Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows

We consider a planar waveguide modeled by the Laplacian in a straight infinite strip with the Dirichlet boundary condition on the upper boundary and with frequently alternating boundary conditions (Dirichlet and Neumann) on the lower boundary. The homogenized operator is the Laplacian subject to the Dirichlet boundary condition on the upper boundary and to the Dirichlet or Neumann condition on the lower one. We prove the uniform resolvent convergence for the perturbed operator in both cases and obtain the estimates for the rate of convergence. Moreover, we construct the leading terms of the asymptotic expansions for the first band functions and the complete asymptotic expansion for the bottom of the spectrum. Bibliography: 17 titles. Illustrations: 3 figures.     

On a Waveguide with Frequently Alternating Boundary Conditions: Homogenized Neumann Condition

We consider a waveguide modeled by the Laplacian in a straight planar strip. The Dirichlet boundary condition is taken on the upper boundary, while on the lower boundary we impose periodically alternating Dirichlet and Neumann condition assuming the period of alternation to be small. We study the case when the homogenization gives the Neumann condition instead of the alternating ones. We establish the uniform resolvent convergence and the estimates for the rate of convergence. It is shown that the rate of the convergence can be improved by employing a special boundary corrector. Other results are the uniform resolvent convergence for the operator on the cell of periodicity obtained by the Floquet–Bloch decomposition, the two terms asymptotics for the band functions, and the complete asymptotic expansion for the bottom of the spectrum with an exponentially small error term.     

Boundary homogenization in domains with randomly oscillating boundary

We consider a model homogenization problem for the Poisson equation in a domain with a rapidly oscillating boundary which is a small random perturbation of a fixed hypersurface. A Fourier boundary condition with random coefficients is imposed on the oscillating boundary. We derive the effective boundary condition, prove a convergence result, and establish error estimates.     

Asymptotic Solutions of Hamilton-Jacobi Equations with State Constraints

We study Hamilton-Jacobi equations in a bounded domain with the state constraint boundary condition. We establish a general convergence result for viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations with the state constraint boundary condition to asymptotic solutions as time goes to infinity.     

On the uniform convergence of spectral expansions for a spectral problem with boundary conditions of the third kind one of which contains the spectral parameter

We study the uniform convergence, on a closed interval, of spectral expansions of Hölder functions in a given complete and minimal system of eigenfunctions corresponding to a spectral problem with spectral parameter in a boundary condition. We consider boundary conditions of the third kind and subject the function to be expanded to a condition of nonlocal type ensuring the uniform convergence. We prove a theorem stating that expansions in the entire system of eigenfunctions of the problem are possible without any additional conditions.     

A PDE Approach to Large-Time Asymptotics for Boundary-Value Problems for Nonconvex Hamilton–Jacobi Equations

We investigate the large-time behavior of three types of initial-boundary value problems for Hamilton–Jacobi Equations with nonconvex Hamiltonians. We consider the Neumann or oblique boundary condition, the state constraint boundary condition and Dirichlet boundary condition. We establish general convergence results for viscosity solutions to asymptotic solutions as time goes to infinity via an approach based on PDE techniques. These results are obtained not only under general conditions on the Hamiltonians but also under weak conditions on the domain and the oblique direction of reflection in the Neumann case.     

Well-posedness and approximation of a measure-valued mass evolution problem with flux boundary conditions

This Note deals with imposing a flux boundary condition on a non-conservative measure-valued mass evolution problem posed on a bounded interval. To establish the well-posedness of the problem, we exploit particle system approximations of the mass accumulation in a thin layer near the active boundary. We derive the convergence rate for the approximation procedure as well as the structure of the flux boundary condition in the limit problem.     

Spline collocation for strongly elliptic equations on the torus

Summary We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4].     

Analysis of a meshless method for the time fractional diffusion-wave equation

In this paper a numerical technique is proposed for solving the time fractional diffusion-wave equation. We obtain a time discrete scheme based on finite difference formula. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method and the convergence order of the time discrete scheme is \(\mathcal {O}(\tau ^{3-\alpha })\). Firstly, we change the main problem based on Dirichlet boundary condition to a new problem based on Robin boundary condition and then, we consider a semi-discrete scheme with Robin boundary condition and show when \(\beta \rightarrow +\infty \) solution of the main semi-discrete problem with Dirichlet boundary condition is convergent to the solution of the new semi-discrete problem with Robin boundary condition. We consider the new semi-discrete problem with Robin boundary condition and use the meshless Galerkin method to approximate the spatial derivatives. Finally, we obtain an error bound for the new problem. We prove that convergence order of the numerical scheme based on Galekin meshless is \(\mathcal {O}(h)\). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. The main aim of the current paper is to obtain an error estimate for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme.     

On the convergence of solutions and eigenelements of a boundary value problem in a domain perforated along the boundary

We consider a boundary value problem in a model domain periodically perforated along the boundary. We assume that the homogeneous Neumann condition is posed on the external boundary and the homogeneous Dirichlet condition is posed on the boundary of the cavities. A limit (homogenized) problem is obtained. We prove the convergence of the solutions, eigenvalues, and eigenfunctions of the original problem to the solutions, eigenvalues, and eigenfunctions, respectively, of the limit problem.     

