三维Helmholtz方程边值问题的新型边界积分-微分方程及其边界元解法

用双层位势表示的二维Neumann边值问题的边界归化方法,将原始问题归化为新型边界积分-微分方程,由此导出一种新的既能保持原始问题的自伴性,又具有可积弱奇性积分核的边界变分方程.本文将此法推广到三维Helmholtz方程Neumann边值问题,并给出最优能量模误差估计和内部最大模超收敛估计.     

A new boundary integral equation formulation for plane orthotropic elastic media

A new boundary integral equation formulation for solving plane elasticity problems involving orthotropic media is presented in this paper. Based on the real variable fundamental solutions of the considered problems, a limit theorem for the transformation from domain integral equations into boundary integral equations (BIEs) and a novel decomposition technique to the fundamental solutions, the regularized BIEs with indirect unknowns, which do not involve the direct calculation of CPV and HFP integrals, are established. The limiting process is done in global coordinates and no separate numerical treatment for strong and weak singular integrals was necessary. The current method does not need to transform the considered problems into isotropic ones as is normally done in the existing literature, so no inverse transform is required. The numerical implementation is carried out using both discontinuous quadratic elements and exact elements, which is developed to model its boundary with negligible error. The validity of the proposed scheme is demonstrated by three numerical examples. Excellent agreement between the numerical results and exact solutions was obtained even with using small amounts of element.     

A method for solving the linear boundary value problem for an integro-differential equation

A method for solving the linear boundary value problem for an integro-differential equation is proposed that is based on interval partition and the introduction of additional parameters. Necessary and sufficient conditions for the solvability of the problem are obtained.     

A parametrix for the fundamental solution of the Klein-Gordon equation on asymptotically de Sitter spaces

In this paper we construct a parametrix for the forward fundamental solution of the wave and Klein-Gordon equations on asymptotically de Sitter spaces without caustics. We use this parametrix to obtain asymptotic expansions for solutions of (□−λ)u=f and to obtain a uniform Lp estimate for a family of bump functions traveling to infinity.     

A method of constructing integrable linear equations and its application to Hill's equation

Starting with a given equation of the form $$\ddot x + [\lambda + \varepsilon f(t)] x = 0$$ , where λ > 0 and ? ? l is a small parameter [heref(t) may be periodic, and so Hill's equation is included], we construct an equation of the form y + [λ + ?f (t) + ?² _g (t)]y = 0, integrable by quadratures, close in a certain sense to the original equation. For x₀ = y₀ and x ₀ ^′ = y ₀ ^′ , an upper bound is obtained for ¦y—x¦ on an interval of length Δt.     

A linear wave equation with a nonlinear boundary condition of viscoelastic type

The paper is devoted to the study of a linear wave equation with a nonlinear boundary condition of viscoelastic type. The existence of a weak solution is proved by using the Faedo–Galerkin method. The uniqueness, regularity and asymptotic expansion of the solution are also discussed.     

A boundary function equation and its numerical solution

We consider the asymptotic solution of the plasma-sheath integro-differential equation, which is singularly perturbed due to the presence of a small coefficient multiplying the highest order (second) derivative. The asymptotic solution is obtained by the boundary function method. A second-order differential equation is derived describing the behavior of the zeroth-order boundary functions. A numerical algorithm for this equation is discussed. __________ Translated from Prikladnaya Matematika i Informatika, No. 24, pp. 24–34, 2006.     

An inverse problem for a linear diffusion equation with non-linear boundary condition

In this paper we consider the determination of an unknown radiation term in the non-linear boundary condition of a linear diffusion equation from an overspecified condition. It is shown that the solutions of this inverse problem is unique and stable.     

On boundary uniqueness theorems for a linear elliptic equation with constant coefficients

For the solutions of an elliptic equation with constant coefficients, we prove uniqueness theorems that generalize the classical boundary uniqueness theorems for analytic functions.     

