Hopf Equation Research Articles

Short branch cut approximation in two-dimensional hydrodynamics with free surface

A potential motion of ideal incompressible fluid with a free surface and infinite depth is considered in two-dimensional geometry. A time-dependent conformal mapping of the lower complex half-plane of the auxiliary complex variable w into the area filled with fluid is performed with the real line of w mapped into the free fluid’s surface. The fluid dynamics can be fully characterized by the motion of the complex singularities in the analytical continuation of both the conformal mapping and the complex velocity. We consider the short branch cut approximation of the dynamics with the small parameter being the ratio of the length of the branch cut to the distance between its centre and the real line of w . We found that the fluid dynamics in that approximation is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including moving square root branch points and poles. These solutions involve practical initial conditions resulting in jets and overturning waves. The solutions are compared with the simulations of the fully nonlinear Eulerian dynamics giving excellent agreement even when the small parameter approaches about one.

A Correlation-Less Approach Towards Adaptive Channel Equalizer Based on Wiener–Hopf Equation

Wiener, the well-known adaptive filter is seen in almost all the applications of adaptive signal processing. It gets the weight updating by solving the Wiener –Hopf equation (ωopt = R−1P). When implemented on the state of the art hardware, like ACIS or FPGA, this filter results in substantial resource consumption. This paper proposes updating the filter weights by reducing the need for an auto-correlation matrix (R) and calculating the filter weights based on cross-correlation vector (P) only. This approach would result in a noticeable resource reduction by compromising some of the performance that may be acceptable in voice or video being less sensitive. The proposed correlation-less approach and the conventional method are first simulated in Matlab and then is implemented in Xilinx Vertex 7 FPGA for one to one comparison. The results indicate that the proposed approach reduces a significant amount of resources along with acceptable performance characteristics. Therefore, it may be incorporated in various applications of mobile communication, especially the adaptive channel equalizer.

Simple solutions of the Burgers and Hopf equations

Описаны все решения уравнения Бюргерса, имеющие аналитическую сложность не выше чем один. Оказалось, что все такие решения распадаются в четыре семейства, чьи размерности не превосходят $3$ и которые представлены элементарными функциями. Приведен пример семейства решений уравнения Бюргерса сложности $2$. Аналогичная задача решена и для уравнения Хопфа. Оказалось, что все решения уравнения Хопфа сложности один - это $2$-параметрическое семейство дробно-линейных функций, которое совпадает с одним из семейств решений уравнения Бюргерса. Библиография: 6 наименований.

Mutual influence of the faces contact area and the pre-fracture zone near the tip of the interfacial crack

BACKGROUND: Under plane strain conditions, a crack model was developed on a plane interface between two different materials, which assumes the existence near its tip of the faces contact area and a narrow lateral pre-fracture zone in a less crack-resistant material of the composite compound. The pre-fracture zone is modeled by the line of normal displacement rupture, on which the normal stress is equal to the tensile strength of the material. Assuming that the dimensions of the pre-fracture zone and the contact zone have the same order of magnitude and are significantly smaller than the crack length, the problem is reduced to the vector Wiener–Hopf equation. METHODS: An approximate method for solving the vector Wiener–Hopf equation was developed, which was used to obtain the equations for determining the sizes of the pre-fracture zone and the contact faces area. The pre-fracture zone orientation was determined from the condition of the potential energy maximum accumulated in the zone. Numerical calculations of the indicated parameters and analysis of their dependences on the configuration and module of external load are executed. RESULTS: A significant mutual influence of the pre-fracture zone and crack faces contact on their sizes and orientation of the zone was revealed.

Asymptotic expansions for Wiener–Hopf equations

Wiener–Hopf Equations are of the form [Formula: see text] These equations arise in many physical problems such as radiative transport theory, reflection of an electromagnetive plane wave, sound wave transmission from a tube, and in material science. They are also known as the renewal equations on the half-line in Probability Theory. In this paper, we present a method of deriving asymptotic expansions for the solutions to these equations. Our method makes use of the Wiener–Hopf technique as well as the asymptotic expansions of Stieltjes and Hilbert transforms.