Asymptotics and Estimates of the Convergence Rate in a Three-Dimensional Boundary-Value Problem with Rapidly Alternating Boundary Conditions

We consider a singularly perturbed boundary-value eigenvalue problem for the Laplace operator in a cylinder with rapidly alternating type of the boundary condition on the lateral surface. The change of the boundary conditions is realized by splitting the lateral surface into many narrow strips on which the Dirichlet and Neumann conditions alternate. We study the case in which the averaged problem contains the Dirichlet boundary condition on the lateral surface. In the case of strips with slowly varying width we construct the first terms of the asymptotic expansions of eigenfunctions; moreover, in the case of strips with rapidly varying width we obtain estimates for the convergence rate.     

Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics

We consider a magnetic Schrödinger operator in a planar infinite strip with frequently and non-periodically alternating Dirichlet and Robin boundary conditions. Assuming that the homogenized boundary condition is the Dirichlet or the Robin one, we establish the uniform resolvent convergence in various operator norms and we prove the estimates for the rates of convergence. It is shown that these estimates can be improved by using special boundary correctors. In the case of periodic alternation, pure Laplacian, and the homogenized Robin boundary condition, we construct two-terms asymptotics for the first band functions, as well as the complete asymptotics expansion (up to an exponentially small term) for the bottom of the band spectrum.     

硅体中磷反应扩散系统解的整体存在性及渐近性

本文用上下解方法，研究了一类带混合非齐次边界条件的化学反应──硅体中磷反应扩散系统，采用避开△的特征函数构造上下解的方法，得到一类充分条件，以使其整体解渐近趋向于Ｓｔｅａｄｙ－ｓｔａｔｅ解．     

Generalized quasilinearization method for reaction-diffusion equations under nonlinear and nonlocal flux conditions

In this paper we consider an initial boundary value problem for a reaction-diffusion equation under nonlinear and nonlocal Robin type boundary condition. Assuming the existence of an ordered pair of upper and lower solutions we establish a generalized quasilinearization method for the problem under consideration whose characteristic feature consists in the construction of monotone sequences converging to the unique solution within the interval of upper and lower solutions, and whose convergence rate is quadratic. Thus this method provides an efficient iteration technique that produces not only improved approximations due to the monotonicity of its iterates, but yields also a measure of the convergence rate.     

Vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier–Stokes equations with a slip boundary condition

In this paper, we investigate the vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier–Stokes equations with a slip boundary condition. We establish the local well‐posedness of the strong solutions for initial boundary value problems for such systems. Furthermore, the vanishing viscosity limit process is established, and a strong rate of convergence is obtained as the boundary of the domain is flat. In addition, it is needed to add some additional condition for density to match well the boundary condition. Copyright © 2017 John Wiley & Sons, Ltd.     

Biharmonic equation in a highly oscillating domain and homogenization of an associated control problem

We consider a Dirichlet boundary control problem posed in an oscillating boundary domain governed by a biharmonic equation. Homogenization of a PDE with a non-homogeneous Dirichlet boundary condition on the oscillating boundary is one of the hardest problems. Here, we study the homogenization of the problem by converting it into an equivalent interior control problem. The convergence of the optimal solution is studied using periodic unfolding operator.     

Homogenization of spectral problem for locally periodic elliptic operators with sign-changing density function

The paper deals with homogenization of a spectral problem for a second order self-adjoint elliptic operator stated in a thin cylinder with homogeneous Neumann boundary condition on the lateral boundary and Dirichlet condition on the bases of the cylinder. We assume that the operator coefficients and the spectral density function are locally periodic in the axial direction of the cylinder, and that the spectral density function changes sign. We show that the behavior of the spectrum depends essentially on whether the average of the density function is zero or not. In both cases we construct the effective 1-dimensional spectral problem and prove the convergence of spectra.     

On the uniform convergence in <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> of Fourier series for a spectral problem with squared spectral parameter in a boundary condition

We study the uniform convergence in C ¹ of the Fourier series of a Hölder function in a system of eigenfunctions corresponding to a spectral problem with squared spectral parameter in a boundary condition. We preliminarily study one more spectral problem with a spectral parameter in a boundary condition.     

Application and Implementation of Incorporating Local Boundary Conditions into Nonlocal Problems

We study nonlocal equations from the area of peridynamics, an instance of nonlocal wave equation, and nonlocal diffusion on bounded domains whose governing equations contain a convolution operator based on integrals. We generalize the notion of convolution to accommodate local boundary conditions. On a bounded domain, the classical operator with local boundary conditions has a purely discrete spectrum, and hence, provides a Hilbert basis. We define an abstract convolution operator using this Hilbert basis, thereby automatically satisfying local boundary conditions. The main goal in this paper is twofold: apply the concept of abstract convolution operator to nonlocal problems and carry out a numerical study of the resulting operators. We study the corresponding initial value problems with prominent boundary conditions such as periodic, antiperiodic, Neumann, and Dirichlet. To connect to the standard convolution, we give an integral representation of the abstract convolution operator. For discretization, we use a weak formulation based on a Galerkin projection and use piecewise polynomials on each element which allows discontinuities of the approximate solution at the element borders. We study convergence order of solutions with respect to polynomial order and observe optimal convergence. We depict the solutions for each boundary condition.     

LINEARIZATION OF A NONLINEAR PERIODIC BOUNDARY CONDITION RELATED TO CORROSION MODELING

We study galvanic currents on a heterogeneous surface. In electrochemistry, the oxidation-reduction reaction producing the current is commonly modeled by a nonlinear elliptic boundary value problem. The boundary condition is of exponential type with periodically varying parameters. We construct an approximation by first homogenizing the problem, and then linearizing about the homogenized solution. This approximation is far more accurate than both previous approximations or direct linearization. We establish convergence estimates for both the two and three-dimensional case and provide two-dimensional numerical experiments.