Fractional and Wiener–Hopf factorizations

A connection between Wiener–Hopf factorizations of an analytic matrix function a(t) and fractional factorizations of the rational part of a⁻¹(t) is obtained. The result is applied to an explicit construction of Wiener–Hopf factorizations of a(t).     

A Wiener–Hopf approximation technique for a multiple plate diffraction problem

An approximation method is derived for the computation of the acoustic field between a series of parallel plates, subject to a time periodic incident field. The method is based on the Wiener–Hopf method of factorization, with computations involving orthogonal bases of functions that are analytic in the complex half‐plane. Copyright 2004 John Wiley & Sons, Ltd.     

Numerical solution of the Helmholtz equation in an infinite strip by Wiener‐Hopf factorization

We describe an algorithm to compute numerically the solution of the Helmholtz equation: Δu + κu = f, u ∈ H₀¹(S), where S is an infinite strip and κ a given bounded function. By using the finite difference approximation on the entire strip, we are led to solve an infinite linear system. When κ is constant the associated matrix is block Toeplitz and banded and the system can be solved using a Wiener‐Hopf factorization. This approach can also be adapted to deal with the case when κ is constant outside a bounded domain of the strip. Numerical results are given to assess the performance of our method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Surface wave scattering by a rectangular impedance cylinder located on a reactive plane

The electromagnetic scattering of the surface wave by a rectangular impedance cylinder located on an infinite reactive plane is considered for the case that the impedances of the horizontal and vertical sides of the cylinder can have different values. Firstly, the diffraction problem is reduced into a modified Wiener–Hopf equation of the third kind and then solved approximately. The solution contains branch‐cut integrals and two infinite sets of constants satisfying two infinite systems of linear algebraic equations. The approximate analytical or numerical evaluations of corresponding integrals and numerical solution of the linear algebraic equation systems are obtained for various values of parameters such as the surface reactance of the plane, the vertical and horizontal wall impedances, the width and the height of the cylinder. Copyright © 2004 John Wiley & Sons, Ltd.     

Generalized factorization for N×N Daniele–Khrapkov matrix functions

A generalization to N×N of the 2×2 Daniele–Khrapkov class of matrix‐valued functions is proposed. This class retains some of the features of the 2×2 Daniele–Khrapkov class, in particular, the presence of certain square‐root functions in its definition. Functions of this class appear in the study of finite‐dimensional integrable systems. The paper concentrates on giving the main properties of the class, using them to outline a method for the study of the Wiener–Hopf factorization of the symbols of this class. This is done through examples that are completely worked out. One of these examples corresponds to a particular case of the motion of a symmetric rigid body with a fixed point (Lagrange top). Copyright © 2001 John Wiley & Sons, Ltd.     

Diffraction of sound waves by a rigid cylindrical cavity of finite length with an internal impedance surface

A rigorous solution is presented for the problem of diffraction of plane harmonic sound waves by a cavity formed by a terminated rigid cylindrical waveguide of finite length whose interior surface is lined by an acoustically absorbent material. The solution is obtained by a modification of the Wiener-Hopf technique and involve an infinite series of unknowns, which are determined from an infinite system of linear algebraic equations. Numerical solution of this system is obtained for various values of the parameters of the problem and their effects on the diffraction phenomenon are shown graphically.Received: December 12, 2001     

Reflection and transmission of plane acoustic waves in an infinite annular duct with a finite gap on the inner wall

The diffraction of acoustic waves by an infinitely long annular duct having a finite gap on the inner wall is investigated rigorously. The related boundary‐value problem is formulated into a modified Wiener–Hopf equation, which is then reduced to a pair of simultaneous Fredholm integral equations of the second kind. At the end of the analysis, numerical results illustrating the effects of the width of the coaxial cylindrical waveguide and the gap length on the diffraction phenomenon are presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Effect of cold plasma permittivity on the radiation of the dominant TEM‐wave by an impedance loaded parallel‐plate waveguide radiator

The effects of wall impedances on the radiation of the dominant transverse electromagnetic wave by an impedance loaded parallel‐plate waveguide radiator immersed in a cold plasma have been analyzed. The solution to the governing mathematical model in cold plasma is determined while using the Wiener–Hopf technique. It is observed that the amplitude of the radiated field increases with increasing permittivity of the plasma. The work presented may be of great interest to quantify the effects of ionosphere plasma on the communicating signals between Earth station and an artificial satellite in the Earth's atmosphere. Copyright © 2015 John Wiley & Sons, Ltd.     

An operator approach for an oblique derivative boundary‐transmission problem

We consider a boundary‐transmission problem for the Helmholtz equation, in a Bessel potential space setting, which arises within the context of wave diffraction theory. The boundary under consideration consists of a strip, and certain conditions are assumed on it in the form of oblique derivatives. Operator theoretical methods are used to deal with the problem and, as a consequence, several convolution type operators are constructed and associated to the problem. At the end, the well‐posedness of the problem is shown for a range of non‐critical regularity orders of the Bessel potential spaces, which include the finite energy norm space. In addition, an operator normalization method is applied to the critical orders case. Copyright © 2004 John Wiley & Sons, Ltd.     

