This paper is concerned with initial-boundary-valueproblems (IBVPs) for a class of nonlinear Schrodinger equations posed either on a half line R+ or on a bounded interval (0, L) with nonhomogeneous boundary condition...
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This paper is concerned with initial-boundary-valueproblems (IBVPs) for a class of nonlinear Schrodinger equations posed either on a half line R+ or on a bounded interval (0, L) with nonhomogeneous boundary conditions. For any s with 0 <= s < 5/2 and s not equal 1/2, it is shown that the relevant IBVPs are locally well-posed if the initial data lie in the L-2-based Sobolev spaces H-s (R+) in the case of the half line and in H-s (0, L) on a bounded interval, provided the boundary data are selected from H-loc((2s + 1) / 4) (R+) and H-loc((s+1)/2)(R+), respectively. (For s > 1/2, compatibility between the initial and boundary conditions is also needed.) Global well-posedness is also discussed when s >= 1. From the point of view of the well-posedness theory, the results obtained reveal a significant difference between the IBVP posed on R+ and the IBVP posed on (0, L). The former is reminiscent of the theory for the pure initial-value problem (IVP) for these Schrodinger equations posed on the whole line R while the theory on a bounded interval looks more like that of the pure IVP posed on a periodic domain. In particular, the regularity demanded of the boundary data for the IBVP on R+ is consistent with the temporal trace results that obtain for solutions of the pure IVP on R, while the slightly higher regularity of boundary data for the IBVP on (0, L) resembles what is found for temporal traces of spatially periodic solutions. Published by Elsevier Masson SAS.
An initial-boundaryvalue problem with homogeneous Dirichlet boundary conditions for three-dimensional Zakharov-Kuznetsov equation is considered. Results on global existence, uniqueness and large-time decay of weak so...
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An initial-boundaryvalue problem with homogeneous Dirichlet boundary conditions for three-dimensional Zakharov-Kuznetsov equation is considered. Results on global existence, uniqueness and large-time decay of weak solutions in certain weighted spaces are established. (C) 2015 Elsevier Inc. All rights reserved.
We prove a theorem about existence, uniqueness and regularity of the solution to an initial-boundaryvalue problem for a nonlinear coupled parabolic system consisting of two equations. Such a system appears in the the...
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We prove a theorem about existence, uniqueness and regularity of the solution to an initial-boundaryvalue problem for a nonlinear coupled parabolic system consisting of two equations. Such a system appears in the thermodiffusion in solid body. In our proof we use an energy method, methods of Sobolev spaces, semigroup theory and the Banach fixed point theorem.
The stability theory for difference approximations of hyperbolic initial boundary value problems is based on normal mode analysis. To perform such a stability investigation analytically is very difficult even for low-...
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The stability theory for difference approximations of hyperbolic initial boundary value problems is based on normal mode analysis. To perform such a stability investigation analytically is very difficult even for low-order approximations of scalar problems. For more complicated cases some numerical technique must be used. Here, a numerical algorithm designed for this purpose is presented. It can handle one- and two-dimensional problems and can easily be extended to higher-dimensional cases. The algorithm is justified by a theoretical analysis and experiments show that it is reliable and efficient.
Artificial boundary conditions for the linearized incompressible Navier–Stokes equations are designed by approximating the symbol of the transparent operator. The related initial boundary value problems are well pose...
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Artificial boundary conditions for the linearized incompressible Navier–Stokes equations are designed by approximating the symbol of the transparent operator. The related initial boundary value problems are well posed in the same spaces as the original Cauchy problem. Furthermore, error estimates for small viscosity are proved.
We consider the problem of constructing stable difference methods for the initialboundaryvalue problem for the linearized equations of gas dynamics in one space dimension using the implicit time differencing methods...
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We consider the problem of constructing stable difference methods for the initialboundaryvalue problem for the linearized equations of gas dynamics in one space dimension using the implicit time differencing methods considered by Beam and Warming [2]. Centered spacial differences are used in the interior. We investigate the stability of this class with two forms of extrapolation for the scalar outflow problem. (We consider the problem of specifying data in the primitive variables and computing in terms of the conservative variables in the interior.) We show that the whole class of methods is stable for the subsonic inflow and outflow problems with various data specifications and extrapolation methods. We also show that the methods considered are stable for the solid wall boundary problem when we set u=0
gas dynamics
initial boundary value problems
difference methods
In this paper we study the order reduction, caused by the presence of time dependent boundary conditions, in the integration of linear parabolic problems whose coefficients may depend on time by means of fractional st...
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In this paper we study the order reduction, caused by the presence of time dependent boundary conditions, in the integration of linear parabolic problems whose coefficients may depend on time by means of fractional step Runge-Kutta methods. This kind of methods includes most of classical splitting, alternating direction or fractional step schemes, as well as new methods of the same types but with higher orders of accuracy. It is proven that such order reduction can be avoided by modifying suitably the commonly chosen boundaryvalues for the calculus of the internal stages (or the intermediate fractionary steps) of the method. Some numerical examples are presented in order to show the practical implications of the theoretical achievements. (C) 2003 IMACS. Published by Elsevier B.V. All rights reserved.
We consider a nonlinear coupled system of evolution equations, the simplest of which models a thermoelastic plate. Smoothing and decay properties of solutions are investigated as well as the local well-posedness and t...
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We consider a nonlinear coupled system of evolution equations, the simplest of which models a thermoelastic plate. Smoothing and decay properties of solutions are investigated as well as the local well-posedness and the global existence of solutions. For the system of standard thermoelasticity it is proved that there is no similar smoothing effect.
The work is devoted to the study of the solvability of initial-boundaryvalueproblems for differential equations h(t)u(tt) - ( alpha partial derivative/partial derivative t + b ) Delta u + cu = f(x, t) (Delta is the ...
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The work is devoted to the study of the solvability of initial-boundaryvalueproblems for differential equations h(t)u(tt) - ( alpha partial derivative/partial derivative t + b ) Delta u + cu = f(x, t) (Delta is the Laplace operator in space variables) with a non-negative function h(t). Similar equations are called pseudohyperbolic equations in the literature. The aim of the work is to prove the existence and uniqueness of regular solutions of the problems under study-solutions that have all S.L. Sobolev derivatives included in the equation.
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