DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

JUSTIFICATION OF QUADRATURE-DIFFERENCE METHODS
FOR SINGULAR INTEGRODIFFERENTIAL EQUATIONS
A. FEDOTOV
Here we propose and justify quadrature-difference methods for different kinds (linear,
nonlinear, and multidimensional) of periodic singular integrodifferential equations.
Copyright © 2006 A. Fedotov. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
In Section 2 we propose and justify quadrature-difference methods for linear and nonlinear
singular integrodifferential equations with Hölder-continuous coefficients and righthand
sides. In Section 3 the same method is justified for linear singular integrodifferential
equations with discontinuous coefficients and right-hand sides. In Section 4 we propose
and justify cubature-difference method for multidimensional singular integrodifferential
equations in Sobolev space.
2. Linear and nonlinear singular integrodifferential equations with
continuous coefficients
Let us consider the linear singular integrodifferential equation
m∑ (
aν (t)x (ν) (t)+b ν (t) ( Jx (ν)) (t)+ ( J 0 h ν x (ν)) (t) ) = y(t) (2.1)
ν=0
and the nonlinear singular integrodifferential equation
F ( t,x (m) (t),...,x(t), ( Jx (m)) (t),...,(Jx)(t), ( J 0 h m x (m)) (t),..., ( J 0 h 0 x ) (t) ) = y(t), (2.2)
where x(t) isadesiredunknown,a ν (t), b ν (t),h ν (t,τ),ν = 0,1,...,m − 1, y(t), and F(t,
u m ,...,u 0 ,v m ,...,v 0 ,w m ,...,w 0 )aregivencontinuous2π-periodic by the variables t,τ
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 403–411