Журнал Сибирского федерального университета
Journal of Siberian Federal University
Математика и Физика
Mathematics & Physics
Редакционный совет:
академик РАН
Е.А. Ваганов
академик РАН
И.И. Гительзон
академик РАН
А.Г. Дегерменджи
академик РАН
В.Ф.Шабанов
чл.-корр. РАН,
д-р физ.-мат. наук
В.В. Зуев
чл.-корр. РАН,
д-р физ.-мат. наук
В.Л. Миронов
чл.-корр. РАН,
д-р техн. наук
Г.Л. Пашков
чл.-корр. РАН,
д-р физ.-мат. наук
В.В.Шайдуров
Editorial Advisory Board
Chairman:
Eugene A. Vaganov
Members:
Josef J. Gitelzon
Vasily F. Shabanov
Andrey G. Degermendzhy
Vladimir V. Zuev
Valery L. Mironov
Gennady L. Pashkov
Vladimir V. Shaidurov
Свидетельство
о регистрации СМИ
ПИ №ФС77 - 28724
от 29.06.2007 г.
Yu.Ya.Belov,
V.G.Kopylova
A.A. Imomov
R.M.Khakimov
G.Kh.Khudayberganov,
U.S.Rakhmonov
M.S.Kildyushov
V.A. Nikishkin
K.A.Kirillov
CONTENTS
Determination of Source Functions in
Composite Type System of Equations
275–288
Limit Theorem for the Joint Distribution
in the Q-processes
289-296
New Periodic Gibbs Measures for q-state
Potts Model on a Cayley Tree
297-304
The Bergman and Cauchy–Szego Kernels
for Matrix Ball of the Second Type
305-310
Asymptotic Behavior at Infinity of the
Dirichlet Problem Solution of the 2k
Order Equation in a Layer
311-317
On Minimal Cubature Formulas Exact
for Haar Polynomials of Low Degrees in
the Two-Dimensional Case
318-323
A.V.Koptev
Generator of Solutions for 2D NavierStokes
Equations
324-330
Редакторы: В.Е.Зализняк, А.В.Щуплев
Компьютерная верстка: Г.В.Хрусталева
Подписано в печать 10.07.14 г. Формат 84×108/16. Усл.печ. л. 12,1.
Уч.-изд. л. 11,7. Бумага тип. Печать офсетная.
Тираж 1000 экз. Заказ 1762
Отпечатано ПЦ БИК СФУ. 660041 Красноярск, пр. Свободный, 82.
2014 7(3)
Стр.1
Editorial Board:
Editor-in-Chief :
Michail I. Gladyshev
Founding Editor:
Vladimir I. Kolmakov
Managing Editor:
Olga F. Aleksandrova
Subject Editor for
Mathematics & Physics:
Prof. Alexander M. Kytmanov
(SibFU, Krasnoyarsk, Russia)
Consulting Editors
Mathematics & Physics:
Prof. Lev Aizenberg
(Bar-Ilan Univ., Tel-Aviv, Israel)
Prof. Viktor K. Andreev
(ICM, Krasnoyarsk, Russia)
Prof. Alexander M. Baranov
(KSP Univ., Krasnoyarsk, Russia)
Prof. Yury Ya. Belov
(SibFU, Krasnoyarsk, Russia)
Prof. Sergey S. Goncharov
(IM, Novosibirsk, Russia)
Prof. Vladimir M. Levchuk
(SibFU, Krasnoyarsk, Russia)
Prof. Yury Yu. Loginov
(SibSASU, Krasnoyarsk, Russia)
Prof. Mikhail V. Noskov
(SibFU, Krasnoyarsk, Russia)
Prof. Sergey G. Ovchinnikov
(IPh., Krasnoyarsk, Russia)
Prof. Gennady S. Patrin
(SibFU, Krasnoyarsk)
Prof. Azimbay Sadullaev
(NUUz., Tashkent, Uzbekistan)
Prof. Nikolai Tarkhanov
(Potsdam Univ., Germany)
Prof. Avgust K. Tsikh
(SibFU, Krasnoyarsk, Russia)
Prof. Valery V. Val’kov
(IPh., Krasnoyarsk, Russia)
Prof. Yury V. Zakharov
(SibSTU, Krasnoyarsk, Russia)
A.E. Silaeva
A.A. Shchitnikov
R.F. Shamoyan,
S.M.Kurilenko
M.N.Nazarov
T.S.Krepizina
V.R.Kulikov
Univalent Differentials of Integer Order
on Variable Torus
331-338
