In this paper, we consider a three dimensional quantum Navier-Stokes-Poisson equations. Existence of global weak solutions is obtained, and convergence toward the classical solution of the incompressible Navier-Stokes equation is rigorously proven for well prepared initial data. Furthermore, the associated convergence rates are also obtained. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, the stability problem of impulsive functional differential equations with infinite delays is considered. By using Lyapunov functions and the Razumikhin technique, some new theorems on the uniform stability and uniform asymptotic stability are obtained. The obtained results are milder and more general than several recent works. Two examples are given to demonstrate the advantages of the results. Copyright © 2014 John Wiley & Sons, Ltd.

]]>An inverse spectral problem is considered for Dirac operators with parameter-dependent transfer conditions inside the interval, and parameter appears also in one boundary condition. The approach that was used in the investigation of uniqueness theorems of inverse problems for Weyl function or two eigenvalue sets is employed. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this note, a non-standard finite difference (NSFD) scheme is proposed for an advection-diffusion-reaction equation with nonlinear reaction term. We first study the diffusion-free case of this equation, that is, an advection-reaction equation. Two exact finite difference schemes are constructed for the advection-reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection-reaction equation is combined with a finite difference space-approximation of the diffusion term to provide a NSFD scheme for the advection-diffusion-reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.

]]>A multiple monopole method based on the generalized multipole technique is presented for the calculation of band structures of two-dimensional mixed solid/fluid phononic crystals. In this method, the fields are expanded by using the fundamental solutions with multiple origins. Besides the sources used to expand the wave fields, an extra monopole source is introduced as the external excitation. By varying the frequency of the excitation, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and sweeping the boundary of the irreducible first Brillouin zone, the band structure of the phononic crystals can be obtained. The method can consider the fluid–solid interface conditions and the transverse wave mode in the solid component strictly. Some typical examples are illustrated to discuss the accuracy of the present method. Copyright © 2014 John Wiley & Sons, Ltd.

]]>This paper deals with numerical methods for reconstruction of inhomogeneous conductivities. We use the concept of Generalized Polarization Tensors to do reconstruction. Basic resolution and stability analysis are presented. Least-square norm methods with respect to Generalized Polarization Tensors are used for reconstruction of conductivities. Finally, reconstruction of three different types of conductivities in the plane is demonstrated. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The use of asymptotic limits to model heterogeneous plates can be troublesome, because it requires *a priori* knowledge on the ratio between characteristic lengths of heterogeneities and thickness. Moreover, it also relies on some assumption on the inclusions, like periodicity. We propose and analyze here *hierarchical modeling* techniques and show that such approach not only avoids such pitfalls, but it is actually simpler to obtain, and it provably converges to the correct asymptotic limits. Its derivation does not requires any restrictive assumptions on the heterogeneities. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we investigate Poincaré type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. We find exact and easily computable constants in these inequalities for some basic domains (rectangles, cubes, and right triangles) and discuss applications of the inequalities to quantitative analysis of partial differential equations. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Let (*a*,*b*) be a finite interval and 1/*p*, *q*, *r*∈*L*^{1}[*a*,*b*]. We show that a general solution (in the weak sense) of the equation (*p**u*^{′})^{′}+*q**u* = *λ**r**u* on (*a*,*b*) can be constructed in terms of power series of the spectral parameter *λ*. The series converge uniformly on [*a*,*b*] and the corresponding coefficients are constructed by means of a simple recursive procedure. We use this representation to solve different types of eigenvalue problems. Several numerical tests are discussed. Copyright © 2014 John Wiley & Sons, Ltd.

Analytical expressions are derived for the distribution rates of spatial coincidences in the counting of photons produced by spontaneous parametric down conversion. Gaussian profiles are assumed for the wave function of the idler and signal light created in type-I spontaneous parametric down conversion. The distribution rates describe ellipses on the detection planes that are oriented at different angles according to the photon coincidences in either horizontal–horizontal, vertical–vertical, horizontal–vertical, or vertical–horizontal position variables. The predictions are in agreement with the experimental data obtained with a type-I beta-barium borate crystal that is illuminated by a 100-mW violet pump laser as well as with the results obtained from the geometry defined by the phase-matching conditions. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Understanding animal movements and modeling the routes they travel can be essential in studies of pathogen transmission dynamics. Pathogen biology is also of crucial importance, defining the manner in which infectious agents are transmitted. In this article, we investigate animal movement with relevance to pathogen transmission by physical rather than airborne contact, using the domestic chicken and its protozoan parasite *Eimeria* as an example. We have obtained a configuration for the maximum possible distance that a chicken can walk through straight and nonoverlapping paths (defined in this paper) on square grid graphs. We have obtained preliminary results for such walks which can be practically adopted and tested as a foundation to improve understanding of nonairborne pathogen transmission. Linking individual nonoverlapping walks within a grid-delineated area can be used to support modeling of the frequently repetitive, overlapping walks characteristic of the domestic chicken, providing a framework to model fecal deposition and subsequent parasite dissemination by fecal/host contact. We also pose an open problem on multiple walks on finite grid graphs. These results grew from biological insights and have potential applications. © 2014 The Authors. *Mathematical Methods in the Applied Sciences* published by John Wiley & Sons Ltd.

