Existence and uniqueness of solutions for acoustic scattering over infinite obstacles

The paper considers the solution of the boundary value problem (BVP) consisting of the Helmholtz equation in the region D with a rigid boundary condition on ∂D and its reformulation as a boundary integral equation (BIE), over an infinite cylindrical surface of arbitrary smooth cross-section. A boundary integral equation, which models three-dimensional acoustic scattering from an infinite rigid cylinder, illustrates the application of the above results to prove existence of solution of the integral equation and the corresponding boundary value problem.     

Existence of solution in elastic wave scattering by unbounded rough surfaces

We consider the two‐dimensional problem of the scattering of a time‐harmonic wave, propagating in an homogeneous, isotropic elastic medium, by a rough surface on which the displacement is assumed to vanish. This surface is assumed to be given as the graph of a function ?∈C^1,1(?). Following up on earlier work establishing uniqueness of solution to this problem, existence of solution is studied via the boundary integral equation method. This requires a novel approach to the study of solvability of integral equations on the real line. The paper establishes the existence of a unique solution to the boundary integral equation formulation in the space of bounded and continuous functions as well as in all L^p spaces, p∈[1, ∞] and hence existence of solution to the elastic wave scattering problem. Copyright © 2002 John Wiley & Sons, Ltd.     

On a nonlocal generalization of the biharmonic Dirichlet problem

We consider a mixed problem with the Dirichlet boundary conditions and integral conditions for the biharmonic equation. We prove the existence and uniqueness of a generalized solution in the weighted Sobolev space W ₂². We show that the problem can be viewed as a generalization of the Dirichlet problem.     

Shape Derivatives of Boundary Integral Operators in Electromagnetic Scattering. Part II: Application to Scattering by a Homogeneous Dielectric Obstacle

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a penetrable bounded obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity T H^-\frac12(div_G,G){\mathsf T \mathsf H^{-\frac{1}{2}}({\rm div}_{\Gamma},\Gamma)}. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the Gateaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.     

局部扰动半平面上的散射问题

该文讨论半平面上有局部扰动情况下的散射问题．通过位势理论,应用边界积分方程的方法研究了该问题解的存在与唯一性．主要方法是运用对称反射,使该无界区域上的散射问题变成一个有界区域上的散射问题,只是这一有界区域的边界不光滑．通过仔细分析相应的边界积分算子,作者得到了其解的存在与唯一性．     

On the Neumann problem for the Helmholtz equation in a plane angle

We prove that the solution of the Neumann problem for the Helmholtz equation in a plane angle Ω with boundary conditions from the space H_−1/2(Γ), where Γ is the boundary of Ω, which is provided by the well‐known Sommerfeld integral, belongs to the Sobolev space H₁(Ω) and depends continuously on the boundary values. To this end, we use another representation of the solution given by the inverse two‐dimensional Fourier transform of an analytic function depending on the Cauchy data of the solution. Copyright © 2000 John Wiley & Sons, Ltd.     

Extremal solutions of boundary value problems

To prove the existence of a solution of a two-point boundary value problem for an nth-order operator equation by the a priori estimate method, we study extremal solutions of auxiliary boundary value problems for an nth-order differential equation with simplest right-hand side, which have a unique solution under certain restrictions on the boundary conditions.     

Boundary Integral Equations for Mixed Boundary Value Problems in R3

Both exterior and interior mixed Dirichlet-Neumann problems in R³ for the scalar Helmholtz equation are solved via boundary integral equations. The integral equations are equivalent to the original problem in the sense that the traces of the weak seolution satisfy the integral equations, and, conversely, the solution of the integral equations inserted into Green's formula yields the solution of the mixed boundary value problem. The calculus of pseudodifferential operators is used to prove existence and regularity of the solution of the integral equations. The regularity results — obtained via Wiener-Hopf technique — show the explicit “edge” behavior of the solution near the submanifold which separates the Dirichlet boundary from the Neumann boundary.     

