A survey on free boundary identification of the truncated boundary in numerical BVPs on infinite intervals

A free boundary formulation for the numerical solution of boundary value problems on infinite intervals was proposed recently in Fazio (SIAM J. Numer. Anal. 33 (1996) 1473). We consider here a survey on recent developments related to the free boundary identification of the truncated boundary. The goals of this survey are: to recall the reasoning for a free boundary identification of the truncated boundary, to report on a comparison of numerical results obtained for a classical test problem by three approaches available in the literature, and to propose some possible ways to extend the free boundary approach to the numerical solution of problems defined on the whole real line.     

On the asymptotic behavior of a simple eigenvalue of a boundary value problem in a domain perforated along the boundary

We consider a boundary value problem for the Laplace operator in a model domain periodically perforated along the boundary. We assume that the homogeneous Neumann condition is posed on the exterior boundary and the homogeneous Dirichlet condition is posed on the boundary of the cavities. We construct and justify the asymptotic expansions of eigenelements of the boundary value problem.     

An Iterative FEM for Solving Wave Problems of a Halfspace

An iterative finite element method for solving wave problems of a halfspace is presented in this paper. The halfspace is first truncated by introducing a fictive finite boundary on which some fictive boundary conditions must be imposed. A finite computational domain is in each iteration subjected to actual boundary conditions on real boundary and to fictive Dirichlet or Neumann boundary conditions on the fictive boundary. The radiation condition is satisfied by using DtN operator. The DtN operator is not introduce in the finite element formulation on the fictive boundary so any finite elements can be used. The method is simple and specially useful for computing higher harmonics.     

The nonlinear boundary layer to the Boltzmann equation with mixed boundary conditions for hard potentials

In this paper, the existence of boundary layer solutions to the Boltzmann equation for hard potential with mixed boundary condition, i.e., a linear combination of Dirichlet boundary condition and diffuse reflection boundary condition at the wall, is considered. The boundary condition is imposed on the incoming particles, and the solution is supposed to approach to a global Maxwellian in the far field. As for the problem with Dirichlet boundary condition (Chen et al., 2004 [5]), the existence of a solution highly depends on the Mach number of the far field Maxwellian. Furthermore, an implicit solvability condition on the boundary data which shows the codimension of the boundary data is related to the number of the positive characteristic speeds is also given.     

The Basic Mixed Problem for an Anisotropic Elastic Body

The mixed boundary value problem is considered for an anisotropic elastic body under the condition that a boundary value of the displacement vector is given on some part of the boundary and a boundary value of the generalized stress vector on the remainder. Using the potential method and the theory of singular integral equations with discontinuous coefficients, the existence of a solution of the mixed boundary value problem is proved.     

Micro/nano sliding plate problem with Navier boundary condition

For Newtonian flow through micro or nano sized channels, the no-slip boundary condition does not apply and must be replaced by a condition which more properly reflects surface roughness. Here we adopt the so-called Navier boundary condition for the sliding plate problem, which is one of the fundamental problems of fluid mechanics. When the no-slip boundary condition is used in the study of the motion of a viscous Newtonian fluid near the intersection of fixed and moving rigid plane boundaries, singular pressure and stress profiles are obtained, leading to a non-integrable force on each boundary. Here we examine the effects of replacing the no-slip boundary condition by a boundary condition which attempts to account for boundary slip due to the tangential shear at the boundary. The Navier boundary condition, possesses a single parameter to account for the slip, the slip length ℓ, and two solutions are obtained; one integral transform solution and a similarity solution which is valid away from the corner. For the former the tangential stress on each boundary is obtained as a solution of a set of coupled integral equations. The particular case solved is right-angled corner flow and equal slip lengths on each boundary. It is found that when the slip length is non-zero the force on each boundary is finite. It is also found that for a suffciently large distance from the corner the tangential stress on each boundary is equal to that of the classical solution. The similarity solution involves two restrictions, either a right-angled corner flow or a dependence on the two slip lengths for each boundary. When the tangential stress on each boundary is calculated from the similarity solution, it is found that the similarity solution makes no additional contribution to the tangential stress of that of the classical solution, thus in agreement with the findings of the integral transform solution. Values of the radial component of velocity along the line θ = π /4 for increasing distance from the corner for the similarity and integral transform solutions are compared, confirming their agreement for sufficiently large distances from the corner. (Received: November 9, 2005)     

Exact boundary controllability for 1‐D quasilinear wave equations with dynamical boundary conditions

By equivalently replacing the dynamical boundary condition by a kind of nonlocal boundary conditions, and noting a hidden regularity of solution on the boundary with a dynamical boundary condition, a constructive method with modular structure is used to get the local exact boundary controllability for 1‐D quasilinear wave equations with dynamical boundary conditions. Copyright © 2016 John Wiley & Sons, Ltd.     