In this paper, a diffusive predator–prey system, in which the prey species exhibits herd behavior and the predator species with quadratic mortality, has been studied. The stability of positive constant equilibrium, Hopf bifurcations, and diffusion-driven Turing instability are investigated under the Neumann boundary condition. The explicit condition for the occurrence of the diffusion-driven Turing instability is derived, which is determined by the relationship of the diffusion rates of two species. The formulas determining the direction and the stability of Hopf bifurcations depending on the parameters of the system are derived. Finally, numerical simulations are carried out to verify and extend the theoretical results and show the existence of spatially homogeneous periodic solutions and nonconstant steady states. Copyright © 2014 John Wiley & Sons, Ltd.

]]>This paper is devoted to study the periodic nature of the solution of the following max-type difference equation:

where the initial conditions *x*_{−2},*x*_{−1},*x*_{0} are arbitrary positive real numbers and is a periodic sequence of period two. Copyright © 2014 John Wiley & Sons, Ltd.

Currently, chaotic systems and chaos-based applications are commonly used in the engineering fields. One of the main structures used in these applications is chaotic control and synchronization. In this paper, the dynamical behaviors of a new hyperchaotic system are considered. Based on Lyapunov Theorem with differential and integral inequalities, the global exponential attractive sets and positively invariant sets are obtained. Furthermore, the rate of the trajectories is also obtained. The global exponential attractive sets of the system obtained in this paper also offer theoretical support to study chaotic control, chaotic synchronization for this system. Computer simulation results show that the proposed method is effective. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The classical orthogonal polynomials (COPs) satisfy a second-order differential equation of the form *σ*(*x*)*y*^{′′}+*τ*(*x*)*y*^{′}+*λ**y* = 0, which is called the equation of hypergeometric type (EHT). It is shown that two numerical methods provide equivalent schemes for the discrete representation of the EHT. Thus, they lead to the same matrix eigenvalue problem. In both cases, explicit closed-form expressions for the matrix elements have been derived in terms only of the zeros of the COPs. On using the equality of the entries of the resulting matrices in the two discretizations, unified identities related to the zeros of the COPs are then introduced. Hence, most of the formulas in the literature known for the roots of Hermite, Laguerre and Jacobi polynomials are recovered as the particular cases of our more general and unified relationships. Furthermore, we present some novel results that were not reported previously. Copyright © 2014 John Wiley & Sons, Ltd.

The aim of this article is to propose an optimization strategy for traffic flowon roundabouts using amacroscopic approach. The roundabout is modeled as a sequence of 2 × 2 junctions with one main lane and secondary incoming and outgoing roads. We consider two cost functionals: the total travel time and the total waiting time, which give an estimate of the time spent by drivers on the network section. These cost functionals areminimized with respect to the right ofway parameter of the incoming roads. For each cost functional, the analytical expression is given for each junction. We then solve numerically the optimization problem and show some numerical results. Copyright © 2014 John Wiley & Sons, Ltd.

]]>This paper is devoted to the study of a nonlinear parabolic p(*x*)-Laplace equation with gradient term and L^{1} data. The authors obtain the existence of renormalized solutions via strong convergence of truncation. Copyright © 2014 John Wiley & Sons, Ltd.

In the paper, we consider equilibrium problems for 2D elastic bodies with thin inclusions modeled in the frame of Timoshenko beam theory. It is assumed that a delamination of the inclusion takes place thus providing a presence of cracks between the inclusion and the elastic body. Nonlinear boundary conditions at the crack faces are imposed to prevent a mutual penetration between the faces. Different problem formulations are analyzed: variational and differential. Dependence on physical parameters characterizing the mechanical properties of the inclusion is investigated. The paper provides a rigorous asymptotic analysis of the model with respect to such parameters. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain rigid inclusions and cracks with the non-penetration conditions, respectively. Also anisotropic inclusions with parameters are analyzed when parameters tend to zero and infinity. In particular, in the limit, we obtain the so called semi-rigid inclusions. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, a collocation method is given to solve singularly perturbated two-point boundary value problems. By using the collocation points, matrix operations and the matrix relations of the Bessel functions of the first kind and their derivatives, the boundary value problem is converted to a system of the matrix equations. By solving this system, the approximate solution is obtained. Also, an error problem is constructed by the residual function, and it is solved by the presented method. Thus, the error function is estimated, and the approximate solutions are improved. Finally, numerical examples are given to show the applicability of the method, and also, our results are compared by existing results. Copyright © 2014 JohnWiley & Sons, Ltd.

