Hopf Equation Research Articles

A Class of Projection Methods for General Variational Inequalities

In this paper, we consider and analyze a new class of projection methods for solving pseudomonotone general variational inequalities using the Wiener–Hopf equations technique. The modified methods converge for pseudomonotone operators. Our proof of convergence is very simple as compared with other methods. The proposed methods include several known methods as special cases.

Development of a sixth-order two-dimensional convection–diffusion scheme via Cole–Hopf transformation

In this paper, we develop a two-dimensional finite-difference scheme for solving the time-dependent convection–diffusion equation. The numerical method exploits Cole–Hopf equation to transform the nonlinear scalar transport equation into the linear heat conduction equation. Within the semi-discretization context, the time derivative term in the transformed parabolic equation is approximated by a second-order accurate time-stepping scheme, resulting in an inhomogeneous Helmholtz equation. We apply the alternating direction implicit scheme of Polezhaev to solve the Helmholtz equation. As the key to success in the present simulation, we develop a Helmholtz scheme with sixth-order spatial accuracy. As is standard practice, we validated the code against test problems which were amenable to exact solutions. Results show excellent agreement for the one-dimensional test problems and good agreement with the analytical solution for the two-dimensional problem.

High-frequency diffraction by a rectangular impedance cylinder on an impedance plane

The scattering of a line source field by a rectangular impedance cylinder on an infinite impedance plane is studied. The diffraction problem is formulated into a modified Wiener–Hopf equation of the third kind and then solved approximately. The solution contains two infinite sets of constants satisfying two infinite systems of linear algebraic equations. Numerical solutions of these systems are obtained for various values of the surface reactance, width and height of the rectangular cylinder, whereby the effects of these parameters on the diffraction phenomenon is studied. The results are compared with the corresponding physical optics solution and the agreement is very satisfactory.

Factorization of Conservative Integral Convolution Type Operators with Slowly Decaying Kernels

Factorization of Conservative Integral Convolution Type Operators with Slowly Decaying Kernels L. G. Arabadzhyan Armenian State Pedagogical University, Erevan, Armenia Received September 23, 1999 1◦. A method for solving convolution type equations (in particular, Wiener–Hopf equations) on the basis of the Volterra factorization of integral operators and the analysis of the nonlinear functional factorization equations was suggested in [1–5]. Let R ≡ [0,+∞), and let E (R) be one of the spaces Lp (R), p ≥ 1, and M (R). By J we denote the identity operator in E (R), and by K we denote the Wiener–Hopf integral operator (K f)(x) = ∞ ∫

Wiener–Hopf analysis of the two-dimensional box-like horn radiator

A hybrid method consisting of employing the mode-matching method in conjunction with the Fourier transform technique is used to analyze the radiation of the dominant TEM-wave from a two-dimensional rectangular horn. The hybrid method that we will adopt reduces the related boundary value problem into a scalar modified Wiener–Hopf equation of the third kind. The solution involves branch-cut integrals which are evaluated approximately and infinitely many unknown constants satisfying an infinite system of linear algebraic equations susceptible to a numerical treatment. Some computational results illustrating the effects of various parameters such as waveguide width, horn length, etc., on the radiation phenomenon are also presented. Comparison of the results with those obtained for a similar finite structure is proved satisfactory.

The scattering of a plane wave by two parallel semi-infinite overlapping screens with dielectric loading

The diffraction of a plane electromagnetic wave by two parallel, overlapping, perfectly conducting half-planes with a dielectric loading is considered by using the Fourier transform technique in conjunction with the mode-matching method. This mixed method of formulation enables one to obtain directly two uncoupled modified Wiener–Hopf equations instead of a matrix one. When the overlap length is set to zero and the dielectric loading is removed, the results are shown to coincide exactly with those obtained previously by Abrahams [1]. Numerical results showing the effects of the dielectric loading on the transmitted field are also presented.

