Generalized Wiener-Hopf Equations Research Articles

The generalized Wiener–Hopf equations for the elastic wave motion in angular regions

In this work, we introduce a general method to deduce spectral functional equations in elasticity and thus, the generalized Wiener–Hopf equations (GWHEs), for the wave motion in angular regions filled by arbitrary linear homogeneous media and illuminated by sources localized at infinity. The work extends the methodology used in electromagnetic applications and proposes for the first time a complete theory to get the GWHEs in elasticity. In particular, we introduce a vector differential equation of first-order characterized by a matrix that depends on the medium filling the angular region. The functional equations are easily obtained by a projection of the reciprocal vectors of this matrix on the elastic field present on the faces of the angular region. The application of the boundary conditions to the functional equations yields GWHEs for practical problems. This paper extends and applies the general theory to the challenging canonical problem of elastic scattering in angular regions.

The generalized Wiener-Hopf equations for wave motion in angular regions: electromagnetic application.

In this work, we introduce a general method to deduce spectral functional equations and, thus, the generalized Wiener–Hopf equations (GWHEs) for wave motion in angular regions filled by arbitrary linear homogeneous media and illuminated by sources localized at infinity with application to electromagnetics. The functional equations are obtained by solving vector differential equations of first order that model the problem. The application of the boundary conditions to the functional equations yields GWHEs for practical problems. This paper shows the general theory and the validity of GWHEs in the context of electromagnetic applications with respect to the current literature. Extension to scattering problems by wedges in arbitrarily linear media in different physics will be presented in future works.

Convergence and stability analysis of iterative algorithm for a generalized set-valued mixed equilibrium problem

In this paper, we consider a generalized set-valued mixed equilibrium problem (in short, GSMEP) in real Hilbert space. Related to GSMEP, we consider a generalized Wiener-Hopf equation problem (in short, GWHEP) and show an equivalence relation between them. Further, we give a fixed-point formulation of GWHEP and construct an iterative algorithm for GWHEP. Furthermore, we extend the notion of stability given by Harder and Hick [3] and prove the existence of a solution of GWHEP and discuss the convergence and stability analysis of the iterative algorithm. Our results can be viewed as a refinement and improvement of some known results in the literature.

The Electromagnetic Field for a PEC Wedge Over a Grounded Dielectric Slab: 2. Diffraction, Modal Field, Surface Waves, and Leaky Waves

AbstractTogether with Part 1, Part 2 describes the theory, the validation, and the application of the generalized Wiener‐Hopf technique (GWHT) to complex scattering problems constituted of wedges and layers that may interact at near field. In particular, we present the full analysis of the canonical problem where a perfectly electrically conducting (PEC) wedge is lying on a grounded dielectric slab bounded by a PEC plane. The structure proposed in the two papers is of interest in antenna technologies and electromagnetic compatibility (tip on a substrate with guiding and antenna properties). While the scope of Part 1 has been to present the method and its validation applied to the problem, this paper (Part 2) focuses on the properties of the solution and it presents physical and engineering insights. Part 2 starts from the solution of generalized Wiener‐Hopf equations (GWHEs) through Fredholm factorization in terms of an analytical element of the spectral unknowns (Part 1), and it illustrates how to estimate physical and engineering characteristics of the problem: geometrical theory of diffraction/uniform theory of diffraction (GTD/UTD) coefficients, total far fields, modal fields, and excitation of surface waves and leaky waves for different excitations (modal source and/or plane wave source).

Wiener–Hopf equation whose kernel is a probability distribution

We prove the existence of a solution of an inhomogeneous generalized Wiener–Hopf equation whose kernel is a probability distribution on R generating a random walk drifting to +∞, while the inhomogeneous term f of the equation belongs to the space L 1(0,∞) or L ∞(0,∞). We establish the asymptotic properties of the solution of this equation under various assumptions about the inhomogeneity f.

Arbitrarily Oriented Perfectly Conducting Wedge Over a Dielectric Half-Space: Diffraction and Total Far Field

Complex scattering targets are often made by structures constituted by wedges and penetrable substrates which may interact at near field. In this paper, we describe a complete procedure to study this problem with possible developments in radar technologies (like GPR), antenna development, or electromagnetic compatibility (tips near substrates). The diffraction of an incident plane wave by a perfectly conducting (PEC) wedge over a dielectric half-space is studied using generalized Wiener–Hopf equations (GWHEs), and the solution is obtained using analytical and numerical–analytical approaches that reduce the Wiener-Hopf (WH) factorization to integral equations (IEs). The mathematical aspects are described in a unified and consistent theory for angular and layered region problems. The proposed procedure is valid for the general case and the paper focuses on E-polarization at normal incidence. The solutions are given in terms of geometrical/uniform theory of diffraction (GTD/UTD) diffraction coefficients and total far fields for engineering applications. This paper presents several numerical test cases that show the validity of the proposed method.

