Problems In Unbounded Domains Research Articles

Equivalence of Entropy and Renormalized Solutions of Anisotropic Elliptic Problem in Unbounded Domains with Measure Data

Equivalence of Entropy and Renormalized Solutions of Anisotropic Elliptic Problem in Unbounded Domains with Measure Data

Efficient Solution of Two-Dimensional Wave Propagation Problems by CQ-Wavelet BEM: Algorithm and Applications

In this paper we consider wave propagation problems in two-dimensional unbounded domains, including dissipative effects, reformulated in terms of space-time boundary integral equations. For their s...

Noncoercive nonlinear Dirichlet problems in unbounded domains

In this paper we study noncoercive nonlinear Dirichlet problems with discontinuous coefficients in unbounded domains. An existence result is proved exploiting suitable coercive nonlinear approximate Dirichlet problems.

On the Mixed Steklov—Neumann and Steklov-type Biharmonic Problems in Unbounded Domains

We study the properties of generalized solutions in unbounded domains and the asymptotic behavior of solutions of elliptic boundary value problems at infinity. Moreover, we study the unique solvability of the mixed Steklov–Neumann and Steklov-type biharmonic problems in unbounded domains with a compact and non-compact boundaries (in particular, the exterior of a compact set, half-space, the domain with conical points) under the assumption that generalized solutions of these problems has a bounded Dirichlet (energy) integral with weight |x|a. Depending on the value of the parameter a, we obtained uniqueness (non-uniqueness) theorems of these problems or present exact formulas for the dimension of the space of solutions.

Generating efficient basis sets for unbounded domains

The treatment of the problems involving unbounded domains (UDs) with vanishing boundary conditions is always challenging. For spectral methods, in particular, very limited basis sets are commonly used for such domains, in which the ranges of the decay rates with acceptable computational efficiency, are very small. Furthermore, maintaining high level of analyticity becomes burdensome. Developing efficient mapped basis tailored for such problem is one of the main strategies to overcome these difficulties. In this work, we present a technique to generate efficient basis sets for UDs. This approach allows using basis sets defined for bounded domains (BDs) for problems in UDs, and hence, providing more freedom to choose from a variety of basis sets. To ensure computational efficiency, the designed transformations cover a wide range of decay rates and allow solving integrals analytically. The method is applied to solve many differential equations encountered frequently in many physics related problems. The results illustrate the efficiency of the developed technique and mapped basis sets.

Existence and uniqueness results in weighted spaces for Dirichlet problem in unbounded domains

We study the Dirichlet problem for a second order linear elliptic partial differential equation with discontinuous coefficients in unbounded domains. We establish an existence and uniqueness result.

Mathematical analysis of Ziolkowski’s PML model with application for wave propagation in metamaterials

In this paper we investigate one Perfectly Matched Layer (PML) model proposed by Ziolkowski in 1999. Various schemes for solving this PML model have been developed and have been shown to be effective in absorbing outgoing waves when the wave propagation in unbounded domain problem is reduced to a bounded domain problem. However, a rigorous analysis of this model is lacking. In this paper we establish the stability of this PML model and propose a fully-discrete finite element scheme to solve this model with edge elements. Discrete stability and optimal error estimate of this scheme are proved. Numerical results justifying the analysis and demonstrating the effectiveness of this PML model are presented.

A local collocation method to construct Dirichlet-type absorbing boundary conditions for transient scalar wave propagation problems

