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International Conference: 2004 - Dynamical Systems and Applications
July 5-10, 2004
Antalya, Turkey |
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Organizers
Akca Haydar, Basem S. Attili, Boucherif Abdelkader, Cho Yeol Je, Covachev Valery, Gyori Istvan, Maksimov Vyacheslav, Stavroulakis Ioannis P.
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On periodic solutions problem for quasi-linear system of hyperbolic equations
by
Anar T. Asanova
Institute of Mathematics MES Republic Kazakhstan
|
On periodic solutions problem |
|
for quasi-linear system of hyperbolic
equations |
|
Institute of
Mathematics of MES of Republic of Kazakhstan, Almaty |
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e-mail (anar@math.kz, dzhumabaev@list.ru ) |
The periodic problem for quasi-linear system of hyperbolic
equations of the second order with two independent variables is
considered on [`(\Omega)] = [0, T]×[0, \omega]
|
|
\partial2u
\partialt \partialx
|
= A(t, x) |
\partialu
\partialx
|
+f(t, x, u, |
\partialu
\partialt
|
), u in Rn, |
| (1) |
|
u(t, 0) = \psi(t), t in [0, T], |
| (2) |
|
u(0, x) = u(T, x), x in [0, \omega], |
| (3) |
where
matrix A(t, x) is continuous on [`(\Omega)] and A(0, x) = A(T, x), function \psi(t) is continuously differentiable on
[0, T], \psi(0) = \psi(T), [(\psi)\dot](0) = [(\psi)\dot](T),
||u(t, x)|| = maxi=[`1, n]|ui(t, x)|.
C([`(\Omega)], Rn) - is space of continuous on [`(\Omega)]
functions u: [`(\Omega)] --> Rn with norm ||u||1 = max(t, x) in [`(\Omega)]||u(t, x)||;
S1(\psi, r) = { u in Rn : ||u -\psi(t)|| < r}, S2([(\psi)\dot], r) = { u in Rn : ||u -[(\psi)\dot](t)|| < r }.
The function f(t, x, u, w) is continuous in [`(\Omega)]×S1(\psi, r) ×S2([(\psi)\dot], r) and satisfies Lipschitz
condition with respect to the last two arguments.
The necessary and sufficient conditions of the unique solvability
of the linear problem (1)-(3) are obtained in the terms of
coefficients and initial dates in [1-3].
The sufficient conditions of the existence and uniqueness of the
periodic solution problem (1)- (3) are established in the terms of
data and algorithm finding its solution are proposed.
1. Asanova A. T., Dzhumabaev D. S. Unique Solvability of the
Boundary Value Problem for Systems of Hyperbolic Equations with
Data on the Characteristics. Computational Mathematics
and Mathematical Physics, 42 (11) (2002), 1609-1621.
2. Asanova A. T., Dzhumabaev D. S. Unique Solvability of the
Nonlocal boundary Value Problems for Systems of Hyperbolic
Equations. Differential Equations, 39 (10) (2003),
1343-1354.
3. Asanova A. T., Dzhumabaev D. S. Correct Solvability of a
Nonlocal Boundary Value Problem for Systems of Hyperbolic
Equations. Doklady Mathematics, 68 (1) (2003), 46-49.
Date received: March 11, 2004
Copyright © 2004 by the author(s).
The author(s) of this document and the organizers of the conference
have granted their consent to include this abstract in
Atlas Conferences Inc.
Document # canu-32.