ON BOUNDARY CONDITIONS FOR FIRST-ORDER SYMMETRIC HYPERBOLIC SYSTEMS WITH CONSTRAINTS

2013 ◽  
Vol 10 (04) ◽  
pp. 725-734 ◽  
Author(s):  
NICOLAE TARFULEA
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Hyperbolic Systems ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
First Order ◽  
The Cauchy Problem ◽  
Initial Boundary Value ◽  
Well Posed

The Cauchy problem for many first-order symmetric hyperbolic (FOSH) systems is constraint preserving, i.e. the solution satisfies certain spatial differential constraints whenever the initial data does. Frequently, artificial space cut-offs are performed for such evolution systems, usually out of the necessity for finite computational domains. However, it may easily happen that boundary conditions at the artificial boundary for such a system lead to an initial boundary value problem which, while well-posed, does not preserve the constraints. Here we consider the problem of finding constraint-preserving boundary conditions for constrained FOSH systems in the well-posed class of maximal non-negative boundary conditions. Based on a characterization of maximal non-negative boundary conditions, we discuss a systematic technique for finding such boundary conditions that preserve the constraints, pending that the constraints satisfy a FOSH system themselves. We exemplify this technique by analyzing a system of wave equations in a first-order formulation subject to divergence constraints.

scholarly journals Countable hyperbolic systems in the theory of nonlinear oscillations

2016 ◽  
Vol 8 (1) ◽  
pp. 163-171
Author(s):  
T.I. Firman
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Order Differential Equation ◽  
Nonlinear Oscillations ◽  
Hyperbolic Systems ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
First Order ◽  
Initial Boundary Value ◽  
Fourth Order Differential Equation

In this article a model example of a mixed problem for a fourth-order differential equation reduced to initial-boundary value problem for countable hyperbolic system of first order coherent differential equations.

WELL-POSEDNESS OF AN INITIAL BOUNDARY VALUE PROBLEM FROM LASER DYNAMICS

2002 ◽  
Vol 12 (04) ◽  
pp. 593-606 ◽  
Author(s):  
F. JOCHMANN ◽  
L. RECKE
Keyword(s):  
Boundary Conditions ◽  
Semiconductor Lasers ◽  
Dynamical Behavior ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Weak Formulation ◽  
First Order ◽  
Laser Dynamics ◽  
Regularity Properties ◽  
Initial Boundary Value

In this paper a mathematical model, consisting of nonlinear first-order ordinary and partial differential equations with initial and boundary conditions, for the dynamical behavior of multisection DFB (distributed feedback) semiconductor lasers is investigated. We introduce a suitable weak formulation and prove existence, uniqueness and regularity properties of the solutions. The assumptions on the data are quite general, in particular, the physically relevant case of piecewise smooth, but discontinuous with respect to space and time coefficients in the equations and in the boundary conditions is included.

scholarly journals WELL-POSED INITIAL-BOUNDARY VALUE PROBLEM FOR A CONSTRAINED EVOLUTION SYSTEM AND RADIATION-CONTROLLING CONSTRAINT-PRESERVING BOUNDARY CONDITIONS

2007 ◽  
Vol 04 (04) ◽  
pp. 587-612 ◽  
Author(s):  
ALEXANDER M. ALEKSEENKO
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value ◽  
Time Derivative ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Evolution System ◽  
Constrained Evolution ◽  
Initial Boundary Value ◽  
Well Posed

A well-posed initial-boundary value problem is formulated for the model problem of the vector wave equation subject to the divergence-free constraint. Existence, uniqueness and stability of the solution is proved by reduction to a system evolving the constraint quantity statically, i.e. the second time derivative of the constraint quantity is zero. A new set of radiation-controlling constraint-preserving boundary conditions is constructed for the free evolution problem. Comparison between the new conditions and the standard constraint-preserving boundary conditions is made using the Fourier–Laplace analysis and the power series decomposition in time. The new boundary conditions satisfy the Kreiss condition and are free from the ill-posed modes growing polynomially in time.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

scholarly journals An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

2021 ◽  
Vol 5 (3) ◽  
pp. 63
Author(s):  
Emilia Bazhlekova
Keyword(s):  
Boundary Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability Estimates ◽  
Inverse Source Problem ◽  
Time Data ◽  
Final Time ◽  
Generalized Eigenfunction ◽  
The One ◽  
Initial Boundary Value

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.

scholarly journals Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains

2020 ◽  
Vol 40 (6) ◽  
pp. 725-736
Author(s):  
Mitsuhiro Nakao
Keyword(s):  
Existence And Uniqueness ◽  
Wave Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Energy ◽  
Time Dependent ◽  
Energy Solution ◽  
Noncylindrical Domain ◽  
Finite Energy Solution ◽  
Initial Boundary Value

We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain \(\bigcup_{0\leq t \lt\infty} \Omega(t)\times\{t\} \subset \mathbb{R}^N\times \mathbb{R}\). We are interested in finite energy solution. We derive an exponential decay of the energy in the case \(\Omega(t)\) is bounded in \(\mathbb{R}^N\) and the estimate \[\int\limits_0^{\infty} E(t)dt \leq C(E(0),\|u(0)\|)< \infty\] in the case \(\Omega(t)\) is unbounded. Existence and uniqueness of finite energy solution are also proved.

A Comprehensive Energy Formulation for General Nonlinear Material Continua

1997 ◽  
Vol 64 (2) ◽  
pp. 353-360 ◽  
Author(s):  
A. Carini ◽  
O. De Donato
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Small Strain ◽  
Constitutive Law ◽  
Nonlinear Material ◽  
Total Potential ◽  
Initial Boundary Value ◽  
Energy Formulation ◽  
The Continuum

By specialization to the continuum problem of a general formulation of the initial/boundary value problem for every nonpotential operator (Tonti, 1984) and by virtue of a suitable choice of the “integrating operator,” a comprehensive energy formulation is established. Referring to the small strain and displacement case in the presence of any inelastic generally nonlinear constitutive law, provided that it is differentiable, this formulation allows us to derive extensions of well-known principles of elasticity (Hu-Washizu, Hellinger-Reissner, total potential energy, and complementary energy). An illustrative example is given. Peculiar properties of the formulation are the energy characterization of the functional and the use of Green functions of the same problem in the elastic range for every inelastic, generally nonlinear material considered.

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