scholarly journals Loss of derivatives for hyperbolic boundary problems with constant coefficients

2018 ◽  
Vol 23 (3) ◽  
pp. 1347-1361
Author(s):  
Matthias Eller ◽  
Keyword(s):  
Boundary Problems ◽  
Loss Of Derivatives ◽  
Constant Coefficients ◽  
Hyperbolic Boundary

Correct Multilevel Discrete-Continual Finite Element Method of Structural Analysis

2014 ◽  
Vol 1040 ◽  
pp. 664-669 ◽  
Author(s):  
Pavel A. Akimov ◽  
Alexandr M. Belostosky ◽  
Marina L. Mozgaleva ◽  
Mojtaba Aslami ◽  
Oleg A. Negrozov
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Structural Analysis ◽  
Analytical Solutions ◽  
Computational Stability ◽  
Boundary Problems ◽  
Piecewise Constant ◽  
Computer Realization ◽  
Constant Coefficients ◽  
Element Method

The distinctive paper is devoted to correct multilevel discrete-continual finite element method (DCFEM) of structural analysis based on precise analytical solutions of resulting multipoint boundary problems for systems of ordinary differential equations with piecewise-constant coefficients. Corresponding semianalytical (discrete-continual) formulations are contemporary mathematical models which currently becoming available for computer realization. Major peculiarities of DCFEM include universality, computer-oriented algorithm involving theory of distributions, computational stability, optimal conditionality of resulting systems and partial Jordan decompositions of matrices of coefficients, eliminating necessity of calculation of root vectors.

Weakly stable hyperbolic boundary problems with large oscillatory coefficients: Simple cascades

2020 ◽  
Vol 17 (01) ◽  
pp. 141-183 ◽  
Author(s):  
Mark Williams
Keyword(s):  
Exact Solutions ◽  
Approximate Solutions ◽  
Energy Estimates ◽  
Time Interval ◽  
Geometric Optics ◽  
Zeroth Order ◽  
Boundary Problems ◽  
First Order ◽  
Weakly Stable ◽  
Hyperbolic Boundary

We prove energy estimates for exact solutions to a class of linear, weakly stable, first-order hyperbolic boundary problems with “large”, oscillatory, zeroth-order coefficients, that is, coefficients whose amplitude is large, [Formula: see text], compared to the wavelength of the oscillations, [Formula: see text]. The methods that have been used previously to prove useful energy estimates for weakly stable problems with oscillatory coefficients (e.g. simultaneous diagonalization of first-order and zeroth-order parts) all appear to fail in the presence of such large coefficients. We show that our estimates provide a way to “justify geometric optics”, that is, a way to decide whether or not approximate solutions, constructed for example by geometric optics, are close to the exact solutions on a time interval independent of [Formula: see text]. Systems of this general type arise in some classical problems of “strongly nonlinear geometric optics” coming from fluid mechanics. Special assumptions that we make here do not yet allow us to treat the latter problems, but we believe the present analysis will provide some guidance on how to attack more general cases.

Correct Method of Analytical Solution of Multipoint Boundary Problems of Structural Analysis for Systems of Ordinary Differential Equations with Piecewise Constant Coefficients

2011 ◽  
Vol 250-253 ◽  
pp. 3652-3655 ◽  
Author(s):  
Pavel A. Akimov ◽  
Vladimir N. Sidorov
Keyword(s):  
Structural Analysis ◽  
Analytical Solution ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Computational Stability ◽  
Boundary Problems ◽  
Piecewise Constant ◽  
Multipoint Boundary ◽  
Correct Method ◽  
Constant Coefficients

This paper is devoted to correct method of analytical solution of multipoint boundary problems of structural analysis for systems of ordinary differential equations with piecewise constant coefficients. Its major peculiarities include universality, computer-oriented algorithm involving theory of distributions, computational stability, optimal conditionality of resultant systems and partial Jordan decomposition of matrix of coefficients, eliminating necessity of calculation of root vectors.

scholarly journals WKB expansions for weakly well-posed hyperbolic boundary value problems in a strip: Time depending loss of derivatives

2021 ◽  
Vol 18 (03) ◽  
pp. 557-608
Author(s):  
Antoine Benoit
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Step Function ◽  
Geometric Optics ◽  
Loss Of Derivatives ◽  
Hyperbolic Boundary ◽  
Well Posed

We are interested in geometric optics expansions for linear hyperbolic systems of equations defined in the strip [Formula: see text]. More precisely the aim of this paper is to describe the influence of the boundary conditions on the behavior of the solution. This question has already been addressed in [A. Benoit, Wkb expansions for hyperbolic boundary value problems in a strip: Selfinteraction meets strong well-posedness, J. Inst. Math. Jussieu 19(5) (2020) 1629–1675] for stable boundary conditions. Here we do not require that the boundary conditions lead to strongly well-posed problems but only to weakly well-posed problems (that is loss(es) of derivatives are possible). The question is thus to determine what can be the minimal loss of derivatives in the energy estimate of the solution. The main result of this paper is to show, thanks to geometric optics expansions, that if the strip problem admits a boundary in the so-called [Formula: see text]-class of [S. Benzoni-Gavage, F. Rousset, D. Serre and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems, Proc. Roy. Soc. Edinburgh Sect. A 132(5) (2002) 1073–1104] then the loss of derivatives shall be at least increasing with the time of resolution. More precisely this loss is bounded by below by a step function increasing with respect to time which depends on the minimal time needed to perform a full regeneration of the wave packet.

Correct Wavelet-based Multilevel Discrete-Continual Methods for Local Solution of Boundary Problems of Structural Analysis

2013 ◽  
Vol 353-356 ◽  
pp. 3224-3227 ◽  
Author(s):  
Pavel A. Akimov ◽  
Marina L. Mozgaleva
Keyword(s):  
Structural Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Problem ◽  
Local Solution ◽  
Wavelet Basis ◽  
Boundary Problems ◽  
Piecewise Constant ◽  
Constant Coefficients ◽  
Operational Formulation

The distinctive paper is devoted to correct wavelet-based multilevel discrete-continual methods for local solution of boundary problems of structural analysis. Initial discrete-continual operational formulation of the considering problem and corresponding operational formulation with the use of wavelet basis are presented. Due to special algorithms of averaging within multigrid approach, reduction of the problem is provided. Resultant multipoint boundary problem of structural mechanics for system of ordinary differential equations with piecewise-constant coefficients is given.

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