scholarly journals Finite-dimensional attractors for a general class of nonautonomous wave equations

2000 ◽  
Vol 13 (5) ◽  
pp. 17-22 ◽  
Author(s):  
A. Eden ◽  
V. Kalantarov ◽  
A. Miranville
Keyword(s):  
General Class ◽  
Wave Equations ◽  
Finite Dimensional

Solving linear systems from dynamical energy analysis - using and reusing preconditioners

2021 ◽  
Vol 263 (5) ◽  
pp. 1041-1052
Author(s):  
Martin Richter ◽  
Gregor Tanner ◽  
Bruno Carpentieri ◽  
David J. Chappell
Keyword(s):  
Wave Equations ◽  
Energy Analysis ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Iterative Solvers ◽  
Computational Method ◽  
Computational Time ◽  
Shell Elements ◽  
Finite Dimensional ◽  
Large Matrix

Dynamical energy analysis (DEA) is a computational method to address high-frequency vibro-acoustics in terms of ray densities. It has been used to describe wave equations governing structure-borne sound in two-dimensional shell elements as well as three-dimensional electrodynamics. To describe either of those problems, the wave equation is reformulated as a propagation of boundary densities. These densities are expressed by finite dimensional approximations. All use-cases have in common that they describe the resulting linear problem using a very large matrix which is block-sparse, often real-valued, but non-symmetric. In order to efficiently use DEA, it is therefore important to also address the performance of solving the corresponding linear system. We will cover three aspects in order to reduce the computational time: The use of preconditioners, properly chosen initial conditions, and choice of iterative solvers. Especially the aspect of potentially reusing preconditioners for different input parameters is investigated.

scholarly journals Existence and approximation of attractors for nonlinear coupled lattice wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lianbing She ◽  
Mirelson M. Freitas ◽  
Mauricio S. Vinhote ◽  
Renhai Wang
Keyword(s):  
Asymptotic Behavior ◽  
Global Attractor ◽  
Wave Equations ◽  
Upper Semicontinuity ◽  
Asymptotic Behavior Of Solutions ◽  
Well Posedness ◽  
Asymptotic Compactness ◽  
Finite Dimensional ◽  
Lattice Wave ◽  
Infinite Lattices

<p style='text-indent:20px;'>This paper is concerned with the asymptotic behavior of solutions to a class of nonlinear coupled discrete wave equations defined on the whole integer set. We first establish the well-posedness of the systems in <inline-formula><tex-math id="M1">\begin{document}$ E: = \ell^2\times\ell^2\times\ell^2\times\ell^2 $\end{document}</tex-math></inline-formula>. We then prove that the solution semigroup has a unique global attractor in <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula>. We finally prove that this attractor can be approximated in terms of upper semicontinuity of <inline-formula><tex-math id="M3">\begin{document}$ E $\end{document}</tex-math></inline-formula> by a finite-dimensional global attractor of a <inline-formula><tex-math id="M4">\begin{document}$ 2(2n+1) $\end{document}</tex-math></inline-formula>-dimensional truncation system as <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula> goes to infinity. The idea of uniform tail-estimates developed by Wang (Phys. D, 128 (1999) 41-52) is employed to prove the asymptotic compactness of the solution semigroups in order to overcome the lack of compactness in infinite lattices.</p>

scholarly journals A GENERALIZATION OF THE THEORY OF STANDARDLY STRATIFIED ALGEBRAS I: STANDARDLY STRATIFIED RINGOIDS

2020 ◽  
pp. 1-36
Author(s):  
O. MENDOZA ◽  
M. ORTÍZ ◽  
C. SÁENZ ◽  
V. SANTIAGO
Keyword(s):  
Classical Theory ◽  
General Class ◽  
General Setting ◽  
Finite Dimensional ◽  
Classical Notion ◽  
Hereditary Algebras

Abstract We extend the classical notion of standardly stratified k-algebra (stated for finite dimensional k-algebras) to the more general class of rings, possibly without 1, with enough idempotents. We show that many of the fundamental results, which are known for classical standardly stratified algebras, can be generalized to this context. Furthermore, new classes of rings appear as: ideally standardly stratified and ideally quasi-hereditary. In the classical theory, it is known that quasi-hereditary and ideally quasi-hereditary algebras are equivalent notions, but in our general setting, this is no longer true. To develop the theory, we use the well-known connection between rings with enough idempotents and skeletally small categories (ringoids or rings with several objects).

CONTROLLING IDEAL TURBULENCE IN TIME-DELAYED CHUA'S CIRCUIT: STABILIZATION AND SYNCHRONIZATION

2010 ◽  
Vol 20 (05) ◽  
pp. 1351-1363 ◽  
Author(s):  
MASAYASU SUZUKI ◽  
NOBORU SAKAMOTO
Keyword(s):  
Wave Equations ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Distributed Parameter ◽  
Distributed Parameter System ◽  
Parameter System ◽  
Equilibrium Solutions ◽  
Control Laws ◽  
Finite Dimensional ◽  
Steady Solutions

We try to stabilize steady solutions of a physical model described by wave equations with nonlinear boundary conditions. This system is a distributed parameter system in which ideal turbulence, introduced by Sharkovsky et al., occurs. Although the behavior of the system is quite intricate both in time and space, by using d'Alembert's solution, the analysis of the dynamic characteristics can be reduced to that of a finite-dimensional difference equation. In this report, based on this analytical method using d'Alembert's solution, we design control laws to stabilize steady solutions (equilibrium solutions and periodic solutions) and synchronize a pair of identical systems.

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