]]>We give some alternative forms of the generating functions for the Bernstein basis functions. Using these forms,we derive a collection of functional equations for the generating functions. By applying these equations, we prove some identities for the Bernstein basis functions. Integrating these identities, we derive a variety of identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients, Pascal's rule, Vandermonde's type of convolution, the Bernoulli polynomials, and the Catalan numbers. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, a humoral and cellular immunity virus dynamics model with the Beddington-DeAngelis incidence rate is set up. We derive the basic reproductive number R_{0}, the cytotoxic T lymphocytes immune response reproductive number R_{1}, the humoral immune response reproductive number R_{2}, humoral immune response competitive reproductive number R_{3}, and cytotoxic T lymphocytes immune response competitive reproductive number R_{4}, and a full description of the relation between the existence of the equilibria and reproductive numbers is given. The global properties of the five equilibria are obtained by constructing Lyapunov functions. Copyright © 2014 John Wiley & Sons, Ltd.

This work provides a mathematical model for a predator-prey system with general functional response and recruitment, which also includes capture on both species, and analyzes its qualitative dynamics. The model is formulated considering a population growth based on a general form of *recruitment* and *predator functional response*, as well as the *capture* on both prey and predators at a rate proportional to their populations. In this sense, it is proved that the dynamics and bifurcations are determined by a two-dimensional threshold parameter. Finally, numerical simulations are performed using some ecological observations on two real species, which validate the theoretical results obtained. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we propose a mathematical model to study the dynamics of anorexic and bulimic populations. The model proposed takes into account, among other things, the effects of peers' influence, media influence, and education.

We prove the existence of three possible equilibria that without media influences are disease-free, bulimic-endemic, and endemic. Neglecting media and education effects, we investigate the stability of such equilibria, and we prove that under the influence of media, only one of such equilibria persists and becomes a global attractor. Which of the three equilibria becomes global attractor depends on the other parameters. Copyright © 2014 John Wiley & Sons, Ltd.

We apply the trial method for the solution of Bernoulli's free boundary problem when the Dirichlet boundary condition is imposed for the solution of the underlying Laplace equation, and the free boundary is updated according to the Neumann boundary condition. The Dirichlet boundary value problem for the Laplacian is solved by an exponentially convergent boundary element method. The update rule for the free boundary is derived from the linearization of the Neumann data around the actual free boundary. With the help of shape sensitivity analysis and Banach's fixed-point theorem, we shed light on the convergence of the respective trial method. Especially, we derive a stabilized version of this trial method. Numerical examples validate the theoretical findings.Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we study an optimal control problem for the mixed boundary value problem for an elastic body with quasistatic evolution of an internal damage variable. We suppose that the evolution of microscopic cracks and cavities responsible for the damage is described by a nonlinear parabolic equation. A density of surface traction **p** acting on a part of boundary of an elastic body Ω is taken as a boundary control. Because the initial boundary value problem of this type can exhibit the Lavrentieff phenomenon and non-uniqueness of weak solutions, we deal with the solvability of this problem in the class of weak variational solutions. Using the convergence concept in variable spaces and following the direct method in calculus of variations, we prove the existence of optimal and approximate solutions to the optimal control problem under rather general assumptions on the quasistatic evolution of damage. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we study a class of nonlinear obstacle problems with nonstandard growth. We obtain the L∞ estimate on the solutions and prove the existence and uniqueness of solutions to such problems. Our results are generalizations of the corresponding results in the constant exponent case. Copyright © 2014 John Wiley & Sons, Ltd.

]]>On the basis of zero curvature equations from semi-direct sums of Lie algebras, we construct integrable couplings of the Giachetti–Johnson hierarchy of soliton equations. We also establish Hamiltonian structures of the resulting integrable couplings by the variational identity. Moreover, we obtain bi-integrable couplings of the Giachetti–Johnson hierarchy and their Hamiltonian structures by applying a class of non-semisimple matrix loop algebras consisting of triangular block matrices. Copyright © 2014 John Wiley & Sons, Ltd.

