Stabilized dual-mixed method for the problem of linear elasticity with mixed boundary conditions

We extend the applicability of the augmented dual-mixed method introduced recently in Gatica (2007), Gatica et al. (2009) to the problem of linear elasticity with mixed boundary conditions. The method is based on the Hellinger–Reissner principle and the symmetry of the stress tensor is imposed in a weak sense. The Neumann boundary condition is prescribed in the finite element space. Then, suitable Galerkin least-squares type terms are added in order to obtain an augmented variational formulation which is coercive in the whole space. This allows to use any finite element subspaces to approximate the displacement, the Cauchy stress tensor and the rotation.     

The singular sources method for an inverse problem with mixed boundary conditions

We use the singular sources method to detect the shape of the obstacle in a mixed boundary value problem. The basic idea of the method is based on the singular behavior of the scattered field of the incident point-sources on the boundary of the obstacle. Moreover we take advantage of the scattered field estimate by the backprojection operator. Also we give a uniqueness proof for the shape reconstruction.     

Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions

Equations of linear and nonlinear infinitesimal elasticity with mixed boundary conditions are considered. The bounded domain is assumed to have a Lipschitz boundary and to satisfy additional regularity assumptions. W1,p regularity for the displacements and Lp regularity for the stresses are proved for some p>2.     

A plane problem of uncoupled thermomagnetoelasticity for an infinite,elliptical cylinder carrying a steady axial current by a boundary integral method

A boundary integral method earlier proposed by two of the authors is used to solve a problem of uncoupled magnetothermoelasticity for an infinite, elliptical cylindrical conductor carrying a steady axial, uniform electric current. The cylinder is placed in a variable ambient temperature and is allowed to exchange heat with the surrounding medium.     

Solvability of the boundary value problem for stationary magnetohydrodynamic equations under mixed boundary conditions for the magnetic field

The global solvability of the boundary value problem for stationary magnetohydrodynamic equations under the Dirichlet boundary condition for the velocity and mixed boundary conditions for the magnetic field is proved.     

Boundary feedback stabilization of the Boussinesq system with mixed boundary conditions

We study the feedback stabilization of the Boussinesq system in a two dimensional domain, with mixed boundary conditions. After ascertaining the precise loss of regularity of the solution in such models, we prove first Green's formulas for functions belonging to weighted Sobolev spaces and then correctly define the underlying control system. This provides a rigorous mathematical framework for models studied in the engineering literature. We prove the stabilizability by extending to the linearized Boussinesq system a local Carleman estimate already established for the Oseen system. Then we determine a feedback control law able to stabilize the linearized system around the stationary solution, with any prescribed exponential decay rate, and able to stabilize locally the nonlinear system.     

Controllability results for the two-dimensional heat equation with mixed boundary conditions using Carleman inequalities: a linear and a semilinear case

In this paper, we prove controllability results for a two-dimensional semilinear heat equation with mixed boundary conditions. It is well-known that mixed boundary conditions can present a singular behaviour of the solution. First, we will prove global Carleman estimates then we will use these inequalities to obtain controllability results.     

Numerical solution of the Goursat problem on a triangular domain with mixed boundary conditions

For the Goursat problem, we consider a triangular domain with mixed Dirichlet and impedance boundary conditions imposed on it. We develop an algorithm for its numerical solution mainly based on Runge-Kutta method and trapezoidal formula. Iterative techniques are constructed to compute some data for the nonlinear part of the differential equation and the impedance boundary condition. Error estimates are derived. Examples are presented to illustrate the effectiveness of the method.     

On the solutions of a boundary value problem of linear thermoelasticity system with nonlocal conditions

This paper studies a boundary value problem with nonlocal conditions for a coupled system of linear thermoelasticity in one-dimensional case. Using an a priori estimate, we prove the uniqueness of the solution. Also, some explicit solutions are obtained by using the separation method.     

ON THE FUNDAMENTAL PROBLEM FOR AN INFINITE ELASTIC PLANE BONDED BY DIFFERENT ANISOTROPIC MATERIALS WITH CRACKS

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An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions

In this paper, the inverse problem of finding the time‐dependent coefficient of heat capacity together with the solution of heat equation with nonlocal boundary and overdetermination conditions is considered. The existence, uniqueness and continuous dependence upon the data are studied. Some considerations on the numerical solution for this inverse problem are presented with the examples. Copyright © 2010 John Wiley & Sons, Ltd.     