Time-dependent boundary conditions for the 2-D linear anisotropic-viscoelastic wave equation

Wave propagation simulation requires a correct implementation of boundary conditions to avoid numerical instabilities. A boundary treatment based on characteristics, which includes as special cases more simple rheologies involving isotropy and elastic behavior, is applied to the anisotropic-viscoelastic wave equation. The method introduces the boundary conditions by specifying the values of the incoming variables, which depend on the solution outside the model volume. The formulation ends up with a wave equation for the boundaries that implicitly includes the boundary conditions. The examples illustrate common problems in geophysical modeling, including free surface and nonreflecting conditions. © 1994 John Wiley & Sons, Inc.     

Multiple solutions for an asymptotically linear Duffing equation with Newmann boundary value conditions

We investigate multiple solutions of an asymptotically linear Duffing equation satisfying Newmann boundary value conditions with resonance using index theory and Morse theory and obtain some new results.     

Existence results for a linear equation with reflection,non-constant coefficient and periodic boundary conditions

This work is devoted to the study of first order linear problems with involution and periodic boundary value conditions. We first prove a correspondence between a large set of such problems with different involutions to later focus our attention to the case of the reflection. We study then different cases for which a Green?s function can be obtained explicitly and derive several results in order to obtain information about its sign. Once the sign is known, maximum and anti-maximum principles follow. We end this work with more general existence and uniqueness of solution results.     

The Riesz basis property of discrete operators and application to a Euler-Bernoulli beam equation with boundary linear feedback control

In this paper, we give an abstract condition of Riesz basisgeneration for discrete operators in Hilbert spaces, from whichwe show that the generalized eigenfunctions of a Euler–Bernoullibeam equation with boundary linear feedback control form a Rieszbasis for the state Hilbert space. As an consequence, the asymptoticexpression of eigenvalues together with exponential stabilityare readily presented.     

A boundary integral equation for the transmission eigenvalue problem for Maxwell equation

We propose a new integral equation formulation to characterize and compute transmission eigenvalues in electromagnetic scattering. As opposed to the approach that was recently developed by Cakoni, Haddar and Meng (2015) which relies on a two‐by‐two system of boundary integral equations, our analysis is based on only one integral equation in terms of the electric‐to‐magnetic boundary trace operator that results in a simplification of the theory and in a considerable reduction of computational costs. We establish Fredholm properties of the integral operators and their analytic dependence on the wave number. Further, we use the numerical algorithm for analytic nonlinear eigenvalue problems that was recently proposed by Beyn (2012) for the numerical computation of the transmission eigenvalues via this new integral equation.     

A boundary integral equation procedure for the mixed boundary value problem of the vector helmholtz equation

A boundary integral method is developed for the mixed boundary value problem for the vector Helmholtz equation in R³. The obtained boundary integral equations for the unknown Cauchy data build a strong elliptic system of pseudodifferential equations which can therefore be used for numerical computations using Galerkin's procedure. We show existence, uniqueness and regularity of the solution of the integral equations. Especially we give the local "edge" behavior of the solution near the submanifold which divides the Dirichlet boundary from the Neumann boundary     

On universal solvability of the first boundary value problem for linear difference equation with constant coefficients

The concept of a linear operator with constant coefficients is introduced and investigated, for which the corresponding first boundary problem is uniquely solvable ( or stable uniquely sovable) for any finite net set.     

The singular perturbation of boundary value problem for third-order nonlinear vector integro-differential equation and its application

In this paper, the singular perturbation of boundary value problem to a class of third-order nonlinear vector integro-differential equation is studied. Using the method of differential inequalities, under certain conditions, the existence of perturbed solution is proved, the uniformly valid asymptotic expansion for arbitrary order and the estimation of remainder term are given. Finally, the results are applied to study singularly perturbed boundary value problem to a nonlinear vector fourth-order differential equation. The existence of solution and its asymptotic estimation can be obtained conveniently.