On the Relationship Between the Factorization Problem in the Wiener Algebra and the Truncated Wiener–Hopf Equation

On the Relationship Between the Factorization Problem in the Wiener Algebra and the Truncated Wiener–Hopf Equation

Simple solutions of the Burgers and Hopf equations

We describe all solutions of the Burgers equation of analytic complexity not exceeding . It turns out that all such solutions fall into four families of dimensions not exceeding that are represented by elementary functions. An example of a family of solutions of the Burgers equation of complexity is given. A similar problem is also solved for the Hopf equation. It turns out that all solutions to the Hopf equation of complexity form a two-parameter family of fractional-linear functions which coincides with one of the families of solutions of the Burgers equation.

Diffraction by a rigid strip in a plate modelled by Mindlin theory.

We consider a plane flexural wave incident on a semi-infinite rigid strip in a Mindlin plate. The boundary conditions on the strip lead to three Wiener-Hopf equations, one of which decouples, leaving a scalar problem and a 2 × 2 matrix problem. The latter is solved using a simple method based on quadrature. The far-field diffraction coefficient is calculated and some numerical results are presented. We also show how the results reduce to the simpler Kirchhoff model in the low-frequency limit.

On the Wiener-Hopf solution of water-wave interaction with a submerged elastic or poroelastic plate.

A solution to the problem of water-wave scattering by a semi-infinite submerged thin elastic plate, which is either porous or non-porous, is presented using the Wiener-Hopf technique. The derivation of the Wiener-Hopf equation is rather different from that which is used traditionally in water-waves problems, and it leads to the required equations directly. It is also shown how the solution can be computed straightforwardly using Cauchy-type integrals, which avoids the need to find the roots of the highly non-trivial dispersion equations. We illustrate the method with some numerical computations, focusing on the evolution of an incident wave pulse which illustrates the existence of two transmitted waves in the submerged plate system. The effect of the porosity is studied, and it is shown to influence the shorter-wavelength pulse much more strongly than the longer-wavelength pulse.

Wiener–Hopf approach for the coaxial waveguide with an impedance-coated groove on the inner wall

In the present work, the band-pass filter characteristics of a coaxial waveguide with an impedance coated groove on the inner wall and perfectly conducting outer wall is analyzed rigorously through the Wiener–Hopf technique. By using the direct Fourier transform, the related boundary value problem is reduced to the Wiener–Hopf equation whose solution contains infinitely many constants satisfying an infinite system of linear algebraic equations. These equations are solved numerically and the field terms, which depends on the solution obtained numerically, are derived explicitly. The problem is also analyzed by applying a mode-matching technique and the results are compared numerically. Besides, some computational results illustrating the effects of parameters such as depth of the groove, surface impedances and radii of the walls are also presented.

The Wiener–Hopf equation with probability kernel of oscillating type

The Wiener–Hopf equation with probability kernel of oscillating type

Truncated Wiener–Hopf equation and matrix function factorization

Truncated Wiener–Hopf equation and matrix function factorization

Inner symmetries of the spatially singular part of the solutions of the Burgers equation and their Lie representations

We describe two new discrete symmetries of the inviscid Burgers (or Riemann–Hopf) equation ut+uux=0. We derived both of them using a local, formal approach of Hopf algebraic renormalization, a tool recently used in algorithmic computations. We prove that one of them is a Lie point transformation. Symmetries generate new exact solutions from the known solutions and provide useful frames of reference in the study of shock wave formation.

On Simple Solutions of Some Equations of Mathematical Physics

All solutions to the Burgers, Hopf, Helmholtz, Klein–Gordon, sine-Gordon, Schrodinger, and Monge–Ampere equations having analytical complexity one (simple solutions) are described. It turns out that all simple solutions of the Burgers and Hopf equation are represented by elementary functions. An example of a family of solutions of complexity two to the Burgers equation is presented. Simple solutions to the Helmholtz (or Klein–Gordon) equation are expressed in terms of Bessel functions and elementary functions. For the Laplace and wave equations, an explicit description is given for the simple solutions that are expressed in terms of Jacobi elliptic functions. Open problems of the theory of analytic complexity (the analytical spectrum of an equation) are discussed.

Radiation of sound by a coaxial waveguide with semi-infinite perforated duct

ABSTRACT Radiation of sound waves by a coaxial duct is considered. The coaxial duct has an inner rigid wall which is infinite, while the outer wall is rigid for and perforated for which makes the problem more interesting. Applying the Fourier transform technique, the mixed boundary value problem is described by a Wiener–Hopf equation and then the solution of the problem is found by means of the Wiener–Hopf (WH) technique. The influence of the parameters of the problem like inner and outer duct radii, frequency and porosity on the sound pressure level is displayed graphically. This study is important because of its applications in noise reduction in modern aircraft jet and turbofan engines. The current problem is also compared with subproblems with and without perforated and very good agreement is observed.