On the numerical solution of a boundary integral equation for the exterior Neumann problem on domains with corners

The authors propose a “modified” Nyström method to approximate the solution of a boundary integral equation connected with the exterior Neumann problem for Laplace's equation on planar domains with corners. They prove the convergence and the stability of the method and show some numerical tests.     

Method of boundary integral equations as applied to the numerical solution of the three-dimensional Dirichlet problem for the laplace equation in a piecewise homogeneous medium

A Dirichlet problem is considered in a three-dimensional domain filled with a piecewise homogeneous medium. The uniqueness of its solution is proved. A system of Fredholm boundary integral equations of the second kind is constructed using the method of surface potentials, and a system of boundary integral equations of the first kind is derived directly from Green’s identity. A technique for the numerical solution of integral equations is proposed, and results of numerical experiments are presented.     

The effect of boundary conditions on the mesoscopic lattice Boltzmann method: Case study of a reaction-diffusion based model for Min-protein oscillation

Min-protein oscillation in Escherichia coli has an essential role in controlling the accurate placement of the cell division septum at the middle-cell zone of the bacteria. This biochemical process has been successfully described by a set of reaction-diffusion equations at the macroscopic level. The lattice Boltzmann method (LBM) has been used to simulate Min-protein oscillation and proved to recover the correct macroscopic equations. In this present work, we studied the effects of LBM boundary conditions (BC) on Min-protein oscillation. The impact of diffusion and reaction dynamics on BCs was also investigated. It was found that the mirror-image BC is a suitable boundary treatment for this Min-protein model. The physical significance of the results is extensively discussed.     

Approximate method for solving the boundary value problem for a parabolic equation with inhomogeneous transmission conditions of nonideal contact type

A linear parabolic equation in a disconnected domain with inhomogeneous transmission conditions of the nonideal contact type is studied. A generalized formulation of the problem is considered. An analogue of the Galerkin method is proposed for solving the problem, and the stability of the method is investigated. This makes it possible to prove existence and uniqueness theorems for the equation under different assumptions on the data smoothness.     

On the convergence rate and optimization of a numerical method with splitting of boundary conditions for the stokes system in a spherical layer in the axisymmetric case: Modification for thick layers

The convergence rate of a fast-converging second-order accurate iterative method with splitting of boundary conditions constructed by the authors for solving an axisymmetric Dirichlet boundary value problem for the Stokes system in a spherical gap is studied numerically. For R/r exceeding about 30, where r and R are the radii of the inner and outer boundary spheres, it is established that the convergence rate of the method is lower (and considerably lower for large R/r) than the convergence rate of its differential version. For this reason, a really simpler, more slowly converging modification of the original method is constructed on the differential level and a finite-element implementation of this modification is built. Numerical experiments have revealed that this modification has the same convergence rate as its differential counterpart for R/r of up to 5 × 10³. When the multigrid method is used to solve the split and auxiliary boundary value problems arising at iterations, the modification is more efficient than the original method starting from R/r ～ 30 and is considerably more efficient for large values of R/r. It is also established that the convergence rates of both methods depend little on the stretching coefficient η of circularly rectangular mesh cells in a range of η that is well sufficient for effective use of the multigrid method for arbitrary values of R/r smaller than ～ 5 × 10³.     

A collocation method based on the Bessel functions of the first kind for singular perturbated differential equations and residual correction

In this paper, a collocation method is given to solve singularly perturbated two‐point boundary value problems. By using the collocation points, matrix operations and the matrix relations of the Bessel functions of the first kind and their derivatives, the boundary value problem is converted to a system of the matrix equations. By solving this system, the approximate solution is obtained. Also, an error problem is constructed by the residual function, and it is solved by the presented method. Thus, the error function is estimated, and the approximate solutions are improved. Finally, numerical examples are given to show the applicability of the method, and also, our results are compared by existing results. Copyright © 2014 JohnWiley & Sons, Ltd.