]]>We describe some approaches to construct soliton-type asymptotic solutions for non-integrable equations, including multisoliton case. As an example, we use the generalized Korteweg-de Vries-4 equation with small dispersion. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this article, the sub-equation method is presented for finding the exact solutions of a nonlinear fractional partial differential equations. For this, the fractional complex transformation method has been used to convert fractional-order partial differential equation to ordinary differential equation. The fractional derivatives are described in Jumarie's the modified Riemann–Liouville sense. We apply to this method for the nonlinear time fractional differential equations. With the aid of symbolic computation, a variety of exact solutions for them are obtained. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The short-time Fourier transform has been shown to be a powerful tool for non-stationary signals and time-varying systems. This paper investigates the signal moments in the Hardy–Sobolev space that do not usually have classical derivatives. That is, signal moments become valid for non-smooth signals if we replace the classical derivatives by the Hardy–Sobolev derivatives. Our work is based on the extension of Cohen's contributions to the local and global behaviors of the signal. The relationship of the moments and spreads of the signal in the time, frequency and short-time Fourier domain are established in the Hardy–Sobolev space. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The concept of mono-component is widely used in nonstationary signal processing and time-frequency analysis. In this paper, we construct several classes of complete rational function systems in the Hardy space, whose boundary values are mono-components. Then, we propose a best approximation algorithm (BAA) based on optimal selections of two parameters in the orthonormal bases according to the approximation error. Effectiveness of BAA is evaluated by comparison experiments with the classical Fourier decomposition algorithm. It is also shown that BAA has promising results for filtering out noises and dealing with real-world signals. Copyright © 2014 John Wiley & Sons, Ltd.

]]>We consider the two-dimensional convection–diffusion equation with a fractional Laplacian, supplemented with step-like initial conditions. We show that the large time behavior of solutions to this IVP is described either by rarefaction waves, or diffusion waves, or suitable self-similar solutions, depending on the order of the fractional dissipation and on a direction of a convective nonlinearity. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we consider a discrete fractional boundary value problem of the form:

where 0 < *α*,*β*≤1, 1 < *α* + *β*≤2, *λ* and *ρ* are constants, *γ* > 0, , is a continuous function, and *E*_{β}*x*(*t*) = *x*(*t* + *β* − 1). The existence and uniqueness of solutions are proved by using Banach's fixed point theorem. An illustrative example is also presented. Copyright © 2014 John Wiley & Sons, Ltd.

By means of a non-exact controllability result, we show the necessity of the conditions of compatibility for the exact synchronization by two groups for a coupled system of wave equations with Dirichlet boundary controls. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The boundary function method [A. B. Vasil'eva, V. F. Butuzov, and L. V. Kalachev, *The boundary function method for singular perturbation problems*, SIAM Studies in Applied Mathematics, Philadelphia, 1995] is used to build an asymptotic expansion at any order of accuracy of a scalar time-harmonic wave scattered by a perfectly reflecting doubly periodic surface with oscillations at small and large scales. Error bounds are rigorously established, in particular in an optimal way on the relevant part of the field. It is also shown how the maximum principle can be used to design a homogenized surface whose reflected wave yields a first-order approximation of the actual one. The theoretical derivations are illustrated by some numerical experiments, which in particular show that using the homogenized surface outperforms the usual approach consisting in setting an effective boundary condition on a flat boundary. Copyright © 2014 John Wiley & Sons, Ltd.

The FFT-based homogenization method of Moulinec–Suquet has recently attracted attention because of its wide range of applicability and short computational time. In this article, we deduce an optimal *a priori* error estimate for the homogenization method of Moulinec–Suquet, which can be interpreted as a spectral collocation method. Such methods are well-known to converge for sufficiently smooth coefficients. We extend this result to rough coefficients. More precisely, we prove convergence of the fields involved for Riemann-integrable coercive coefficients without the need for an *a priori* regularization.

We show that our *L*^{2} estimates are optimal and extend to mildly nonlinear situations and *L*^{p} estimates for *p* in the vicinity of 2. The results carry over to the case of scalar elliptic and curl − curl-type equations, encountered, for instance, in stationary electromagnetism. Copyright © 2014 John Wiley & Sons, Ltd.

This work deals with the existence and uniqueness of a nontrivial solution for the third-order *p*-Laplacian *m*-point eigenvalue problems on time scales. We find several sufficient conditions of the existence and uniqueness of nontrivial solution of eigenvalue problems when *λ* is in some interval. The proofs are based on the nonlinear alternative of Leray–Schauder. To illustrate the results, some examples are included. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, sufficient conditions are established for the existence and uniqueness of global solutions to stochastic impulsive systems with expectations in the nonlinear terms. The maximal interval and the estimate of mild solutions are also discussed. These results are obtained by using the fixed point theorem, interval partition, and Lyapunov-like technique. Finally, examples are given to illustrate the theory. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, by incorporating latencies for both human beings and female mosquitoes to the mosquito-borne diseases model, we investigate a class of multi-group dengue disease model and study the impacts of heterogeneity and latencies on the spread of infectious disease. Dynamical properties of the multi-group model with distributed delays are established. The results showthat the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium depends only on the basic reproduction number. Our proofs for global stability of equilibria use the classical method of Lyapunov functions and the graph-theoretic approach for large-scale delay systems. Copyright © 2014 John Wiley & Sons, Ltd.

