Page 644 - Handbook Of Integral Equations

12.4-4. The General Equation of the First Kind With Cauchy Kernel

Consider the general equation of the first kind with Cauchy kernel
1 M(t, τ)
ϕ(τ) dτ = f(t), (16)
πi τ – t
L
where the integral is understood in the sense of the Cauchy principal value and is taken over a closed
or nonclosed contour L. As usual, the functions a(t), f(t), and M(t, τ)on L are assumed to satisfy
the H¨ older condition, where the last function satisfies this condition with respect to both variables.
We perform the following manipulation with the kernel:
M(t, τ) M(t, τ) – M(t, t) M(t, t)
= +
τ – t τ – t τ – t
and write
1 M(t, τ) – M(t, t)
M(t, t)= b(t), = K(t, τ). (17)
πi τ – t
We can rewrite Eq. (16) in the form
b(t) ϕ(τ)
dτ + K(t, τ)ϕ(τ) dτ = f(t). (18)
πi L τ – t L
It follows from formulas (17) that the function b(t) satisfies the H¨ older condition on the entire
contour L and K(t, τ) satisfies this condition everywhere except for the points with τ = t at which
this function satisfies the estimate
A
|K(t, τ)| < , 0 ≤ λ <1.
|τ – t| λ
The general singular integral equation of the first kind with Cauchy kernel is frequently written in
the form (18).
The general singular integral equation of the first kind is a special case of the complete singular
integral equation whose theory is treated in Chapter 13. In general, it cannot be solved in a closed
form. However, there are some cases in which such a solution is possible.
Let the function M(t, τ) in Eq. (16), which satisfies the H¨ older condition with respect to both
variables on the smooth closed contour L by assumption, have an analytic continuation to the
+
domain Ω with respect to each of the variables. If M(t, t) ≡ 1, then the solution of Eq. (16) can
be obtained by means of the Poincar´ e–Bertrand formula (see Subsection 12.2-6). This solution is
given by the relation
1 M(t, τ)
ϕ(t)= f(τ) dτ. (19)
πi L τ – t
Eq. (16) can be solved without the assumption that the function M(t, τ) satisfies the condition
+
M(t, t) ≡ 1. Namely, assume that the function M(t, τ) has the analytic continuation to Ω with
respect to each of the variables and that M(z, z) ≠ 0 for z ∈ Ω . In this case, the solution of Eq. (16)
+
has the form
1 1 M(t, τ) f(τ)
ϕ(t)= dτ. (20)
πi M(t, t) L M(τ, τ) τ – t
In Section 12.5, a numerical method for solving a special case of the general equation of the first
kind is given, which is of independent interest from the viewpoint of applications.
Remark 1. The solutions of complete singular integral equations that are constructed in Sub-
section 12.4-4 can also be applied for the case in which the contour L is a collection of finitely many
disjoint smooth closed contours.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 627