Page 614 - Handbook Of Integral Equations

Chapter 12

Methods for Solving Singular

Integral Equations of the First Kind

12.1. Some Definitions and Remarks

12.1-1. Integral Equations of the First Kind With Cauchy Kernel
A singular integral equation of the first kind with Cauchy kernel has the form

1 ϕ(τ) 2
dτ = f(t), i = –1, (1)
πi L τ – t
where L is a smooth closed or nonclosed contour in the complex plane of the variable z = x + iy,
1
t and τ are the complex coordinates on L, ϕ(t) is the unknown function, is the Cauchy kernel,
τ – t
and f(t) is a given function, which is called the right-hand side of Eq. (1). The integral on the
left-hand side only exists in the sense of the Cauchy principal value (see Subsection 12.2-5).
A singular integral equation in which L is a smooth closed contour, as well as an equation of
the form
1 ∞ ϕ(t)
dt = f(x), –∞ < x < ∞, (2)
π t – x
–∞
on the real axis and an equation with Cauchy kernel
1 b ϕ(t)
dt = f(x), a ≤ x ≤ b, (3)
π a t – x
on a finite interval, are special cases of Eq. (1).
A general singular integral equation of the first kind with Cauchy kernel has the form
1 M(t, τ)
ϕ(τ) dτ = f(t), (4)
πi L τ – t
where M(t, τ) is a given function. This equation can also be rewritten in a different (equivalent)
form, which is given in Subsection 12.4-4.
Assume that all functions in Eqs. (1)–(4) satisfy the H¨ older condition (Subsection 12.2-2) and
the function M(t, τ) satisfies this condition with respect to both variables.

12.1-2. Integral Equations of the First Kind With Hilbert Kernel
The simplest singular integral equation of the first kind with Hilbert kernel has the form
2π

1 ξ – x
cot ϕ(ξ) dξ = f(x), (5)
2π 0 2

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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