RECONSTRUCTION OF ACOUSTIC TRANSFER MATRICES BY INSTATIONARY COMPUTATIONAL FLUID DYNAMICS

Thermoacoustic combustion instabilities are a frequently encountered problem in the operation of combustion equipment. The “brute-force” application of computational fluid dynamics to the analysis of thermoacoustic instabilities is estimated to be forbiddingly expensive for many systems of technical interest due to the high computational demands of a time- and space-accurate simulation of a (low Mach number) compressible reacting flow in a complex geometry. Thermoacoustic systems can be modelled efficiently as networks of acoustic multi-ports, where each multi-port corresponds to a certain component of the system, e.g., air or fuel supply, burner, flame, combustor and suitable terminations, and is represented mathematically by its transfer matrix. For some multi-ports, the transfer matrix can be derived analytically from first principles: i.e., the equations of fluid motions and suitable approximations. However, the acoustic behavior of more complicated components, e.g., a burner or a flame, has to be determined by empirical methods, by using a “black box” approach common in communications engineering. In this work, a method is introduced which allows one to reconstruct the transfer matrix of an acoustic two-port from an instationary computation of the response of the two-port to an imposed perturbation of the steady state. Firstly, from the time series data of fluctuating velocity and pressure on both sides of the two-port, the auto- and cross-correlations of the fluctuations are estimated. Then, the unit impulse responses of the multi-port are computed by inverting the Wiener–Hopf equation. Finally, the unit impulse responses are z -transformed to yield the coefficients of the transfer matrix. The method is applied to the one-dimensional model of a heat source with time delay placed in a low-Mach-number compressible flow, for which an analytical description can be derived from first principles. Computational predictions of the transfer matrix have been validated successfully against these analytical results. Furthermore, a comparison of the novel approach presented in this paper with a method for computing the frequency response of a flame by Laplace-transforming its step response is carried out.

Slip boundary condition on an idealized porous wall

Slip boundary condition on a porous wall is investigated by considering a viscous flow near an idealized porous wall based on the Stokes’ approximation. The idealized porous wall is composed of a large number of parallel and equidistant thin semi-infinite plates. The flow is assumed to be a simple shear flow far from the plates and stagnant deep inside the channels of the plates. The slip velocity on the idealized porous wall is calculated by solving a Wiener–Hopf equation. From the streamline patterns shown, it is found that Moffatt’s infinite sequence of viscous eddies developed between the two adjacent plates. Pressure and shear stress distributions on the plates are shown and local flow near the edge of the plate is discussed. Laminar shear flow at the farfield unidirectional along the edges of the plates is also considered.

Slow viscous flow in a partitioned channel

Two-dimensional slow viscous flow in a partitioned channel is investigated based on the Stokes approximation. The partitioned channel is composed of two infinite parallel planes and a semi-infinite plate located midway between the infinite planes. The flow is allowed to turn around the semi-infinite plate by a proper pressure gradient. By solving a Wiener–Hopf equation, an exact analytic solution for the stream function is obtained. From the streamline patterns shown, it is found that a Moffatt’s infinite sequence of viscous eddies develops in the channel. The pressure and shear stress distribution on the boundaries are illustrated, and a local analysis for the flow near the edge of the plate is performed. By superposing the symmetric and the anti-symmetric flow, general flows in this partitioned channel are obtained and typical flow patterns are shown.

Approximate factorization of Wiener–Hopf kernels

The key step in the solution of a Wiener–Hopf equation is the factorization of the Fourier transform of the kernel, K(α) (here, α is the Fourier variable). Exact factorizations are seldom useful, either because they do not exist or because they are too cumbersome to evaluate accurately. It is desirable, therefore, to construct approximations of K(α) which are both easy to factorize and simple to evaluate. Recently, Crighton examined the case that K depends on a small parameter, ε; i.e., K=K(α;ε). Crighton showed that matched asymptotic expansions (MAE) can be used to construct a composite asymptotic approximation to K for ε→0, valid uniformly over the full required range of α. The composite approximation comprises several relatively simple factors in typical applications, and these factors are then decomposed into their Wiener–Hopf factors by standard techniques. Abrahams has introduced an alternative approximation method for Wiener–Hopf kernels based on Padé approximants. Padé approximants give explicit uniform approximations to K valid in the full relevant range of α, do not require a small expansion parameter, and can be computed with relatively little effort. Although Crighton’s and Abrahams’s approaches seem at first glance unrelated, this talk will show that under certain circumstances, they are closely related.

