Page 651 - Handbook Of Integral Equations

on the left-hand side of Eq. (4) form the characteristic part or the characteristic of the complete
singular equation and the third summand is called the regular part, and the function K(t, τ) is called
the kernel of the regular part. It follows from the above estimate for the kernel of the regular part
that K(t, τ) is a Fredholm kernel.
For Eqs. (1) and (4) we shall use the operator notation

K[ϕ(t)] = f(t), (5)

where the operator K is called a singular operator.
The equation
b(t) ϕ(τ)
◦
K [ϕ(t)] ≡ a(t)ϕ(t)+ dτ = f(t) (6)
πi L τ – t
is called the characteristic equation corresponding to the complete equation (4), and the operator K ◦
is called the characteristic operator.
For the regular part of the equation we introduce the notation

K r [ϕ(t)] ≡ K(t, τ)ϕ(τ) dτ,
L
where the operator K r is called a regular (Fredholm) operator, and we rewrite the complete singular
equation in another operator form:

K[ϕ(t)] ≡ K [ϕ(t)] + K r [ϕ(t)] = f(t), (7)
◦
which will be used in what follows.
The equation
1 b(τ)ψ(τ)
K [ψ(t)] ≡ a(t)ψ(t) – dτ + K(τ, t)ψ(τ) dτ = g(t), (8)
∗
πi L τ – t L
obtained from Eq. (4) by transposing the variables in the kernel is said to be transposed to (4). The
∗
operator K is said to be transposed to the operator K.
In particular, the equation
1 b(τ)
K [ψ(t)] ≡ a(t)ψ(t) – ψ(τ) dτ = g(t) (9)
◦∗
πi L τ – t
is the equation transposed to the characteristic equation (6). It should be noted that the operator K ◦∗
transposed to the characteristic operator K differs from the operator K ∗◦ that is characteristic for
◦
the transposed equation (9). The latter is defined by the formula
b(t) ψ(τ)
∗◦
K [ψ(t)] ≡ a(t)ψ(t) – dτ. (10)
πi L τ – t
Throughout the following we assume that in the general case the contour L consists of m +1
closed smooth curves L = L 0 + L 1 + ··· + L m . For equations with nonclosed contours, see, for
example, the books by F. D. Gakhov (1977) and N. I. Muskhelishvili (1992).
Remark 1. The above relationship between Eqs. (1) and (4) that involves the properties of these
equations is violated if we modify the condition and assume that in Eq. (1) the function M(t, τ)
satisfies the H¨ older condition everywhere on the contour except for finitely many points at which M
has jump discontinuities. In this case, the complete singular integral equation must be represented
in the form (4) with separated characteristic and regular parts in some way that differs from the
transformation (2) and (3) because the above transformation of Eq. (1) does not lead to the desired
decomposition. For equations with discontinuous coefficients, see the cited books.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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