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◦
2 . Right Regularization. On replacing in Eq. (19) the desired function by the expression (20),
˜
ϕ(t)= K[ω(t)], (22)
where ω(t) is a new unknown function, we arrive at the integral equation
˜
KK[ω(t)] = f(t), (23)
which is a Fredholm equation as well. Thus, from the singular integral equation (19) for the unknown
function ϕ(t) we passed to the Fredholm integral equation for the new unknown function ω(t).
On solving the Fredholm equation (23), we find a solution of the original equation (19) by
formula (22). The application of formula (22) requires integration only (a proper integral and a
singular integral must be found).
This is the second method of the regularization, which is called right regularization.

13.4-4. The Problem of Equivalent Regularization
In the reduction of a singular integral equation to a regular one we perform a functional transformation
over the corresponding equation. In general, this transformation can either introduce new irrelevant
solutions that do not satisfy the original equation or imply a loss of some solutions. Therefore, in
general, the resultant equation is not equivalent to the original equation. Consider the relationship
between the solutions of these equations and find out in what cases these equations are equivalent.
◦
1 . Left Regularization. Consider a singular equation
K[ϕ(t)] = f(t) (24)

and the corresponding regular equation
˜
˜
KK[ϕ(t)] = K[f(t)]. (25)
Let us write out Eq. (25) in the form
˜

K K[ϕ(t)] – f(t) = 0. (26)
˜
Since the operator K is homogeneous, it follows that each solution of the original equation (24)
(a function that vanishes the expression K[ϕ(t)] – f(t)) satisfies Eq. (26) as well. Hence, the left
regularization implies no loss of solutions. However, a solution of the regularized equation need not
be a solution of the original equation.
Consider the singular integral equation corresponding to the regularizer
˜
K[ω(t)] = 0. (27)
Let ω 1 (t), ... , ω p (t) be a complete system of its solutions, i.e., a maximal collection of linearly
˜
independent eigenfunctions of the regularizer K.
We regard Eq. (26) as a singular equation of the form (27) with the unknown function ω(t)=
K[ϕ(t)] – f(t). We obtain
p

K[ϕ(t)] – f(t)= α j ω j (t), (28)
j=1
where the α j are some constants.
We see that the regularized equation is equivalent to Eq. (28) rather than the original equation (24).
Thus, Eq. (25) is equivalent to Eq. (28) in which α j are arbitrary or definite constants. It may
occur that Eq. (28) is solvable only under the assumption that all α j satisfy the condition α j =0.
In this case, Eq. (25) is equivalent to the original equation (24), and the regularizer defines an
equivalent transformation. In particular, if the regularizer has no eigenfunctions, then the right-hand
side of Eq. (28) is identically zero, and it must be equivalent. This operator certainly exists for
ν ≥ 0. For instance, we can take the regularizer K , which has no eigenfunctions for the case under
∗◦
consideration because the index of the regularizer K ∗◦ is equal to –ν ≤ 0.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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