A Method for Solving the Hopf Equation and Generalization of Sobolev–Schwartz Distributions

We introduce distributions that are functionals with values in a nonarchimedian field of Laurent series. These objects naturally generalize Sobolev–Schwartz distributions, and we use them to find generalized solutions of the Hopf equation in the form of infinitely narrow solitons and shock waves. We propose a method for computation of the profile of an infinitely narrow soliton and of a shock wave described by the Hopf equation.

Singular Solution of the Liouville Equation under Perturbation

The Cauchy problem for the Liouville equation with a small perturbation is considered. We are interested in the asymptotics of the perturbed solution under the assumption that one has singularity. The main goal is to study both the asymptotic approximation of the singular lines and the asymptotic approximation of the solution everywhere outside the narrow neighbourhood of the singular lines. The Cauchy problem for the Liouville equation with a small perturbation ∂ 2 t u−∂ 2 u+8 exp u = eF[u], 0 <e � 1, (0.1) u|t=0 = ψ0(x; e) ,∂ tu|t=0 = ψ1(x; e) ,x ∈ R (0.2) is considered. We are interesting for asymptotics of the perturbed solution u(x, t; e )a s e → 0. Perturbation theory for integrable equations remains a very attractive task. As a rule a perturbation of smooth solutions such as a single soliton were usually considered. We intend here to discuss a perturbation of a singular solution under assumption that the perturbed solution has singularities as well. A simple well known instance of this kind is a chock wave under weak perturbation as given by the Hopf equation with a small perturbation term ut + uux = ef (u), 0 <e � 1. In this paper we consider a more complicated problem namely a perturbation of the singular solution of Liouville equation. We deal with the singularities studied by Pogrebkov and Polivanov [1].

Some Relationships in the Wiener–Hopf Equation and an Algorithm for its Solution

When designing multidimensional optimal systems of information processing and control it is necessary to solve Wiener–Hopf matrix equations. The present article suggests a simple method for their solution.

Analysis of an Algorithm for Solving the Wiener–Hopf Equation

A simple proof is given of the validity of one of the algorithms for solving the Wiener–Hopf equation.

Noise and speckle reduction in synthetic aperture radar imagery by nonparametric Wiener filtering

We present a Wiener filter that is especially suitable for speckle and noise reduction in multilook synthetic aperture radar (SAR) imagery. The proposed filter is nonparametric, not being based on parametrized analytical models of signal statistics. Instead, the Wiener-Hopf equation is expressed entirely in terms of observed signal statistics, with no reference to the possibly unobservable pure signal and noise. This Wiener filter is simple in concept and implementation, exactly minimum mean-square error, and directly applicable to signal-dependent and multiplicative noise. We demonstrate the filtering of a genuine two-look SAR image and show how a nonnegatively constrained version of the filter substantially reduces ringing.

Generalized quasi-variational inequalities for fuzzy mappings

In this paper, we use the projection technique to prove that generalized quasi-variational inequality for fuzzy mappings are equivalent to the implicit Wiener–Hopf equations for fuzzy mappings. This equivalence is used to analyze a number of iterative algorithms for solving variational inequalities for fuzzy mappings. We also study the convergence of these iterative algorithms.

Remarkable statistical behavior for truncated Burgers-Hopf dynamics.