This paper introduces an effective way to equip the standard finite element method (FEM) for the solution of transient scalar wave propagation problems in unbounded domains. Similar to many other methods, we truncate the unbounded domain at an artificial boundary and convert the problem into a bounded one by prescribing appropriate absorbing boundary conditions (ABCs) at the truncating boundary. In the present method, the ABCs are time-dependent, and they are constructed by a simple collocation approach which is local in space and time. Therefore, the method does not make use of any routine schemes such as Fourier and Laplace transform. We shall show that the method is simple, and it can be easily applied to an explicit time domain FEM approach so that the sparsity of the FEM scheme (as well as its efficiency) can be preserved. The proposed method does not require any auxiliary variables as well as any approximating differential operators. This feature roots from the fact that here the ABCs are Dirichlet-type (or first-type) and thus they can be easily imposed to the corresponding boundaries. Therefore, the method shares some similarities with the conventional 1st and 2nd order ABC methods in terms of the simplicity of implementation. The method employs basis functions that exactly satisfy the governing and dispersion wave equations. The basis function can be easily adjusted to act as outgoing waves transmitting energy from the interior domain (near field) towards exterior domain (far field); i.e, they can cope with satisfaction of radiation boundary conditions. Several numerical examples are presented to evaluate the performance and to demonstrate the effectiveness of the approach. We shall show that the present method is capable of yielding results with a proper level of accuracy, similar to that of the perfectly matched layers method (PMLs), and it performs stably even in the case of long-term computations.

Correction to 'The unified transform for mixed boundary condition problems in unbounded domains'.

[This corrects the article DOI: 10.1098/rspa.2018.0605.].

The unified transform for mixed boundary condition problems in unbounded domains

This paper implements the unified transform to problems in unbounded domains with solutions having corner singularities. Consequently, a wide variety of mixed boundary condition problems can be solved without the need for the Wiener-Hopf technique. Such problems arise frequently in acoustic scattering or in the calculation of electric fields in geometries involving finite and/or multiple plates. The new approach constructs a global relation that relates known boundary data, such as the scattered normal velocity on a rigid plate, to unknown boundary values, such as the jump in pressure upstream of the plate. By approximating the known data and the unknown boundary values by suitable functions and evaluating the global relation at collocation points, one can accurately obtain the expansion coefficients of the unknown boundary values. The method is illustrated for the modified Helmholtz and Helmholtz equations. In each case, comparisons between the traditional Wiener-Hopf approach, other spectral or boundary methods and the unified transform approach are discussed.

On well-posedness of time-harmonic problems in an unbounded strip for a thin plate model

We study the propagation of elastic waves in the time-harmonic regime in a waveguide which is unbounded in one direction and bounded in the two other (transverse) directions. We assume that the waveguide is thin in one of these transverse directions, which leads us to consider a Kirchhoff-Love plate model in a locally perturbed 2D strip. For time harmonic scattering problems in unbounded domains, well-posedness does not hold in a classical setting and it is necessary to prescribe the behaviour of the solution at infinity. This is challenging for the model that we consider and constitutes our main contribution. Two types of boundary conditions are considered: either the strip is simply supported or the strip is clamped. The two boundary conditions are treated with two different methods. For the simply supported problem, the analysis is based on a result of Hilbert basis in the transverse section. For the clamped problem, this property does not hold. Instead we adopt the Kondratiev's approach, based on the use of the Fourier transform in the unbounded direction, together with techniques of weighted Sobolev spaces with detached asymptotics. After introducing radiation conditions, the corresponding scattering problems are shown to be well-posed in the Fredholm sense. We also show that the solutions are the physical (outgoing) solutions in the sense of the limiting absorption principle.

Solving acoustic scattering problem by the meshless method based on radial basis function

In this paper, radial point collocation method (RPCM) is introduced to solve the acoustic scattering problem. This is a mathematically simple, easy-to-program and truly meshless method, which has been successfully applied to solve the solid mechanics and convection diffusion problems. However, application of this method to investigate acoustic problems, in particle the acoustic scattering problem is relatively new. The main advantage of this method is its mathematically simple, easy to program, and truly meshless. A Hermite-type interpolation method is employed to improve the solution accuracy while the Neumann boundary conditions exist. In addition, acoustic scattering problem is a typical unbounded domain problem, in order to solve it with RPCM, the domain is truncated to a finite region and an artificial boundary condition (ABC) is imposed. Finally, numerical example is presented to validate the accuracy and effectiveness of RPCM. In the future, the extension of RPCM to more complex and practical problems, especially the three-dimensional situations need to be investigated in more detail.