]]>A new variety of (3 + 1)-dimensional Burgers equations is presented. The recursion operator of the Burgers equation is employed to establish these higher-dimensional integrable models. A generalized dispersion relation and a generalized form for the one kink solutions is developed. The new equations generate distinct solitons structures and distinct dispersion relations as well. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Solvability of Cauchy's problem in for fractional Hamilton–Jacobi equation (1.1) with subcritical nonlinearity is studied here both in the classical Sobolev spaces and in the locally uniform spaces. The first part of the paper is devoted to the global in time solvability of subcritical equation (1.1) in *locally uniform phase space*, a generalization of the standard Sobolev spaces. Subcritical growth of the nonlinear term with respect to the gradient is considered. We prove next the global in time solvability in classical Sobolev spaces, in Hilbert case. Regularization effect is used there to guarantee global in time extendibility of the local solution. Copyright © 2014 John Wiley & Sons, Ltd.

By the means of a differential inequality technique, we obtain a lower bound for blow-up time if p and the initial value satisfy some conditions. Also, we establish a blow-up criterion and an upper bound for blow-up time under some conditions as well as a nonblow-up and exponential decay under some other conditions. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this work, we deal with the existence of the fractional integrable equations involving two generalized symmetries compatible with nonlinear systems. The method used is based on the Bä cklund transformation or B-transformation. Furthermore, we shall factorize the fractional heat operator in order to yield the fractional Riccati equation. This is done by utilizing matrix transform Miura type and matrix operators, that is, matrices whose entries are differential operators of fractional order. The fractional differential operator is taken in the sense of Riemann–Liouville calculus. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we are concerned with the Cauchy problem for the density-dependent incompressible flow of liquid crystals in thewhole space (N ≥ 2).We prove the localwell-posedness for large initial velocity field and director field of the system in critical Besov spaces if the initial density is close to a positive constant. We show also the global well-posedness for this system under a smallness assumption on initial data. In particular, this result allows us to work in Besov space with negative regularity indices, where the initial velocity becomes small in the presence of the strong oscillations. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we study the existence of solutions with prescribed *L*^{2}-norm for the following Kirchhoff type equation:

where *N*≤3,*a*,*b* > 0 are constants, *p*∈(2,2*),2*=6 if *N* = 3, and 2*=+*∞* if *N* = 1,2. We obtain the sharp existence of global constraint minimizers for . In the case , the existence of solutions with prescribed *L*^{2}-norm is obtained for all *L*^{2}-norm. The key point is the analysis of excluding the dichotomy of the minimizing sequences for the related constrained minimization problem. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper we construct two distinct generalized holomorphic orthogonal function systems over infinite cylinders in . Explicit representation formulae and properties of the obtained basis functions are given. As an application, we present some plot simulations that illustrate the results of this work. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we consider a one-dimensional linear Bresse system with infinite memories acting in the three equations of the system. We establish well-posedness and asymptotic stability results for the system under some conditions imposed into the relaxation functions regardless to the speeds of wave propagations. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Previous analysis and research on the power option – one of the exotic options – have focused on the interest rate of the stock and its volatility as constant parameters throughout the run of execution. In this paper, we attempt to extend these results to the more practical and realistic case of when these parameters are time dependent. By making no *ansatz* or relying on *ad hoc* methods, we are able to achieve this via an algorithmic method – the Lie group approach – leading to exact solutions for the power option problem. Copyright © 2014 John Wiley & Sons, Ltd.

Let *u*_{ϵ} be the solution of the Poisson equation in a domain perforated by thin tubes with a nonlinear Robin-type boundary condition on the boundary of the tubes (the flux here being *β*(*ϵ*)*σ*(*x*,*u*_{ϵ})), and with a Dirichlet condition on the rest of the boundary of Ω. *ϵ* is a small parameter that we shall make to go to zero; it denotes the period of a grid on a plane where the tubes/cylinders have their bases; the size of the transversal section of the tubes is *O*(*a*_{ϵ}) with *a*_{ϵ}≪*ϵ*. A certain nonperiodicity is allowed for the distribution of the thin tubes, although the perimeter is a fixed number *a*. Here, is a strictly monotonic function of the second argument, and the adsorption parameter *β*(*ϵ*) > 0 can converge toward infinity. Depending on the relations between the three parameters *ϵ*, *a*_{ϵ}, and *β*(*ϵ*), the effective equations in volume are obtained. Among the multiple possible relations, we provide *critical relations*, which imply different averages of the process ranging from linear to nonlinear. All this allows us to derive spectral convergence as *ϵ*0 for the associated spectral problems in the case of *σ* a linear function of *u*_{ϵ}. Copyright © 2014 John Wiley & Sons, Ltd.