Solutions to Painleve III and other nonlinear equations by a generalized Cole–Hopf transformation

In this paper, we introduce a generalized form of Cole‐Hopf transformation and apply it to find new closed‐form (analytic) solutions to Painleve III equation. The same transformation is used then to find analytic solutions for the van der Pol and other nonlinear convective equations. These solutions provide analytic insights to some practical problems and might be used also to test the accuracy of numerical solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation of a delayed reaction–diffusion neural network

In this paper, a delayed reaction–diffusion neural network with Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of the trivial uniform steady state is discussed. The existence of Hopf bifurcation at the trivial steady state is established. Using the normal form theory and the center manifold reduction of partial function differential equations, explicit formulae are derived to determine the direction and stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation of a delayed Cohen–Grossberg neural network with diffusion

In this paper, a delayed Cohen–Grossberg neural network with diffusion under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equation, the local stability of the trivial uniform steady state and the existence of Hopf bifurcation at the trivial steady state are established, respectively. By using the normal form theory and the center manifold reduction of partial function differential equations, formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results. Copyright © 2016 John Wiley & Sons, Ltd.     

On the new solutions of the generalized Burgers–Huxley equation

In this paper, we investigate a class of generalized Burgers–Huxley equation by employing the bifurcation method of planar dynamical systems. Firstly, we reduce the equation to a planar system via the traveling wave solution ansatz; then by computing the singular point quantities, we obtain the conditions of integrability and determine the existence of one stable limit cycle from Hopf bifurcation in the corresponding planar system. From this, some new exact solutions and a special periodic traveling wave solution, which is isolated as a limit, are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

A nonhomogeneous boundary value problem for the Kuramoto–Sivashinsky equation in a quarter plane

We study the initial boundary value problem for the one‐dimensional Kuramoto–Sivashinsky equation posed in a half line with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results of the Cauchy problem of the Kuramoto–Sivashinsky equation posed on the whole line , the initial boundary value problem of the Kuramoto–Sivashinsky equation is shown to be globally well‐posed in Sobolev space for any s >?2. Copyright © 2017 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation of a delayed virus infection model with Beddington–DeAngelis infection function and cytotoxic T‐lymphocyte immune response

In this paper, a class of virus infection model with Beddington–DeAngelis infection function and cytotoxic T‐lymphocyte immune response is investigated. Time delay in the immune response term is incorporated into the model. We show that the dynamics of the model are determined by the basic reproduction number and the immune response reproduction number . If , then the infection‐free equilibrium is globally asymptotically stable. If , then the immune‐free equilibrium is globally asymptotically stable. If , then the stability of the interior equilibrium is investigated. We conclude that Hopf bifurcation occurs as the time delay passes through a critical value. Numerical simulations are carried out to support our theoretical conclusion well. Copyright © 2015 John Wiley & Sons, Ltd.     

Energy decay estimates for the dissipative wave equation with space–time dependent potential

We study the asymptotic behavior of solutions of dissipative wave equations with space–time‐dependent potential. When the potential is only time‐dependent, Fourier analysis is a useful tool to derive sharp decay estimates for solutions. When the potential is only space‐dependent, a powerful technique has been developed by Todorova and Yordanov to capture the exact decay of solutions. The presence of a space–time‐dependent potential, as in our case, requires modifications of this technique. We find the energy decay and decay of the L² norm of solutions in the case of space–time‐dependent potential. Copyright © 2010 John Wiley & Sons, Ltd.     

Exact travelling wave solutions for the generalized nonlinear Schrödinger (GNLS) equation with a source by Extended tanh–coth, sine–cosine and Exp-Function methods

The capability of Extended tanh–coth, sine–cosine and Exp-Function methods as alternative approaches to obtain the analytic solution of different types of applied differential equations in engineering mathematics has been revealed. In this study, the generalized nonlinear Schrödinger (GNLS) equation is solved by three different methods. To obtain the single-soliton solutions for the equation, the Extended tanh–coth and sine–cosine methods are used. Furthermore, for this nonlinear evolution equation the Exp-Function method is applied to derive various travelling wave solution. Results show that while the first two procedures easily provide a concise solution, the Exp-Function method provides a powerful mathematical means for solving nonlinear evolution equations in mathematical physics.     

Mixed boundary value problems of diffraction by a half‐plane with an obstacle perpendicular to the boundary

The paper is devoted to the analysis of wave diffraction problems modeled by classes of mixed boundary conditions and the Helmholtz equation, within a half‐plane with a crack. Potential theory together with Fredholm theory, and explicit operator relations, are conveniently implemented to perform the analysis of the problems. In particular, an interplay between Wiener–Hopf plus/minus Hankel operators and Wiener–Hopf operators assumes a relevant preponderance in the final results. As main conclusions, this study reveals conditions for the well‐posedness of the corresponding boundary value problems in certain Sobolev spaces and equivalent reduction to systems of Wiener–Hopf equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Modified ‐expansion method for finding exact wave solutions of the coupled Klein–Gordon–Schrödinger equation

The coupled Klein–Gordon–Schrödinger equation is reduced to a nonlinear ordinary differential equation (ODE) by using Lie classical symmetries, and various solutions of the nonlinear ODE are obtained by the modified ‐expansion method proposed recently. With the aid of solutions of the nonlinear ODE, more explicit traveling wave solutions of the coupled Klein–Gordon–Schrödinger equation are found out. The traveling wave solutions are expressed by the hyperbolic functions, trigonometric functions, and rational functions. Copyright © 2012 John Wiley & Sons, Ltd.     

Qualitative behavior and exact travelling wave solutions of the Zhiber–Shabat equation

In this paper, the qualitative behavior and exact travelling wave solutions of the Zhiber–Shabat equation are studied by using qualitative theory of polynomial differential system. The phase portraits of system are given under different parametric conditions. Some exact travelling wave solutions of the Zhiber–Shabat equation are obtained. The results presented in this paper improve the previous results.