Characterizations of boundary pluripolar hulls

We present some basic properties of the so-called boundary relative extremal function and discuss boundary pluripolar sets and boundary pluripolar hulls. We show that for B-regular domains the boundary pluripolar hull is always trivial on the boundary of the domain and present a “boundary version” of Zeriahi’s theorem on the completeness of pluripolar sets.     

Partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls

In this paper, we propose the concept of partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls, and make a deep discussion on it. We analyze the relation-ship between the partial approximate boundary synchronization and the partial exact boundary synchronization, and obtain sufficient conditions to realize the partial approximate boundary synchronization and necessary conditions of Kalman's criterion. In addition, with the help of partial synchronization decomposition, a condition that the approximately synchronizable state does not depend on the sequence of boundary controls is also given.     

AN ARTIFICIAL BOUNDARY CONDITION FOR THE INCOMPRESSIBLE VISCOUS FLOWS IN A NO-SLIP CHANNEL

1.IntroductionWhencomputingthenumericals0luti0nsofviscousfluidfl0wproblemsinallun-boundedd0main,0neoftenintroducesartificialboundaries,andsetsupanartificialbopundarycondition0nthem;thenthe0riginalproblemisreducedtoaproblemonab0undedc0mputationald0main.InordertoIimitthecomputatio11alcost,theseboundariesmustnotbet00farfromthedomainofinterest.Theref0re,theartificialboundaryc0nditi0nsmustbegoodapprotimationt0the"exact"boundaryconditions(sothatthes0lutionoftheproblemintheboundeddonlainisequaltothes…     

含开边界二维Stokes问题的Galerkin边界元解法

本文推导了含有开边界的二维有限域上Stokes问题的边界积分方程, 得出基于单层位势的第一类间接边界积分方程.对与之等价的边界变分方程用Galerkin边界元求解以得出单层位势的向量密度. 对于含有开边界端点的边界单元,采用特别的插值函数, 以模拟其固有的奇异性.论文用若干数值算例模拟了含有开边界的有限区域上不可压缩粘性流体的绕流.       

Adjoint and self-adjoint boundary value problems on a geometric graph

We consider boundary value problems of arbitrary order for linear differential equations on a geometric graph. Solutions of boundary value problems are coordinated at the interior vertices of the graph and satisfy given conditions at the boundary vertices. For considered boundary value problems, we construct adjoint boundary value problems and obtain a self-adjointness criterion. We describe the structure of the solution set of homogeneous self-adjoint boundary value problems with alternating coefficients of a differential equation and obtain nondegeneracy conditions for these boundary value problems.     

Well‐posedness of a parabolic free boundary problem driven by diffusion and surface tension

Well‐posedness and regularity results are shown for a class of free boundary problems consisting of diffusion on a free domain where the boundary movement depends on its mean curvature of the boundary and the diffusion on the boundary, and initial conditions are radially symmetric. Short‐time existence and uniqueness of solutions in a suitable Sobolev space are shown using a fixed‐point argument. Higher regularity is a posteriori. Finally, it is shown that solutions exist globally in time and converge to equilibrium if the boundary movement depends on the mean curvature of the boundary and diffusion in a specific way. A mathematical model describing the swelling of a cell due to osmosis is treated as an example. Copyright © 2014 John Wiley & Sons, Ltd.     

Limit Behaviour of Solutions to a Class of Equivalued Surface Boundary Value Problems for Parabolic Equations

In this paper, we discuss the limit behaviour of solutions for a class of equivalued surface boundary value problems for parabolic equations. When the equivalued surface boundary \overline{\Gamma}^\varepsilon_1 shrinks to a fixed point on boundary \Gamma_1, only homogeneous Neumann boundary conditions or Neumann boundary conditions with Dirac function appear on \Gamma_1.     