Boundary integral equations for the Helmholtz equation: The third boundary value problem

In this paper we study the application of boundary integral equation methods for the solution of the third, or Robin, boundary value problem for the exterior Helmholtz equation. In contrast to earlier work, the boundary value problem is interpreted here in a weak sense which allows data to be specified in L_∞ (?D), ?D being the boundary of the exterior domain which we assume to be Lyapunov of index 1. For this exterior boundary value problem, we employ Green's theorem to derive a pair of boundary integral equations which have a unique simultaneous solution. We then show that this solution yields a solution of the original exterior boundary value problem.     

Solvability of a nonlinear model Boltzmann equation in the problem of a plane shock wave

We consider a nonlinear system of integral equations describing the structure of a plane shock wave. Based on physical reasoning, we propose an iterative method for constructing an approximate solution of this system. The problem reduces to studying decoupled scalar nonlinear and linear integral equations for the gas temperature, density, and velocity. We formulate a theorem on the existence of a positive bounded solution of a nonlinear equation of the Uryson type. We also prove theorems on the existence and uniqueness of bounded positive solutions for linear integral equations in the space L ₁[?r, r] for all finite r < +∞. For a more general nonlinear integral equation, we prove a theorem on the existence of a positive solution and also find a lower bound and an integral upper bound for the constructed solution.     

Polarized light transfer above a reflecting surface

The existence and uniqueness are established for the solution of the equation of transfer of polarized light in a homogeneous atmosphere of finite optical thickness, assuming reflection by the planetary surface. A general L_p-space formulation is adopted. The boundary value problem is first written as a vector-valued integral equation. Using monotonicity properties of the spectral radii of the integral operators involved as well as recent half-range completeness results for kinetic equations with reflective boundary conditions, the present results follow as a corollary.     

One Goursat problem in a Sobolev space

In this paper we consider a hyperbolic-type differential equation with L _p -coefficients in a three-dimensional space. For this equation we study the Goursat problem with nonclassical boundary constraints not requiringmatched conditions. We prove the equivalence of these boundary conditions to classical ones in the case when one seeks for a solution to the stated problem in an anisotropic space introduced by S. L. Sobolev. In addition, we prove the correct solvability of the Goursat problem by the method of integral equations.     

Mathematical basis of the linear sampling method for partially coated obstacles

We consider the inverse scattering problem of determining the shape of a partially coated obstacle D. To this end, we solve a scattering problem for the Helmholtz equation where the scattered field satisfies mixed Dirichlet–Neumann-impedance boundary conditions on the Lipschitz boundary of the scatterer D. Based on the analysis of the boundary integral system to the direct scattering problem, we propose how to reconstruct the shape of the obstacle D by using the linear sampling method.     

The method of boundary integral equations for a mixed scattering problem

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R² as cross section. To this end, we solve a scattering problem for the Helmholtz equation in R² where the scattering object is a combination of a crack Γ and a bounded obstacle D, and we have Dirichlet-impedance type boundary condition on Γ and Dirichlet boundary condition on ∂D (∂D∈C²). Applying potential theory, the problem can be reformulated as a boundary integral system. We establish the existence and uniqueness of a solution to the system by using the Fredholm theory.     

On the spline collocation method for the single layer equation related to time-fractional diffusion

The boundary element spline collocation method is studied for the time-fractional diffusion equation in a bounded two-dimensional domain. We represent the solution as the single layer potential which leads to a Volterra integral equation of the first kind. We discretize the boundary integral equation with the spline collocation method on uniform meshes both in spatial and time variables. In the stability analysis we utilize the Fourier analysis technique developed for anisotropic pseudodifferential equations. We prove that the collocation solution is quasi-optimal under some stability condition for the mesh parameters. We have to assume that the mesh parameter in time satisfies (h_t=c h^\frac2a)(h_t=c h^{\frac{2}{\alpha}}), where (h) is the spatial mesh parameter.     

