Page 663 - Handbook Of Integral Equations

13.3. Complete Singular Integral Equations Solvable in a
Closed Form

In contrast with characteristic equations and their transposed equations, complete singular
integral equations cannot be solved in the closed form in general. However, there are some cases in
which complete equations can be solved in a closed form.

13.3-1. Closed-Form Solutions in the Case of Constant Coefficients
Consider the complete singular integral equation with Cauchy kernel in the form (see Subsec-
tion 13.1-1)
b(t) ϕ(τ)
a(t)ϕ(t)+ dτ + K(t, τ)ϕ(τ) dτ = f(t), (1)
πi L τ – t L
where L is an arbitrary closed contour. Let us show that Eq. (1) can be solved in a closed form if
a(t)= a and b(t)= b are constants and K(t, τ) is an arbitrary function that has an analytic continuation
+
to the domain Ω with respect to each variable.
Under the above assumptions, Eq. (1) has the form
1 M(t, τ)
aϕ(t)+ ϕ(τ) dτ = f(t), (2)
πi L τ – t
where M(t, τ)= b + πi(t – τ)K(t, τ), so that M(t, t)= b = const. Let b ≠ 0. We write
1 M(t, τ)
ψ(t)= ϕ(τ) dτ. (3)
bπi L τ – t
According to Subsection 12.4-4, the function ϕ(t) can be expressed via ψ(t) and ψ(t) can be
expressed via ϕ(t). Then we rewrite Eq. (2) as follows:
aϕ(t)+ bψ(t)= f(t). (4)

On applying the operation (3) to this equation, we obtain

aψ(t)+ bϕ(t)= w(t), (5)
where
1 M(t, τ)
w(t)= f(τ) dτ.
bπi τ – t
L
By solving system (4), (5) we find ϕ(t):
1 1 M(t, τ)
ϕ(t)= af(t) – f(τ) dτ (6)
2
a – b 2 πi L τ – t
under the assumption that a ≠ ±b.
Thus, for a ≠ ±b and for a kernel K(t, τ) that can be analytically continued, Eq. (1) or (2) is
solvable and has the unique solution given by formula (6).
Equation (1) was studied above for b ≠ 0. This assumption is natural because, for b ≡ 0, Eq. (1)
is no longer singular. However, the Fredholm equation obtained for b = 0, that is,

aϕ(t)+ K(t, τ)ϕ(τ) dτ = f(t), a = const, (7)
L
is solvable in a closed form for a kernel K(t, τ) that has analytic continuation.
+
Let a function K(t, τ) have an analytic continuation to the domain Ω with respect to each of
the variables and continuous for t, τ ∈ L. In this case, the following assertions hold.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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