A simplified one-dimensional model system is introduced and studied here that exhibits intrinsic chaos with many degrees of freedom as well as increased predictability and slower decay of correlations for the large-scale features of the system. These are important features in common with vastly more complex problems involving climate modeling or molecular biological systems. This model is a suitable approximation of the Burgers-Hopf equation involving Galerkin projection on Fourier modes. The model has a detailed mathematical structure that leads to a well-defined equilibrium statistical theory as well as a simple scaling theory for correlations. The numerical evidence presented here strongly supports the behavior predicted from these statistical theories. Unlike the celebrated dissipative and dispersive approximations of the Burgers-Hopf equation, which exhibit exactly solvable and/or completely integrable behavior, these model approximations have strong intrinsic chaos with ergodic behavior.

Large- [formula omitted] limit of the generalized 2DYang–Mills theory on cylinder

Using the collective field theory approach of large- N generalized two-dimensional Yang–Mills theory on cylinder, it is shown that the classical equation of motion of collective field is a generalized Hopf equation. Then, using the Itzykson–Zuber integral at the large- N limit, it is found that the classical Young tableau density, which satisfies the saddle-point equation and determines the large- N limit of free energy, is the inverse of the solution of this generalized Hopf equation, at a certain point.

Características hidráulicas de tubos de bambu gigante

Tendo em vista a falta de estudos e pesquisas que possibilitem o desenvolvimento de tecnologias para o aproveitamento de bambu gigante (Dendrocalamus giganteus) como conduto de água procurou-se, com este trabalho, verificar a validade das equações de Darcy-Weisbach, Hazen-Williams e Manning, na estimativa da perda de carga em tubulações de bambu submetidos a dois processos de remoção de nós dos colmos: ao método de impacto por lâminas circulares e ao de um dispositivo mecânico desenvolvido especialmente para este fim, foram estimados os seguintes parâmetros hidráulicos: coeficientes das equações de Hazen-Williams (C), Manning (η), e rugosidade absoluta (ε), estimada através das equações de Hopf e Colebrook. A análise foi realizada sob as seguintes condições operacionais: pressão de trabalho variando de 146,6 a 195,5 kPa, intervalo de vazão de 3 a 12,9 L s-1 e diâmetros de tubulação de 90 a 130 mm, cujos resultados demonstraram que a remoção dos nós pelo processo mecânico proporcionou melhor acabamento no tubo e, conseqüentemente, menor perda de carga por atrito. Os valores obtidos para os parâmetros hidráulicos foram: ε = 0,0013 m (Hopf), ε= 0,0022 m (Colebrook), C = 89 e η= 0,014, para os tubos perfurados mecanicamente, e, C = 43 e η = 0,027 para os tubos perfurados por impacto. Não se constatou diferença significativa na estimativa da perda de carga para os tubos perfurados mecanicamente, pelas equações de Darcy-Weisbach, Hazen-Williams e Manning.

Optimal reconstruction in multirate transmultiplexer systems under channel noise: Wiener separation filtering approach

The problem of optimal signal reconstruction in transmultiplexer systems under channel noise is considered in this work. The objective of signal reconstruction in transmultiplexer systems is to develop an appropriate separation filter bank design such that the output signals can resemble the original input signals as closely as possible. Based on some multirate digital signal processing techniques, such as polyphase decomposition, noble identity and signal blocking, a transmultiplexer system can be transformed into an equivalent one of single sampling rate, but with blocked input/output signals. It is shown that for a given combining filter bank, a causal and stable separation filter bank is developed by using calculus of variation method and matrix spectral factorization technique, to achieve optimal signal reconstruction of noisy transmultiplexer systems from the viewpoint of Wiener–Hopf equation in the frequency domain. For practical application, a suboptimal FIR separation filter bank is also presented to approximate the proposed IIR Wiener separation filter bank via the least-squares method and maintain the reconstruction performance. Three examples of a two-band, a five-band and a speech transmission transmultiplexer system are provided to illustrate the reconstruction performance of the proposed method.