Conditions for Convergence of the
Mellin-Barnes Integral for a Solution
to a System of Algebraic Equations
339-346
T.A.Kustitskaya
Risk Aversion for Defining Elliptic Acceptance
Sets in the Model of Generalized
Coherent Risk Measures
347-361
V.M.Levchuk,
P.K.Shtukkert
Problems on Structure for Quasifields
of Orders 16 and 32
362-372
New Approaches to the Analysis of
the Elementary Reactions Kinetics
373-382
On a New Embedding Theorem in
Analytic Bergman Type Spaces in
Bounded Strictly Pseudoconvex Domains
of n-dimensional Complex Space
383-388
Determination
of Mechanical Force
between two Planar Inductors in the
Problem of Electrodynamic Excitation
of Seismic Waves
389-397
On Creeping Motion of Binary
Mixture in a Tube with Rectangular
Cross-Section
398-406
Стр.2
Journal of Siberian Federal University. Mathematics & Physics 2014, 7(3), 275–288
УДК 517.9
Determination of Source Functions in Composite Type
System of Equations
Yury Ya.Belov∗
Vera G.Kopylova†
Institute of Mathematics and Computer Science,
Siberian Federal University,
Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 20.04.2014, received in revised form 05.05.2014, accepted 28.05.2014
The problem of identification of the source function for semievolutionary system of two partial differential
equations is considered in the paper. The investigated system of equations is obtained from the original
system by adding the time derivative containing a small parameter ε > 0 to the elliptic equation. The
Cauchy problem and the second boundary-value problem are considered.
Keywords: identification, inverse problem, parabolic equation, the method of weak approximation, small
parameter, convergence.
We obtain a priori (uniform in ε > 0) estimates of solutions of approximate problems. We
prove convergence of solutions approximating the inverse problems to solutions of original problems
when ε → 0 on the basis of the obtained a priori estimates. We obtain that the rate of
convergence of solutions of approximate problems is O(ε1/2) in class of continuous functions.
The case of the first boundary-value problem has been studied by Yu.Ya.Belov.
An identification problem of source functions in the composite type system is considered
in [1–3].
1. Formulation of the problem and reduction it to the direct
problem
Consider in the strip G[0,T] = {(t,x) | 0 t T, x ∈ E1} the problem of determining
real-valued functions ε
u(t,x), ε
εε
v(t,x), ε
ε
ut(t,x)+a11(t)ε
vt(t,x)+a21(t)ε
u(t,x)+a12(t)ε
u(t,x)+a22(t)ε
where constant ε ∈ (0, 1]. Initial conditions are
ε
u(0,x) = u0(x),
and the overdetermination condition are
ε
g(t), satisfying the system of equations
v(t,x) = µ1uxx(t,x)+ ε
ε
v(t,x) = µ2
ε
vxx(t,x)+F(t,x),
ε
v(0,x) = v0(x),
ut,x0 = ϕ(t), ϕ ∈ C2[0,T],
where ϕ(t) is a given function, 0 t T and x0 is some fixed point.
∗ybelov@sfu-kras.ru
†kopylova.vera@mail.ru
- Siberian Federal University. All rights reserved
c
– 275 –
g(t)f(t,x),
(1)
(2)
(3)
Стр.3