共查询到20条相似文献,搜索用时 31 毫秒
1.
Qinghua Wu & Guozheng Yan 《计算数学(英文版)》2015,33(5):517-532
We consider the inverse scattering problem of determining the shape of a thin dielectric infinite cylinder having an open arc as cross section. Assuming that the electric field is polarized in the TM mode, this leads to a mixed boundary value problem for the Helmholtz equation defined in the exterior of an open arc in $R^2$. We suppose that the arc has mixed Dirichlet-impedance boundary condition, and try to recover the shape of the arc through the far field pattern by using the factorization method. However, we are not able to apply the basic theorem introduced by Kirsch to treat the far field operator $F$, and some auxiliary operators have to be considered. The theoretical validation of the factorization method to our problem is given in this paper, and some numerical results are presented to show the viability of our method. 相似文献
2.
We consider the inverse scattering problem of determining the shape and location of a crack surrounded by a known inhomogeneous media. Both the Dirichlet boundary condition and a mixed type boundary conditions are considered. In order to avoid using the background Green function in the inversion process, a reciprocity relationship between the Green function and the solution of an auxiliary scattering problem is proved. Then we focus on extending the factorization method to our inverse shape reconstruction problems by using far field measurements at fixed wave number. We remark that this is done in a non intuitive space for the mixed type boundary condition as we indicate in the sequel. 相似文献
3.
A new approach for symbolically solving linear boundary value problems is presented. Rather than using general-purpose tools for obtaining parametrized solutions of the underlying ODE and fitting them against the specified boundary conditions (which may be quite expensive), the problem is interpreted as an operator inversion problem in a suitable Banach space setting. Using the concept of the oblique Moore—Penrose inverse, it is possible to transform the inversion problem into a system of operator equations that can be attacked by virtue of non-commutative Gröbner bases. The resulting operator solution can be represented as an integral operator having the classical Green’s function as its kernel. Although, at this stage of research, we cannot yet give an algorithmic formulation of the method and its domain of admissible inputs, we do believe that it has promising perspectives of automation and generalization; some of these perspectives are discussed. 相似文献
4.
The present paper is the first of a series of papers in which time varying linear systems with boundary conditions and integral operators with semi-separable kernels are studied. The interplay between the systems and the integral operators is one of the main features. A general theory of systems with boundary conditions is developed which includes a detailed study of minimality and minimal factorization. This first part has mainly an introductory character. It contains for systems with boundary conditions a systematic analysis of the concepts of transfer operator, realization, similarity, cascade connection and factorization. For discrete systems an analogous theory is developed. 相似文献
5.
《Advances in Mathematics》1985,55(3):242-315
The uniform asympotic behavior of the scattering amplitude near the forward peak, in the case of classical scattering of waves by a convex obstacle, is derived. A microlocal model is obtained for the scattering operator. This is achieved by use of a parametrix for diffractive boundary problems and by a new study of a class of Fourier integral operators, those with folding canonical relations. A crucial ingredient consists of putting a Fourier integral operator with folding canonical relation into a normal form. The analysis also gives the asymptotic behavior of the normal derivative of the scattered wave on a neighbourhood of the shadow boundary, thus providing a corrected version of the Kirchhoff approximation. 相似文献
6.
Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems
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O. Chkadua S. E. Mikhailov D. Natroshvili 《Mathematical Methods in the Applied Sciences》2017,40(6):1817-1837
The paper deals with the three‐dimensional Dirichlet boundary value problem (BVP) for a second‐order strongly elliptic self‐adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary‐domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary‐domain integral equation system is studied. We establish that the obtained localized boundary‐domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd. 相似文献
7.
Viorel Barbu 《偏微分方程通讯》2013,38(7):1318-1338
One proves that the generator of the transition semigroup of a stochastic differential equation with boundary reflection on a convex set K is the realization of a second order elliptic operator on K with zero oblique derivative boundary conditions. Several implications to parabolic problems with oblique derivative are also derived. 相似文献
8.
We are interested in solving the inverse problem of acoustic wave scattering to reconstruct the position and the shape of sound-hard obstacles from a given incident field and the corresponding far field pattern of the scattered field. The method we suggest is an extension of the hybrid method for the reconstruction of sound-soft cracks as presented in [R. Kress, P. Serranho, A hybrid method for two-dimensional crack reconstruction, Inverse Problems 21 (2005) 773–784] to the case of sound-hard obstacles. The designation of the method is justified by the fact that it can be interpreted as a hybrid between a regularized Newton method applied to a nonlinear operator equation with the operator that maps the unknown boundary onto the solution of the direct scattering problem and a decomposition method in the spirit of the potential method as described in [A. Kirsch, R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in: Cannon, Hornung (Eds.), Inverse Problems, ISNM, vol. 77, 1986, pp. 93–102. Since the method does not require a forward solver for each Newton step its computational costs are reduced. By some numerical examples we illustrate the feasibility of the method. 相似文献
9.
