ – x
N(x, ξ) cot = N(x, ξ) – N(x, x) cot + N(x, x) cot , (12)
2 2 2
where we write

ξ – x

1
N(x, x)= –b(x), N(x, ξ) – N(x, x) cot = K(x, ξ). (13)
2π 2
In this case, Eq. (11) with regard to (12) and (13) becomes
2π 2π
b(x) ξ – x

a(x)ϕ(x) – cot ϕ(ξ) dξ + K(x, ξ)ϕ(ξ) dξ = f(x). (14)
2π 0 2 0
It follows from formulas (13) that the function b(x) satisfies the H¨ older condition, and the ker-
nel 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 equation in the form (14) is also called a complete singular integral equation with Hilbert
1
kernel. The functions a(x) and b(x) are called the coefficients of Eq. (14), cot (ξ – x) is called the
2
Hilbert kernel, and the known function f(x) is called the right-hand side of the equation. The first
and second summands in Eq. (14) form the so-called characteristic part or the characteristic of the
complete singular equation, and the third summand is called its regular part; the function K(x, ξ)is
called the kernel of the regular part.
The equation
2π

b(x) ξ – x
a(x)ϕ(x) – cot ϕ(ξ) dξ = f(x), (15)
2π 0 2
is called the characteristic equation corresponding to the complete equation (14).
As usual, the above and the forthcoming equations whose right-hand sides are zero everywhere
on their domains are said to be homogeneous, and otherwise they are said to be nonhomogeneous.

13.1-3. Fredholm Equations of the Second Kind on a Contour
Fredholm theory and methods for solving Fredholm integral equations of the second kind presented
in Chapter 11 remain valid if all functions and parameters in the equations are treated as complex ones
and an interval of the real axis is replaced by a contour L. Here we present only some information
and write the Fredholm integral equation of the second kind in the form that is convenient for the
purposes of this chapter.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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