Vol. 48, No.8 where a =1, b = 71. The exact solution is J(8) = 1/s. 3. J a ~ 8 :s b, the right-hand side is. /(q) == .fb2 - {J/(bq2). The meshes in s and q are: 8j = a +jh, ...
Russian Mathematics (t». VUZ) Vol.
IZ1Jesti,'ga YUZ" Matematika
48, No.8, pp.S9-66, 2004 .
uno
517.968:519.642
NUMERICAL SOLUTION OF THE ABELIAN SINGULAR INTEGRAL EQUATION BY THE) GENERALIZED QUADRATURE METHOD V.S. Sizikov, A.V. Smirnov, and B.A. Fedorov
1. Introduction The numerical solution of singular integral equations (8IE) is studied in a number of works ([1]-[6] et all). The following 8IE are studied: a) one-dimensional SIE of the I-st kind with the Cauchy kernel [2], [3], [7], for example ([2], pp.8, 73),
(II y(s)ds
JIJ
= f(:C),
a S s: S i,
(1)
X-8
where y(s) is the desired function;
b) SIEof the I-at kind with the Hilbert kernel [2], [3] or the generalized Hilbert kernel [3J; c) SIE of the I~-nd kind with t~e Cauchy or Hilbert kernels [1], [4], [7], for example, [4],
r:
R(x) x- S 1 A($)Y(x) + ~ 10 ctg -2-y (s)ds+ 271"
1 hex, s)y(s)ds 2
11"
0
= f(:C), 0 < x
< 21r,
(2)
.where A(a;); B(x) are given continuous functions, h(:b; s) is a known 27f-periodic' function; if A(x) - 0 for some values of x, but not for all of them, then equation (2) is one of the III-rd
kind (cf. [81, p.140); d) SIE of the I-st or II-nd kind with logarithmic and other (weakly singular) kernels, for example
([3], p.6), ,
.
r I
I
1 Z - S 1 - 271" 10 In sin-- y(s)ds+ 271" 2
1 hex,
. ' .
2
'1r
0
.
s)y(s)ds = J(x),
0