《高校应用数学学报(英文版)》2016,(4)
Firstly, the Riemann boundary value problem for a kind of degenerate elliptic system of the first order equations in R~4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system's solution, the boundary value problem as stated above is transformed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R~4 are derived. 相似文献
10.
The mixed problem for the Laplace equation outside cuts on the plane is considered. As boundary conditions, the value of the
desired function on one side of each of the cuts and the value of its oblique derivative on the other side are prescribed.
This problem generalizes the mixed Dirichlet-Neumann problem. By using the potential method, the problem reduces to a uniquely
solvable Fredholm integral equation of the second kind.
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 115–135, 2006. 相似文献
11.
In this paper we obtain, by the method of Loewner chains, some sufficient conditions for the analyticity and the univalence of the functions defined by an integral operator. These conditions involves Ruscheweyh and Salagean derivative operator in the open unit disk. In particular cases, we find the well-known conditions for univalency established by Becker [3], Ahlfors [2], Kanas and Srivastava [8] and others for analytic mappings f : U ! C: Also,we obtain the corresponding new, useful and simpler conditions for this integral operator. 相似文献
12.
Haibing Wang Jijun Liu 《高等学校计算数学学报(英文版)》2007,16(3):203-214
We consider the numerical solution for the Helmholtz equation in R~2 with mixed boundary conditions.The solvability of this mixed boundary value problem is estab- lished by the boundary integral equation method.Based on the Green formula,we express the solution in terms of the boundary data.The key to the numerical real- ization of this method is the computation of weakly singular integrals.Numerical performances show the validity and feasibility of our method.The numerical schemes proposed in this paper have been applied in the realization of probe method for inverse scattering problems. 相似文献
13.
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(divG,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. 相似文献
14.
A. A. Polosin 《Differential Equations》2018,54(10):1371-1383
A mixed oblique derivative boundary value problem is considered for the Helmholtz equation in a half-disk. We prove the unique solvability of this problem for sufficiently large values of the parameter occurring in the equation, the leading part of the inverse operator being constructed explicitly. 相似文献
15.
The semilocal convergence for a modified multi-point Jarratt method for solving non-linear equations in Banach spaces is established with the third-order Fréchet derivative of the operator under a general continuity condition. The recurrence relations are derived for the method, and from this, we prove an existence-uniqueness theorem, and give a priori error bounds. The R-order of the method is also analyzed with the third-order Fréchet derivative of the operator under different continuity conditions. Numerical application on non-linear integral equation of the mixed type is given to show our approach. 相似文献
16.
Gunther Schmidt 《Applications of Mathematics》2013,58(3):279-307
The paper is devoted to an integral equation algorithm for studying the scattering of plane waves by multilayer diffraction gratings under oblique incidence. The scattering problem is described by a system of Helmholtz equations with piecewise constant coefficients in ?2 coupled by special transmission conditions at the interfaces between different layers. Boundary integral methods lead to a system of singular integral equations, containing at least two equations for each interface. To deal with an arbitrary number of material layers we present the extension of a recursive procedure developed by Maystre for normal incidence, which transforms the problem to a sequence of equations with 2×2 operator matrices on each interface. Necessary and sufficient conditions for the applicability of the algorithm are derived. 相似文献
17.
We consider acoustic scattering problems described by the mixed boundary value problem for the scalar Helmholtz equation in the exterior of a 2D bounded domain or in the exterior of a crack. The boundary of the domain is assumed to have a finite set of corner points where the scattered wave may have singular behaviour. The paper is concerned with the sensitivity of the far‐field pattern with respect to small perturbations of the shape of the scatterer. Using a modification of the method of adjoint problems, we obtain an integral representation for the Gâteaux derivative which contains only boundary values of functions easily computable by standard BEM and which depends explicitly on the perturbation of the boundary. In some cases, we show the direct influence of the singularities of the solution on the sensitivity of the far‐field pattern. In this way, we generalize the domain sensitivity analysis developed earlier for smooth domains by Hettlich, Kirsch, Kress, Potthast and others. Finally, we show that the same approach can be applied to scattering from 3D domains with smooth edges. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
18.
Maria Giovanna Garroni Jose Luis Menaldi 《Annali di Matematica Pura ed Applicata》1989,154(1):147-222
Summary We construct the Green function for a parabolic- elliptic integro- differential operator of second order with boundary conditions given by a first order operator, both with Hölder continuous coefficients. Also, we prove the existence of a unique invariant density associated with the corresponding reflected diffusion process with jumps. This allows us to study the asymptotic behaviour of the solution for some oblique derivative problems.This research have been supported in part by Ministero P.I. « Progetto Caleolo delle Variazioni » and N.S.F. under grant No. DMS-8702236. 相似文献
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本文处理二阶拟线性混合(椭圆-抛物)型方程在单连通区域上的间断斜微商问题。我们首先导出最简单的混合型方程上述边值问题解的表示式,并证明此边值问题解的唯一性,然后用逐次迭代法证明上述问题解的存在性。本文获得了此间断边值问题的可解性结果,包括有关文献的结果作为特殊情形。 相似文献