The main objective of this research note is to provide an interesting result for the reducibility of the Kampé de Fériet function. The result is derived with the help of two results for the terminating _{3}*F*_{2} series very recently obtained by Rakha *et al*. A few interesting special cases have also been given. Copyright © 2014 John Wiley & Sons, Ltd.

Applying three critical point theorems, we prove the existence of at least three weak solutions for a class of differential equations with *p*(*x*)-Laplacian and subject to small perturbations of nonhomogeneous Neumann conditions. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with asymptotic stability of a Bresse system with two frictional dissipations. Under mathematical condition of equal speed of wave propagation, we prove that the system is exponentially stable. Otherwise, we show that Bresse system is not exponentially stable. Then, in the latter case, by using a recent result in linear operator theory, we prove the solution decays polynomially to zero with optimal decay rate. Better rates of polynomial decay depending on the regularity of initial data are also achieved. Copyright © 2014 John Wiley & Sons, Ltd.

We obtain an explicit formula for the diagonal singularities of the scattering amplitude for the Dirac equation with short-range electromagnetic potentials. Using this expansion we uniquely reconstruct an electric potential and magnetic field from the high-energy limit of the scattering amplitude. Moreover, supposing that the electric potential and magnetic field are asymptotic sums of homogeneous terms we give the unique reconstruction procedure for these asymptotics from the scattering amplitude, known for some energy *E*. Furthermore, we prove that the set of the averaged scattering solutions to the Dirac equation is dense in the set of all solutions to the Dirac equation that are in *L*^{2}(Ω), where Ω is any connected bounded open set in with smooth boundary, and we show that if we know an electric potential and a magnetic field for , then the scattering amplitude, given for some energy *E*, uniquely determines these electric potential and magnetic field everywhere in . Combining this uniqueness result with the reconstruction procedure for the asymptotics of the electric potential and the magnetic field we show that the scattering amplitude, known for some *E*, uniquely determines a electric potential and a magnetic field, that are asymptotic sums of homogeneous terms, which converges to the electric potential and the magnetic field respectively. Moreover, we discuss the symmetries of the kernel of the scattering matrix, which follow from the parity, charge-conjugation and time-reversal transformations for the Dirac operator. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we are interested in looking for multiple solutions for the following system of nonhomogenous Kirchhoff-type equations:

- (1.1)

where constants *a*,*c* > 0;*b*,*d*,*λ*≥0, *N* = 1,2 or 3, *f*,*g*∈*L*^{2}(*R*^{N}) and *f*,*g*≢0, *F*∈*C*^{1}(*R*^{N}×*R*^{2},*R*), , *V*∈*C*(*R*^{N},*R*) satisfy some appropriate conditions. Under more relaxed assumptions on the nonlinear term *F*, the existence of one negative energy solution and one positive energy solution for the nonhomogenous system 2.1 is obtained by Ekeland's variational principle and Mountain Pass Theorem, respectively. Copyright © 2014 John Wiley & Sons, Ltd.

In this note, we study the application of generalized fractional operators to a particular class of nonstandard Lagrangians. These are typical of dissipative systems, and the corresponding Euler–Lagrange and Hamilton equations are analyzed. The dependence of the equation of motion on the generalized kernel permits to obtain a wide range of different configurations of motion. Some examples are discussed and analyzed.Copyright © 2014 John Wiley & Sons, Ltd.

This paper considers the one parameter families of extrinsic differential geometries of Lorentzian hypersurfaces on pseudo *n*-spheres. We give a continuous relationship among the three types Gauss indicatrices by one parameter map. Meanwhile, we also give the singularity analysis of the one parameter Gauss indicatrices of Lorentzian hypersurfaces on pseudo *n*-spheres by the Legendrian singularity theory. Copyright © 2014 John Wiley & Sons, Ltd.