Analyzing the Impact of Sigma-Delta Modulation on Performance Parameters of Adaptive Filters

Sigma-delta modulation (SDM), initially introduced for ADCs and DACs is now well known in an alternative domain of short word-length (SWL) algorithm based digital system design. Its applications start from simple arithmetic operations to very complex circuits, like the adaptive filters. The conventional multi-bit processing systems consume more resources in their implementation on ASICs and FPGAs, whereas, SDM based systems being processing the data in a single-bit result a compact design. This paper is an effort towards SDM based single-bit, design of adaptive filter using Wiener–Hopf equation on FPGA, along with analyzing the impact of SDM on performance parameters, like, Signal to Noise Ratio (SNR), Probability of Error (PE), and Mean Square Error (MSE). Besides, the error graph, showing the difference between the filtered signal in multi-bit and SDM based is also shown and analyzed using Matlab. As, the mobile phone operates in a continuous varying environment (for example, during the travel), and hence the statistical parameters (SNR, MSE, and PE) due to varying channel also vary accordingly in its conventional operations and architectures, but results show that instead if SDM is used, those statistical parameters remain constant, and in comparison to the multi-bit approach the SDM based system outdoes in all means of functionality as well as performance. Hence, it may be concluded that for continuous varying environments of wireless communication, especially the mobile communication systems, the SDM based approach may be adopted for reasons of good performance as well as compact system design.

Radiation of acoustic waves by a partially lined pipe with an interior perforated screen

The paper presents analytical and numerical results of radiation phenomena at the far field and solution of the acoustic wave equation with boundary conditions imposed by the pipe wall. A semi-infinite pipe with partial lining and interior perforated screen is considered. The study is important because of its applications in noise reduction in exhausts of automobile engines, in modern aircraft jet and turbofan engines. This problem is described by a Wiener–Hopf equation and then solved numerically. The solution involves three infinite sets of coefficients satisfying three infinite systems of linear algebraic equations. Numerical solutions of these systems are obtained for various values of the parameters of the problem and their effects on the radiation phenomenon are shown graphically.

EM-Wave Diffraction by a Finite Plate with Neumann Conditions Immersed in Cold Plasma

The present research article presents the investigation of diffraction phenomenon of EM-plane wave by a non-symmetric plate of finite length under the influence of cold plasma. The Wiener–Hopf equation is formulated by boundary value problem related to this model and Fourier transform. The standard way of Wiener–Hopf procedure is used to tackle the resulting equation. Asymptotic expansion and modified stationary phase method are used to find the result for the diffracted wave by finite plate under the assumption of Neumann boundary conditions in an anisotropic medium. The case of an isotropic medium has been obtained by assigning the particular values to elements of permittivity tensor. The high-frequency signal can be assumed only when very large operating frequency is taken into account as compared to the cyclotron frequency. Various physical parameters for isotropic and anisotropic medium are discussed graphically.

A new class of exact solutions in the planar nonstationary problem of motion of a fluid with a free boundary

We consider the classical problem of potential unsteady flow of an ideal incompressible fluid with a free boundary. It was previously discovered that in the absence of external forces and capillarity, a wide class of exact solutions of the problem can be described by the Hopf equation for a complex velocity. We here obtain a new class of solutions described by the Hopf equation for a quantity that is the inverse of the complex velocity. These solutions describe the evolution of two-dimensional perturbations of the free boundary in compression or expansion of a circular cavity (in the unperturbed state) in the fluid.

Diffraction of sound waves by a lined cylindrical cavity

In this paper, diffraction of sound waves through a lined cavity is analyzed rigorously. The inner–outer surfaces of the cavity and the base of the cavity are coated with three different absorbing linings. By using the Fourier transform technique in conjunction with the Mode-Matching method, the related boundary value problem is formulated as a Wiener–Hopf equation. In the solution, two infinite sets of unknown coefficients are involved that satisfy two infinite systems of linear algebraic equations. Numerical solution of this system is obtained for various values of the parameters of the problem. The graphical results are also presented which show that how efficiently the sound diffraction can be reduced by selection of problem parameters.