A Numerical Method for Solving a System of Hypersingular Integral Equations of the Second Kind
- PDF / 212,619 Bytes
- 14 Pages / 594 x 792 pts Page_size
- 70 Downloads / 186 Views
A NUMERICAL METHOD FOR SOLVING A SYSTEM OF HYPERSINGULAR INTEGRAL EQUATIONS OF THE SECOND KIND
UDC 517.698.519.6
O. V. Kostenko
Abstract. A numerical method for solving a system of hypersingular integral equations of the second kind is presented. The theorem on the existence and uniqueness of a solution to such a system is proved. The rate of convergence of an approximate solution to the exact solution is estimated. Keywords: system of integral equations, numerical method, integral in the sense of the Hadamard finite part, existence and uniqueness of solutions, convergence rate. PROBLEM STATEMENT In solving applied problems of mathematics and also in mathematical modeling of physical processes, systems of integral equations occur that are of the following form:
hui ( y) 1 - y 2 +
m
1
1 p
1
ui ( t )
ò (t - y) 2
1 - t 2 dt +
-1
1
ò ln | t - y | ui (t )
1 - t 2 dt
-1
1
å p ò K ik (t, y)ui (t )
k =1 k¹i
a p
1 - t 2 dt = f i ( y) , i = 1, m,
(1)
-1
,a where h and a are given complex and real constants, respectively. We denote by C[r-1 the set of functions that are , 1] continuously r-times differentiable on the interval [ -1, 1] and are such that the rth derivative on the interval [ -1, 1] satisfies the H&& older condition with the index a , 0 < a £ 1. Suppose that, in the system of integral equations (1),
,a ,a functions f i ( y) , i = 1, m, belong to the set C[0-1 , functions K ik ( t, y) , i = 1, m, k = 1, m, belong to the set C[1-1 , 1] , 1]
with respect to each variable uniformly relative to the other variable, and the sought-for functions ui ( t ) , i = 1, m, ,a . The second addend in the left side of the ith (i = 1, m) equation of system (1) is understood belong to the set C[1-1 , 1] in the sense of the finite Hadamard part, and the third addend is an improper integral. This system of equations is obtained by the author of this article in constructing a mathematical model of diffraction scattering of waves by a lattice consisting of a finite number of imperfectly conductive tapes [1], which confirms its urgency. The objective of this article is the presentation of a numerical method for solving the system of integral equations (1) and its substantiation, namely, the proof of a criterion for the existence and uniqueness of a solution, estimation of the norm of the difference between approximate and exact solutions that allows one to characterize the B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Kharkiv, Ukraine, [email protected] and [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, May–June, 2016, pp. 67–82. Original article submitted April 6, 2015. 394
1060-0396/16/5203-0394 ©2016 Springer Science+Business Media New York
convergence rate of the proposed method, demonstration and analysis of the results of its application to the solution of model problems, and also estimation of its numerical convergence. Note that, at the present time, qualitative properties of exact and approximate solutio
Data Loading...