Boundary Element Methods for Maxwell Transmission Problems in Lipschitz Domains

Summary. We consider the Maxwell equations in a domain with Lipschitz boundary and the boundary integral operator A occuring in the Calderón projector. We prove an inf-sup condition for A using a Hodge decomposition. We apply this to two types of boundary value problems: the exterior scattering problem by a perfectly conducting body, and the dielectric problem with two different materials in the interior and exterior domain. In both cases we obtain an equivalent boundary equation which has a unique solution. We then consider Galerkin discretizations with Raviart-Thomas spaces. We show that these spaces have discrete Hodge decompositions which are in some sense close to the continuous Hodge decomposition. This property allows us to prove quasioptimal convergence of the resulting boundary element methods. Mathematics Subject Classification (2000):65N30     

The evolution dam problem for a compressible fluid with nonlinear Darcy's law and Dirichlet boundary condition

In this paper, we consider the evolution dam problem (P) related to a compressible fluid flow governed by a generalized nonlinear Darcy's law with Dirichlet boundary conditions on some part of the boundary. We establish existence of a solution for this problem. We choose a convenient regularized problem (P_?) for which we prove the existence and uniqueness of solution using the comparison Lemma 2.1 and the Schauder fixed‐point theorem. Then, we pass to the limit, when ? goes to 0, to get a solution for our problem. Moreover, we will see another approach for the incompressible case where we pass to the limit in (P), when α goes to 0, to get a solution.     

A nonlocal problem for a mixed-type equation whose order degenerates along the line of change of type

We consider a mixed-type equation whose order degenerates along the line of change of type. For this equation we study the unique solvability of a nonlocal problem with the Saigo operators in the boundary condition. We prove the uniqueness theorem under certain conditions (stated as inequalities) on known functions. To prove the existence of solution to the problem, we equivalently reduce it to a singular integral equation with the Cauchy kernel. We establish a condition ensuring the existence of a regularizer which reduces the obtained equation to a Fredholm equation of the second kind, whose unique solvability follows from that of the problem.     

A fast numerical method for harmonic equation based on natural boundary integral

Based on the coupling of the natural boundary integral method and the finite elements method, we mainly investigate the numerical solution of Neumann problem of harmonic equation in an exterior elliptic. Using our trigonometric wavelets and Galerkin method, there obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. On the other hand, we prove that the numerical solution possesses exponential convergence rate. Especially, examples state that our method still has good accuracy for small j when the solution u ₀(θ) is almost singular.     

The operator equations of Lippmann–Schwinger type for acoustic and electromagnetic scattering problems in L 2

This article is concerned with the scattering of acoustic and electromagnetic time harmonic plane waves by an inhomogeneous medium. These problems can be translated into volume integral equations of the second kind – the most prominent example is the Lippmann–Schwinger integral equation. In this work, we study a particular class of scattering problems where the integral operator in the corresponding operator equation of Lippmann–Schwinger type fails to be compact. Such integral equations typically arise if the modelling of the inhomogeneous medium necessitates space-dependent coefficients in the highest order terms of the underlying partial differential equation. The two examples treated here are acoustic scattering from a medium with a space-dependent material density and electromagnetic medium scattering where both the electric permittivity and the magnetic permeability vary. In these cases, Riesz theory is not applicable for the solution of the arising integral equations of Lippmann–Schwinger type. Therefore, we show that positivity assumptions on the relative material parameters allow to prove positivity of the arising volume potentials in tailor-made weighted spaces of square integrable functions. This result merely holds for imaginary wavenumber and we exploit a compactness argument to conclude that the arising integral equations are of Fredholm type, even if the integral operators themselves are not compact. Finally, we explain how the solution of the integral equations in L ² affects the notion of a solution of the scattering problem and illustrate why the order of convergence of a Galerkin scheme set up in L ² does not suffer from our L ² setting, compared to schemes in higher order Sobolev spaces.