Approximation of Boundary Conditions at Infinity for a Harmonic Equation

Starting from the canonical boundary reduction, this paper studies an approximate differential boundary condition and an approximate integral boundary condition on an artificial boundary for the exterior problem of a harmonic equation, and gives an error estimate for the latter. This estimate reveals the relationship between the error and the approximate grade boundary conditions as well as the radius of the artificial boundary.     

Polarization of vacuum with nontrivial boundary conditions

In the framework of the zeta-regularization approach, we consider the polarization of the scalar field vacuum with nontrivial boundary conditions originating from electrodynamics in the presence of a conducting infinitely thin boundary layer. Boundary conditions of the first type correspond to the case where the field is continuous on the boundary while its derivative has a jump proportional to the boundary value of the field. Boundary conditions of the second type correspond to the case where the field derivative is continuous on the boundary but the field itself has a jump proportional to the field derivative on the boundary. We explicitly obtain the zeta function of the scalar field Laplace operator with the above boundary conditions and calculate all the heat kernel coefficients. We obtain an expression for the energy of the scalar field vacuum fluctuations.     

Uniqueness of analytic solutions for stationary plate oscillations in an annulus

The equations governing the harmonic oscillations of a plate with transverse shear deformation are considered in an annular domain. It is shown that under nonstandard boundary conditions where both the displacements and tractions are zero on the internal boundary curve, the corresponding analytic solution is zero in the entire domain. This property is then used to prove that a boundary value problem with Dirichlet or Neumann conditions on the external boundary and Robin conditions on the internal boundary has at most one analytic solution.     

An Algorithm of Linear Combinations: Thermal Conduction

This paper presents computational algorithms that make it possible to overcome some difficulties in the numerical solving boundary value problems of thermal conduction when the solution domain has a complex form or the boundary conditions differ from the standard ones. The boundary contours are assumed to be broken lines (the 2D case) or triangles (the 3D case). The boundary conditions and calculation results are presented as discrete functions whose values or averaged values are given at the geometric centers of the boundary elements. The boundary conditions can be imposed on the heat flows through the boundary elements as well as on the temperature, a linear combination of the temperature and the heat flow intensity both at the boundary of the solution domain and inside it. The solution to the boundary value problem is presented in the form of a linear combination of fundamental solutions of the Laplace equation and their partial derivatives, as well as any solutions of these equations that are regular in the solution domain, and the values of functions which can be calculated at the points of the boundary of the solution domain and at its internal points. If a solution included in the linear combination has a singularity at a boundary element, its average value over this boundary element is considered.     

Solution for a problem of linear plane elasticity with mixed boundary conditions on an ellipse by the method of boundary integrals

A numerical boundary integral scheme is proposed for the solution of the system of field equations of plane, linear elasticity in stresses for homogeneous, isotropic media in the domain bounded by an ellipse under mixed boundary conditions. The stresses are prescribed on one half of the ellipse, while the displacements are given on the other half. The method relies on previous analytical work within the Boundary Integral Method [1], [2].The considered problem with mixed boundary conditions is replaced by two subproblems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way and the problem at this stage is reduced to the solution of a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution inside the domain, and the unknown boundary values of stresses or displacements on proper parts of the boundary.On the basis of the obtained results, it is inferred that the tangential stress component on the fixed part of the boundary has a singularity at each of the two separation points, thought to be of logarithmic type. A tentative form for the singular solution is proposed to calculate the full solution in bulk directly from the given boundary conditions using the well-known Boundary Collocation Method. It is shown that this addition substantially decreases the error in satisfying the boundary conditions on some interval not containing the singular points.The obtained results are discussed and boundary curves for unknown functions are provided, as well as three-dimensional plots for quantities of practical interest. The efficiency of the used numerical schemes is discussed, in what concerns the number of boundary nodes needed to calculate the approximate solution.     

Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary

We prove that the Cauchy data of Dirichlet or Neumann Δ- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e., a point at which a positive measure of geodesics leaving the point return to the point. In the case of real analytic Riemannian manifolds with real analytic boundary, maximal sup norm bounds on boundary traces of eigenfunctions can only be achieved if there exists a point on the boundary at which all geodesics loop back. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds. The key new ingredient is the Melrose–Taylor diffractive parametrix and Melrose’s analysis of the Weyl law.