Regularity of Nash payoffs of Markovian nonzero-sum stochastic differential games

ABSTRACTIn this paper we deal with the problem of existence of a smooth solution of the Hamilton–Jacobi–Bellman–Isaacs (HJBI for short) system of equations associated with nonzero-sum stochastic differential games. We consider the problem in unbounded domains either in the case of continuous generators or for discontinuous ones. In each case we show the existence of a smooth solution of the system. As a consequence, we show that the game has smooth Nash payoffs which are given by means of the solution of the HJBI system and the stochastic process which governs the dynamic of the controlled system.

Pseudospectral method for non-isotropic heat transfer using mixed Hermite-Legendre interpolation

In this paper, we develop a new mixed pseudospectral method for heat transfer by using generalized Hermite functions and Legendre polynomials in unbounded domains. Fundamental to spectral methods for various unbounded-domain problems, we establish some basic results on the mixed generalized Hermite-Legendre interpolation. As an example, a new mixed generalized Hermite-Legendre pseudospectral scheme is provided for non-isotropic heat transfer. Its convergence is proved. Numerical results demonstrate the spectral accuracy of this approach.

On Some Elliptic Problems in Unbounded Domains

The author presents a method allowing to obtain existence of a solution for some elliptic problems set in unbounded domains, and shows exponential rate of convergence of the approximate solution toward the solution.

Global existence and large-time behavior of solutions to the Cauchy problem of one-dimensional viscous radiative and reactive gas

In this paper we consider the global existence and large-time behavior of solutions of one-dimensional viscous radiative and reactive gas with large initial data. In precedent studies the case for the initial-boundary value problem in bounded domain has been investigated and our main purpose focuses on the corresponding problem in unbounded domain. If the initial data is a large perturbation of a non-vacuum constant equilibrium state and is assumed to be without vacuum, mass concentrations, or vanishing temperatures, then we can show that its Cauchy problem admits a unique global smooth non-vacuum solution which tends time-asymptotically to such an equilibrium state. The key point in the analysis is to deduce the uniform positive lower and upper bounds on the specific volume and the absolute temperature.

On the Steklov-Type Biharmonic Problem in Unbounded Domains

We study the unique solvability of the Steklov–type biharmonic problem in unbounded domains under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight |x| a . Depending on the value of the parameter a, we prove uniqueness theorems or present exact formulas for the dimension of the solution space of the Steklov–type biharmonic problem in the exterior of a compact set and in a half-space.

Existence of Solutions of Anisotropic Elliptic Equations with Variable Exponents in Unbounded Domains

We consider a class of anisotropic elliptic differential equations of second order with divergent form and variable exponents. The corresponding elliptic operators are pseudo-monotone and coercive. We obtain solvability conditions for the Dirichlet problem in unbounded domains Ω ⊂ ℝ n , n ≥ 2. The proof of existence of solutions is free of restrictions on growth of data for |x| → ∞.

On Entropy Solutions of Anisotropic Elliptic Equations with Variable Nonlinearity Indices

For a certain class of second-order anisotropic elliptic equations with variable nonlinearity indices and L1 right-hand side we consider the Dirichlet problem in arbitrary unbounded domains. We prove the existence and uniqueness of entropy solutions in anisotropic Sobolev spaces with variable indices.

On the discretization and application of two space–time boundary integral equations for 3D wave propagation problems in unbounded domains

In this paper, we consider 3D wave propagation problems in unbounded domains, such as those of acoustic waves in non viscous fluids, or of seismic waves in (infinite) homogeneous isotropic materials, where the propagation velocity c is much higher than 1. For example, in the case of air and water c≈343m/s and c≈1500m/s respectively, while for seismic P-waves in linear solids we may have c≈6000m/s or higher. These waves can be generated by sources, possible away from the obstacles. We further assume that the dimensions of the obstacles are much smaller than that of the wave velocity, and that the problem transients are not excessively short.For their solution we consider two different approaches. The first directly uses a known space–time boundary integral equation to determine the problem solution. In the second one, after having defined an artificial boundary delimiting the region of computational interest, the above mentioned integral equation is interpreted as a non reflecting boundary condition to be coupled with a classical finite element method.For such problems, we show that in some cases the computational cost and storage, required by the above numerical approaches, can be significantly reduced by taking into account a property that till now has not been considered. To show the effectiveness of this reduction, the proposed approach is applied to several problems, including multiple scattering.