We develop a numerical predictive tool for multiphase fluid mixtures consisting of biofilms grown in a viscous fluid matrix by implementing a second-order finite difference discretization of the multiphase biofilm model developed recently on a general purpose graphic processing unit. With this numerical tool, we study a 3-D biomass–flow interaction resulting in biomass growth, structure formation, deformation, and detachment phenomena in biofilms grown in a water channel in quiescent state and subject to a shear flow condition, respectively. The numerical investigation is limited in the viscous regime of the biofilm–solvent mixture. In quiescent flows, the model predicts growth patterns consistent with experimental findings for single or multiple adjacent biofilm colonies, the so-called mushroom shape growth pattern. The simulated biomass growth both in density and thickness matches very well with the experimentally grown biofilm in a water channel. When shear is imposed at a boundary, our numerical studies reproduce wavy patterns, pinching, and streaming phenomena observed in biofilms grown in a water channel. Copyright © 2013 John Wiley & Sons, Ltd.

]]>This paper deals with a fully parabolic attraction–repulsion chemotaxis system in two-dimensional smoothly bounded domains. It is shown that the system admits global *bounded* classical solutions whenever the repulsion is dominated. The proof is based on an entropy-like inequality and coupled estimate techniques. Copyright © 2014 John Wiley & Sons, Ltd.

This paper studies the construction and approximation of quasi-interpolation for spherical scattered data. First of all, a kind of quasi-interpolation operator with Gaussian kernel is constructed to approximate the spherical function, and two Jackson type theorems are established. Second, the classical Shepard operator is extended from Euclidean space to the unit sphere, and the error of approximation by the spherical Shepard operator is estimated. Finally, the compact supported kernel is used to construct quasi-interpolation operator for fitting spherical scattered data, where the spherical modulus of continuity and separation distance of scattered sampling points are employed as the measurements of approximation error. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we consider three-dimensional incompressible magnetohydrodynamics equations. By using interpolation inequalities in anisotropic Lebesgue space, we provide regularity criteria involving the velocity or alternatively involving the fractional derivative of velocity in one direction, which generalize some known results. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this article, we give some results on the *S*-essential spectra of a linear operator defined on a Banach space. Furthermore, we apply the obtained results to determine the *S*-essential spectra of an integro-differential operator with abstract boundary conditions in the Banach space *L*_{p}([−*a*,*a*] × [−1,1]),*p* ≥ 1 and *a* > 0. Copyright © 2012 John Wiley & Sons, Ltd.

This paper deals with a discrete Nicholson's blowflies model with linear harvesting term. By using contraction mapping fixed point theorem, we obtain sufficient conditions for the existence of unique almost periodic positive solution. Moreover, we investigate exponential convergence of the almost periodic positive solution by Lyapunov functional. Copyright © 2013 John Wiley & Sons, Ltd.

The aim of this paper is to show that we can extend the notion of *convergence in the norm-resolvent sense* to the case of several unbounded noncommuting operators (and to quaternionic operators as a particular case) using the notion of *S*-resolvent operator. With this notion, we can define bounded functions of unbounded operators using the *S*-functional calculus for *n*-tuples of noncommuting operators. The same notion can be extended to the case of the *F*-resolvent operator, which is the basis of the *F*-functional calculus, a monogenic functional calculus for *n*-tuples of commuting operators. We also prove some properties of the *F*-functional calculus, which are of independent interest. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, we study the approximate controllability of control systems with state and control in Banach spaces and described by a second-order semilinear abstract differential equation. We compare the approximate controllability of the system with the approximate controllability of an associated discrete system. Copyright © 2013 John Wiley & Sons, Ltd.

]]>In this paper, a right-hand side identification problem for a parabolic equation with an overdetermined condition on an observation point is considered. A first and second order of accuracy difference schemes are constructed for obtaining approximate solutions of the problem that arises in two-phase flow in capillaries. Stability estimates and numerical results are also established. Copyright © 2013 John Wiley & Sons, Ltd.

Based on Adomian decomposition method, a new algorithm for solving boundary value problem (BVP) of nonlinear partial differential equations on the rectangular area is proposed. The solutions obtained by the method precisely satisfy all boundary conditions, except the small pieces near the four corners of the rectangular area. A theorem on the boundary error is given. Hence, the Adomian decomposition method is more efficiently applied to BVPs for partial differential equations. Segmented and weighted analytical solutions with a high accuracy for the BVP of nonlinear groundwater equations on a rectangular area are obtained by the present algorithm. Copyright © 2013 John Wiley & Sons, Ltd.

The induction equation of relativistic magnetohydrodynamics is considered as a singular perturbation problem for small magnetic diffusivity. When the quantities depend on a single space variable, the resulting hyperbolic equation may be studied with techniques of asymptotic analysis. Different approximations are found for initial, intermediate, and large times. The last case is the most difficult; the approximate magnetic flux function satisfies a certain parabolic equation. This equation is studied from the viewpoint of energy dissipation, providing clues on the behavior of the electric and magnetic fields. Copyright © 2013 John Wiley & Sons, Ltd.