Sensitivity Analysis for General Nonlinear Nonconvex Variational Inequalities

In this communication, we proved that the parametric general nonlinear nonconvex variational inequalities are equivalent to the parametric general Wiener-Hopf equations. We use this alternative equivalence formulation to studied the sensitivity analysis for general nonlinear nonconvex variational inequalities without assuming the differentiability of the given data.

EXTENDED GENERAL NONLINEAR QUASI-VARIATIONAL INEQUALITIES AND PROJECTION DYNAMICAL SYSTEMS

The aim of this paper is to introduce and study a new class of the extended general nonlinear quasi-variational inequalities and a new class of the extended general Wiener-Hopf equations. The equivalence between the extended general nonlinear quasi-variational inequalities and the fixed point problems, and as well as the extended general Wiener-Hopf equations is established. Then by using these equivalences, we discuss the existence and uniqueness of a solution of the extended general nonlinear quasi-variational inequalities. Applying the equivalent alternative formulation and a nearly uniformly Lipschitzian mapping $S$, we define some new $p$-step projection iterative algorithms with mixed errors for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping $S$ which is also a unique solution of the extended general nonlinear quasi-variational inequalities. The convergence analysis of the suggested iterative schemes under some suitable conditions is studied. We also suggest and analyze a class of extended general projection dynamical systems associated with the extended general nonlinear quasi-variational inequalities. We show that the trajectory of the solution of the extended general projection dynamical system converges globally exponential to a unique solution of the extended general nonlinear quasi-variational inequalities. Results obtained in this paper may be viewed as an refinement and improvement of the previously known results.

The Wiener-Hopf Equation Technique for Solving General Nonlinear Regularized Nonconvex Variational Inequalities

AbstractIn this paper, we introduce and study some new classes of extended general nonlinear regularized non-convex variational inequalities and the extended general nonconvex Wiener-Hopf equations, and by the projection operator technique, we establish the equivalence between the extended general nonlinear regularized nonconvex variational inequalities and the fixed point problems as well as the extended general nonconvex Wiener-Hopf equations. Then by using this equivalent formulation, we discuss the existence and uniqueness of solution of the problem of extended general nonlinear regularized nonconvex variational inequalities. We apply the equivalent alternative formulation and a nearly uniformly Lipschitzian mapping S for constructing some new p-step projection iterative algorithms with mixed errors for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping S which is unique solution of the problem of extended general nonlinear regularized nonconvex variational inequalities. We also consider the convergence analysis of the suggested iterative schemes under some suitable conditions.Mathematics Subject Classification (2010)Primary 47H05; Secondary 47J20, 49J40

The Wiener-Hopf Formulation of the Penetrable Wedge Problem. Part III: The Skew Incidence Case

Earlier works obtained the semi-analytical solution of the generalized Wiener-Hopf equations that formulate the diffraction of a plane wave by a dielectric wedge at normal incidence. This solution permits the separating, recognizing, and understanding of the different components of the dielectric wedge field: reflected and refracted plane waves, surface waves, lateral waves, diffraction coefficients. This article considers the general case of a plane wave impinging on an arbitrary penetrable wedge at skew incidence. Comparing previous findings with the skew incident case, no new conceptual and numerical difficulties are present. However, the involved algebraic manipulations become very cumbersome, and to handle them, the MATHEMATICA (Wolfram Research, Inc., Champaign, Illinois, USA) computer program was used.

The Wiener-Hopf Solution of the Isotropic Penetrable Wedge Problem: Diffraction and Total Field

The diffraction of an incident plane wave by an isotropic penetrable wedge is studied using generalized Wiener-Hopf equations, and the solution is obtained using analytical and numerical-analytical approaches that reduce the Wiener-Hopf factorization to Fredholm integral equations of second kind. Mathematical aspects are described in a unified and consistent theory for angular region problems. The formulation is presented in the general case of skew incidence and several numerical tests at normal incidence are reported to validate the new technique. The solutions consider engineering applications in terms of GTD/UTD diffraction coefficients and total fields.

The Wiener-Hopf Formulation of the Penetrable Wedge Problem: Part II

Solutions of the generalized Wiener-Hopf equations obtained in the companion article (Daniele, 2010) do not provide the global spectra of the unknowns but only some of their analytical elements. It means that this solution is not sufficient to evaluate the dielectric wedge field. To obtain the global spectra, a process of analytical continuation is required. In this article, this is accomplished by using some properties of the Wiener-Hopf equations of the problem. After obtaining the global spectra of the Wiener-Hopf unknowns, the article presents Laplace transform expressions for arbitrary observation direction. In order to evaluate the far field, the Sommerfeld technique is briefly recalled. This technique permits one to separate, recognize, and understand the different components of the dielectric wedge field: reflected and refracted plane waves, surface waves, lateral waves, and diffraction coefficients.

