Page 667 - Handbook Of Integral Equations

◦
◦
the coefficients of the Riemann problems that correspond to the operators K and K . This means
1 2
◦
that the coefficient of the Riemann problem for the operator (K 2 K 1 ) is equal to the product of
◦
◦
the coefficients of the Riemann problems for the operators K and K , and hence the index of the
1 2
product of singular operators is equal to the sum of indices of the factors:
ν = ν 1 + ν 2 . (12)
In its complete form, the operator K 2 K 1 is defined by the expression
b(t) ϕ(τ)
K 2 K 1 [ϕ(t)] ≡ a(t)ϕ(t)+ dτ + K(t, τ)ϕ(τ) dτ,
πi L τ – t L

where a(t) and b(t) are defined by formulas (9). For a regular kernel K(t, τ), on the basis of
formulas (4) we can write out the explicit expression.
For a singular operator K and its transposed operator K (see Subsection 13.1-1), the following
∗
relations hold:

∗
ψ(t)K[ϕ(t)] dt = ϕK [ψ(t)] dt
L L
for any functions ϕ(t) and ψ(t) that satisfy the H¨ older condition, and
∗
∗
∗
(K 2 K 1 ) = K K .
1
2
13.4-2. The Regularizer
The regularization method is a reduction of a singular integral equation to a Fredholm equation. The
reduction process itself is known as regularization.
If a singular operator K 2 is such that the operator K 2 K 1 is regular (Fredholm), i.e., contains no
singular integral (b(t) ≡ 0), then K 2 is called the regularizing operator with respect to the singular
operator K 1 or, briefly, a regularizer. Note that if K 2 is a regularizer, then the operator K 1 K 2 is
regular as well.
Let us find the general form of a regularizer. By definition, the following relation must hold:

b(t)= a 2 (t)b 1 (t)+ b 2 (t)a 1 (t) = 0, (13)

which implies that
a 2 (t)= g(t)a 1 (t), b 2 (t)= –g(t)b 1 (t), (14)
where g(t) is an arbitrary function that vanishes nowhere and satisfies the H¨ older condition.
Hence, if K is a singular operator,

b(t) ϕ(τ)
K[ϕ(t)] ≡ a(t)ϕ(t)+ dτ + K(t, τ)ϕ(τ) dτ, (15)
πi L τ – t L
˜
then, in general, the regularizer K can be expressed as follows:
g(t)b(t) ω(τ)
˜
˜
K[ω(t)] ≡ g(t)a(t)ω(t) – dτ + K(t, τ)ω(τ) dτ, (16)
πi L τ – t L
˜
where K(t, τ) is an arbitrary Fredholm kernel and g(t) is an arbitrary function satisfying the H¨ older
condition.
Since the index of a regular operator (b(t) ≡ 0) is clearly equal to zero, it follows from the
property of the product of operators that the index of the regularizer has the same modulus as the

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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