]]>The present investigation deals with a predator–prey model with disease that spreads among the predator species only. The predator species is split out into two groups—the susceptible predator and the infected predator both of which feeds on prey species. The stability and bifurcation analyses are carried out and discussed at length. On the basis of the normal form theory and center manifold reduction, the explicit formulae are derived to determine stability and direction of Hopf bifurcating periodic solution. An extensive quantitative analysis has been performed in order to validate the applicability of our model under consideration. Copyright © 2013 John Wiley & Sons, Ltd.

]]>In this paper, we study the existence of infinitely many homoclinic solutions for the second-order self-adjoint discrete Hamiltonian system , where , and are unnecessarily positive definites for all . By using the variant fountain theorem, we obtain an existence criterion to guarantee that the aforementioned system has infinitely many homoclinic solutions under the assumption that *W*(*n*,*x*) is asymptotically quadratic as | *x* | + ∞ . Copyright © 2013 John Wiley & Sons, Ltd.

We consider an elastic rod with rounded ends and diameter proportional to a small parameter *h* > 0. The roundness of the ends is described by an exponent *m* ∈ (0,1). We derive for the rod an asymptotically sharp Korn inequality with a special weighted anisotropic norm. Weight factors with *m*-dependent powers of *h* appear both in the anisotropic norm and the Korn inequality itself, and we discover three critical values *m* = 1 ∕ 4, *m* = 1 ∕ 2 and *m* = 3 ∕ 4 at which these exponents are crucially changed. Copyright © 2014 John Wiley & Sons, Ltd.

We consider a kinematic dynamo model in a bounded interior simply connected region Ω and in an insulating exterior region . In the so-called direct problem, the magnetic field **B** and the electric field **E** are unknown and are driven by a given incompressible flow field **w**. After eliminating **E**, a vector and a scalar potential ansatz for **B** in the interior and exterior domains, respectively, are applied, leading to a coupled interface problem. We apply a finite element approach in the bounded interior domain Ω, whereas a symmetric boundary element approach in the unbounded exterior domain Ω^{c} is used. We present results on the well-posedness of the continuous coupled variational formulation, prove the well-posedness and stability of the semi-discretized and fully discretized schemes, and provide quasi-optimal error estimates for the fully discretized scheme. Copyright © 2013 John Wiley & Sons, Ltd.

We study the existence of positive solutions for systems of second-order nonlinear ordinary differential equations, subject to multipoint boundary conditions. Copyright © 2013 John Wiley & Sons, Ltd.

We propose three new discrete variational schemes that capture the conservative-dissipative structure of a generalized Kramers equation. The first two schemes are single-step minimization schemes, whereas the third one combines a streaming and a minimization step. The cost functionals in the schemes are inspired by the rate functional in the Freidlin-Wentzell theory of large deviations for the underlying stochastic system. We prove that all three schemes converge to the solution of the generalized Kramers equation. Copyright © 2013 John Wiley & Sons, Ltd.

]]>This paper deals with the form and the periodicity of the solutions of the max-type system of difference equations

where , and are positive two-periodic sequences and initial values *x*_{0}, *x*_{ − 1}, *y*_{0}, *y*_{ − 1} ∈ (0, + ∞ ). Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we present a method for the construction of a class of multi-step finite differences schemes for solving arbitrary order linear two-point boundary value problems. The construction technique is based on Padé approximant. It is easy to derive multi-step difference schemes, and it includes many existing schemes as its special cases. Numerical experiments show that the proposed schemes are flexible and convergent. Copyright © 2013 John Wiley & Sons, Ltd.

]]>We study in this paper the Dirichlet boundary value problem of second-order differential system in the form

where *u* ∈ *R*^{n},*A*,*B* ∈ *R*^{n},*M* ∈ *R*^{n × n} is a symmetric matrix, *F* : [0,1] × *R*^{n} *R*^{n} such a system comes from a model describing the vibration of a multi-storey building. By using the saddle point theorem, we prove an existence theorem for the solutions to the given system. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we study the following nonlinear problem of Kirchhoff type with critical Sobolev exponent:

where *a*, *b* > 0 are constants. Under certain assumptions on the sign-changing function *f*(*x*,*u*), we prove the existence of positive solutions by variational methods.

Our main results can be viewed as a partial extension of a recent result of He and Zou in [Journal of Differential Equations, 2012] concerning the existence of positive solutions to the nonlinear Kirchhoff problem

where *ϵ* > 0 is a parameter, *V* (*x*) is a positive continuous potential, and with 4 < *p* < 6 and satisfies the Ambrosetti–Rabinowitz type condition. Copyright © 2013 John Wiley & Sons, Ltd.