The Wiener-Hopf Formulation of the Penetrable Wedge Problem: Part I

The diffraction by a dielectric wedge of an arbitrary aperture angle is studied by means of the generalized Wiener-Hopf technique. Generalized Wiener-Hopf equations are deduced that apparently cannot be solved in closed form. The factorization of the kernels can be reduced to Fredholm equations of the second kind by using a standard procedure (Fredholm factorization). The solutions of the Fredholm equations are obtained through a simple quadrature technique that very quickly yields a numerically stable evaluation of the unknowns of the generalized Wiener-Hopf equations. The generalized Wiener-Hopf technique provides a representation of the fields only in certain regions of the spectral domain. To obtain valid solutions everywhere, a process of analytical continuation is required. This latter task is accomplished in the companion article.

An asymptotic property of the solution to the homogeneous generalized Wiener-Hopf equation

We consider the homogeneous generalized Wiener-Hopf equation $$ S(x) = \int\limits_{ - \infty }^x {S(x - y)F(dy),x \geqslant 0,} $$ wehere F is a probability distribution on ℝ with zero mean, finite variance, and infinite moment ∫0∞x3F(dy). Its P*-solution S(x) enjoys the property $$ S(x) - ax \sim b\int\limits_0^x {\int\limits_y^\infty {\int\limits_v^\infty {F((u,\infty ))dudvdy} } asx \to \infty ,} $$ where a and b are explicit positive constants.

On the existence of a solution of a homogeneous system of generalized Wiener-Hopf equations

We prove the existence of a non-decreasing solution of a homogeneous system of generalized Wiener-Hopf equations and establish asymptotic properties of this solution.

Sensitivity analysis for a system of generalized mixed implicit equilibrium problems in uniformly smooth Banach spaces

In this paper, a new system of parametric generalized mixed implicit equilibrium problems involving non-monotone set-valued mappings in real Banach spaces is introduced and studied. We first generalize the notion of the Yosida approximation in Hilbert spaces introduced by Moudafi to reflexive Banach spaces. Further, by using the notion of the Yosida approximation, we consider a system of parametric generalized Wiener–Hopf equation problems and show its equivalence to the system of parametric generalized mixed implicit equilibrium problems. By using a fixed point formulation of the system of parametric generalized Wiener–Hopf equation problems, we study the behavior and sensitivity analysis of a solution set of the system of parametric generalized mixed implicit equilibrium problems. We prove that, under suitable assumptions, the solution set of the system of parametric generalized mixed implicit equilibrium problems is nonempty, closed and Lipschitz continuous with respect to the parameters. Our results are new, and improve and generalize some known results in this field.

Projection iterative methods for extended general variational inequalities

In this paper, we introduce and study a new class of variational inequalities involving three operators, which is called the extended general variational inequality. Using the projection technique, we show that the extended general variational inequalities are equivalent to the fixed point and the extended general Wiener-Hopf equations. This equivalent formulation is used to suggest and analyze a number of projection iterative methods for solving the extended general variational inequalities. We also consider the convergence of these new methods under some suitable conditions. Since the extended general variational inequalities include general variational inequalities and related optimization problems as special cases, results proved in this paper continue to hold for these problems.

Existence and iterative approximation of solutions of generalized mixed equilibrium problems

In this paper, we consider a generalized mixed equilibrium problem involving non-monotone set-valued mappings in real Hilbert space. We extend the notions of the Yosida approximation and its corresponding regularized operator given by Moudafi and Thera [A. Moudafi, M. Thera, Proximal and dynamical approaches to equilibrium problems, in: Lecture Notes in Econom. and Math. System, vol. 477, Springer-Verlag, Berlin, 2002, pp. 187–201] and discuss some of their properties. Further, we consider a generalized Wiener–Hopf equation problem and show its equivalence with the generalized mixed equilibrium problem. Using a fixed point formulation of the generalized Wiener–Hopf equation problem, we construct an iterative algorithm. Furthermore, we extend the notion of stability given by Harder and Hick [A.M. Harder, T.L. Hicks, Stability results for fixed-point iteration procedures, Math. Japonica 33 (5) (1998) 693–706]. We prove the existence of a solution and discuss the convergence and stability of the iterative algorithm for the generalized Wiener–Hopf equation problem. Since the generalized mixed equilibrium problems include variational inequalities as special cases, the results presented in this paper continue to hold for these problems.

General Wiener–Hopf equation technique for nonexpansive mappings and general variational inequalities in Hilbert spaces

In this paper, we show the general variational inequality problems are equivalent to solving the general Wiener–Hopf equations. By using the equivalence, we establish a general iterative algorithm for finding the solution of general variational inequalities, general Wiener–Hopf and the fixed point of nonexpansive mappings. Our results extend and improve the recent corresponding results announced by many others.

General Wiener-Hopf equations and nonexpansive mappings

In this paper, we show that the general variational inequalities are equivalent to a new class of general Wiener-Hopf equations involving the nonexpansive mappings. Using this equivalence, we suggest and analyze an iterative method for finding the common elements of the solution set of the general variational inequalities and the solution set of the fixed-point of the nonexpansive mapping. We also consider the convergence criteria of the proposed method under some mild conditions. Since the general variational inequalities and the Wiener-Hopf equations include several classes of variational inequalities and Wiener-Hopf equations as special cases, our results continue to hold for these problems. Results obtained in this paper may be viewed as a refinement and improvement of the previously known results.