Computational Mathematics and Modeling, Vol. 29, No. 3, July, 2018
II. NUMERICAL METHODS
SOLUTION OF AN INTEGRAL EQUATION OF THE FIRST KIND
WITH A LOGARITHMIC KERNEL
V. I. Dmitriev, I. V. Dmitrieva, and N. A. Osokin
An iterative method is proposed and investigated for the solution of an integral equation with a kernel
with a logarithmic singularity in an arbitrary definition domain.
Keywords: integral equation of the first kind, numerical methods.
Many applications require solving an integral equation of the first kind with a logarithmic kernel:
x − y
⋅ dx = f (y)
y ∈ −1,1
Solution of an integral equation of the first kind is an unstable problem. However, the logarithmic singularity of
the integral-equation kernel can be exploited to obtain a stable solution.
It has been shown  that for the integral equation of the first kind
K(x, y) ⋅ u(x) ⋅ dx = f (y)
y ∈ −1,1
with its kernel representable in the form
K(x, y) = ln
x − y
+ N(x, y)
is differentiable with respect to y and
is the class of continuous functions),
a unique solution exists satisfying the Höelder condition on the closed interval
. A stable method that
solves this equation by separating the kernel singularity has been proposed in .
An iterative method of stable solution of Eq. (1) has been proposed in . The iterative method is con-
structed as follows.
Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia; e-mail: email@example.com.
Translated from Prikladnaya Matematika i Informatika, No. 56, 2017, pp. 61–71.
1046–283X/18/2903–0307 © 2018 Springer Science+Business Media, LLC 307