The initial-boundary value problem for the “good” Boussinesq equation on the bounded domain

The initial-boundary value problem for the “good” Boussinesq equation on the bounded domain is studied in this article. The local and global well-posedness of this initial-boundary value problem is given.     

The displacement initial-boundary value problem in bending of thermoelastic plates weakened by cracks

The displacement initial-boundary value problem for bending of a thermoelastic plate with transverse shear deformation weakened by a crack is studied, and its unique solvability in spaces of distributions is proved by means of a combination of the Laplace transformation and variational methods.     

An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations

The time-harmonic Maxwell equations are considered in the low-frequency case. A finite element domain decomposition approach is proposed for the numerical approximation of the exact solution. This leads to an iteration-by-subdomain procedure, which is proven to converge. The rate of convergence turns out to be independent of the mesh size, showing that the preconditioner implicitly defined by the iterative procedure is optimal. For obtaining this convergence result it has been necessary to prove a regularity theorem for Dirichlet and Neumann harmonic fields.

     

ON AN INVERSE INITIAL-BOUNDARY VALUE PROBLEM FOR A ONE-DIMENSIONAL HYPERBOLIC SYSTEM OF DIFFERENTIAL EQUATIONS

1.IntroductionTheproblemstodeterminethecoefficientsindifferentialequationsfromknownfunctionaloftheirsolutionsareoftencalledinverseproblems.Amongtheseproblemsthesimplestisaboutone-dimensionalwaveequations.Thisproblemcanbediscussedinthetimedomainorinthefreq…     

Numerical simulation for the initial-boundary value problem of the Klein-Gordon-Zakharov equations

In this paper,a new conservative finite difference scheme with a parameter θ is proposed for the initial-boundary problem of the Klein-Gordon-Zakharov (KGZ) equations.Convergence of the numerical solutions are proved with order O(h 2 + τ 2) in the energy norm.Numerical results show that the scheme is accurate and efficient.     

Solvability of initial-boundary value problems for bending of thermoelastic plates with mixed boundary conditions

Initial-boundary value problems for bending of a thermoelastic plate with transverse shear deformation are studied under the assumption that various parts of the boundary are subjected to different types of physical conditions. The unique solvability of these problems is established in spaces of distributions by means of a combination of the Laplace transform and variational methods.     

Initial boundary value problem for modified Zakharov equations

In this paper the authors consider the existence and uniqueness of the solution to the initial boundary value problem for a class of modified Zakharov equations, prove the global existence of the solution to the problem by a priori integral estimates and Galerkin method.     

Local existence of solutions of impedance initial-boundary value problem for non-linear Maxwell equations

Local existence of solutions of a mixed problem for non-linear Maxwell equations is proved. The solution is so regular that its third derivatives are from L₂. However, the considered problem is characteristic we were able to obtain necessary estimate because compatibility conditions were used.     

The initial-boundary value problems for a class of nonlinear wave equations with damping term

In this paper we study the existence and uniqueness of the global generalized solution and the global classical solution, the blowup of the solution and the energy decay of the solutions of the initial-boundary value problems for a class of nonlinear wave equations.     

Initial-boundary value problem and integrodifferential equations of electrodynamics for an inhomogeneous conductive body

The initial-boundary value problem of determining the electromagnetic field in an inhomogeneous conducting sample and the surrounding external medium is solved under certain assumptions on the sample, the external medium, and the external current density. The existence of a classical solution to this problem is proved. The electromagnetic field under small variations in the sample’s electric conductivity is computed by applying perturbation theory.     

Nonlocal overdetermined boundary value problem for stationary Navier-Stokes equations

A stationary system of Stokes and Navier-Stokes equations describing the flow of a homogeneous incompressible fluid in a bounded domain is considered. The vector of the flow velocity and a finite number of nonlocal conditions are defined at a part of the domain boundary. It is proved that, in the linear case, the problem has at least one stable solution. In the nonlinear case, the local solvability of the problem is proved.     

Continuation of a solution to a homogeneous system of Maxwell equations

We consider the problem on the continuation of a solution to a system of Maxwell equations, using its values on a part of the domain boundary.     

Exponential decay for Maxwell equations with a boundary memory condition

We study the asymptotic behavior of the solution of the Maxwell equations with the following boundary condition of memory type:

(0.1)     

Analysis of an optimal control problem for the three-dimensional coupled modified Navier-Stokes and Maxwell equations

The mathematical formulation and analysis of an optimal control problem associated with a viscous, incompressible, electrically conducting fluid in a bounded three-dimensional domain with fixed perfectly conducting boundaries is considered. The objective of control is the matching of the velocity and magnetic fields to given target fields; control is effected through distributed mechanical force and current controls. The existence of optimal solutions is shown, the Gâteaux differentiability for the magnetohydrodynamic system with respect to controls is proved, and the optimality system is obtained.     

A posteriori error estimates for Maxwell equations

Maxwell equations are posed as variational boundary value problems in the function space and are discretized by Nédélec finite elements. In Beck et al., 2000, a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain. In the present paper, we prove the reliability of that error estimator on Lipschitz domains. The key is to establish new error estimates for the commuting quasi-interpolation operators recently introduced in J. Schöberl, Commuting quasi-interpolation operators for mixed finite elements. Similar estimates are required for additive Schwarz preconditioning. To incorporate boundary conditions, we establish a new extension result.

     

An initial boundary-value problem for a system of equations of magnetohydrodynamics

In the present paper we consider one initial boundary-value problem for a system of equations of magnetohydrodynamics in the case where it is necessary to take into account the displacement currents in the Maxwell system of equations. We prove a local (in time) unique solvability of this problem in the Sobolev spaces. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 2, pp. 228–254, April–June, 2000. Translated by R. Lapinskas     

Initial-boundary value problem for the nonlinear string-beam system

This paper deals with a nonlinear string-beam system describing the torsional-vertical oscillations of a suspension bridge. We consider the initial-boundary value problem and study the existence and uniqueness question. We assume time independent right hand sides, but allow quite general nonlinear terms. Using the Faedo-Galerkin method we prove the existence of a unique solution on an arbitrary large time interval.