ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2018, Vol. 12, No. 2, pp. 382–394.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
A.V. Voytishek, 2018, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2018, Vol. XXI, No. 2, pp. 32–45.
Development and Optimization of Randomized Functional
Numerical Methods for Solving the Practically Signiﬁcant
Fredholm Integral Equations of the Second Kind
A. V. Voytishek
Institute of Computational Mathematics and Mathematical Geophysics,
pr. Аkad. Lavrent’eva 6, Novosibirsk, 630090 Russia
Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia
Received September 12, 2017
Abstract—Under study are the randomized algorithms for numerical solution of the Fredholm
integral equations of the second kind (from the viewpoint of their application for the practically
important problems of mathematical physics). The projection, grid and projection-grid methods are
distinguished. Certain advantages of the projection and projection-grid methods are demonstrated
(allowing using them for numerical solution of the equations with the integrable singularities
in kernels and free terms).
Keywords: applied Fredholm integral equations of the second kind, integrable singularities,
numerical randomized functional method, projection, grid, projection-grid functional algo-
1. INTEGRAL EQUATIONS
With the development of computer technology, the interest is growing in the numerical algorithms
for solving applied problems, whereas the computing schemes implemented on modern multiprocessor
devices are of special importance. For this reason, the algorithms of numerical statistical modeling (or
Monte Carlo methods) are promising .
One of the main applications of the algorithms of the Monte Carlo method is solution of the various
integral Fredholm equations of the second kind of the form
+ f(x) or ϕ = Kϕ + f, (1.1)
which are encountered in the study of topical applied problems. Here, ϕ(x) is an unknown approximated
function (solution), whereas the functions k(x
,x) (the kernel of the integral operator K from (1.1)) and
f(x) (the free term of the equation) are given (for example, see [2, p. 221]).
In this article, a special attention is paid to the functional applied algorithms of the Monte Carlo
method [1, 3] used for solution of the problem of approximation of the solution ϕ(x) of (1.1) on a compact
set X ⊆ R
For the sake of further discussion, the two conditions are important:
Condition 1.1. The function k(x, y) can be calculated for all x, y ∈ X ⊆ R
Condition 1.2. The integral operator K is contracting and
< 1, (1.2)