In this paper, we provide a sufficient condition, in terms of the horizontal gradient of two horizontal velocity components and the gradient of liquid crystal molecular orientation field, for the breakdown of local in time strong solutions to the three-dimensional incompressible nematic liquid crystal flows. More precisely, let *T*_{∗} be the maximal existence time of the local strong solution (*u*,*d*), then *T*_{∗}<+*∞* if and only if

where *u*^{h}=(*u*^{1},*u*^{2}), ∇_{h}=(*∂*_{1},*∂*_{2}). This result can be regarded as the generalization of the well-known Beale-Kato-Majda (BKM) type criterion and is even new for the three-dimensional incompressible Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.

Drawing on viral dynamics theory, this paper presents a differential equations model with time delay to investigate the stock investor behavior driven by new product announcement (NPA) signal. Visually, we look upon investors in stock market as cells *in vivo* and the NPA signals as free virus. The potential investors will be ‘infected’ by the dissociative NPA signal and then make investment decisions. In order to better understand the ‘infection’ process, we extract and establish a multi-stage process during which NPA signal is delivered and ‘infects’ the potential investors. A time-delay effect is employed to reflect the evaluation stage at which potential investors comprehensively evaluate and decide whether to invest or not. In addition, we introduce a set of external and internal factors into the model, including information sensitivity and investor sentiment, and so on, which are pivotal for examining investor behavior. Equilibrium analysis and numerical simulations are employed to check out the properties of the model and highlight the practical application values of the model. Copyright © 2016 John Wiley & Sons, Ltd.

Solution of any engineering problem starts with a modelling process, which typically involves a choice among different kinds of models. To create a realistic model, one has to think carefully about the modelling process. Particularly in the case of coupled problems when several models are coupled together to represent a given physical phenomenon. This paper presents an approach based on the category theory that allows to describe this modelling process on a more abstract level. Using the advantages of abstract level, one can describe the coupling process in a concise way and introduce certain criteria to check consistency of a coupled model. The main idea of the proposed approach is to introduce a structure in the modelling process, which allows to see how different models interact without a precise look into them. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Wave propagation in lossy nonlinear metamaterials is analytically investigated by means of perturbation methods. In the left-handed band of the nonlinear metamaterial, a higher-order nonlinear Schrödinger equation is obtained, while in the frequency band gap, a dissipative short-pulse equation is derived. In both cases, dissipation is described by linear terms, which lead to an exponential decay of the solutions. The decay rates, that is, the inverses of the linear loss coefficients in these two models, are found in terms of the dielectric and magnetic properties of the metamaterial. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper presents a cylindrical multipole expansion for periodic sources with applications for three-phase power cables. It is the aim of the contribution to provide some analytical solutions and techniques that can be useful in the calculation of cable losses. Explicit analytical results are given for the fields generated by a three-phase helical current distribution and which can be computed efficiently as an input to other numerical methods such as, for example, the Method of Moments. It is shown that the field computations are numerically stable at low frequencies (such as 50 Hz) as well as in the quasi-magnetostatic limit provided that sources are divergence-free. The cylindrical multipole expansion is furthermore used to derive an efficient analytical model of a measurement coil to measure and estimate the complex valued permeability of magnetic steel armour in the presence of a strong skin-effect. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is devoted to the time-fractional gas dynamics equation with Caputo derivative. Fractional operators are very natural tools to model memory-dependent phenomena. Modified iteration method is proposed to obtain the approximate and analytical solution of the fractional gas dynamics equation. This method is a combined form of the new iteration method and Laplace transform. Modified iteration method really is powerful and simple method compared with other methods. Existence and uniqueness of solution are proven. Numerical results for different cases of the equation are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The well-known low-frequency expansion of the total acoustic field in the exterior of a penetrable spherical scatterer is revisited in view of the Atkinson Wilcox theorem. The corresponding low-frequency approximations of any order are calculated following a purely algebraic algorithmic procedure, based on the spectral decomposition of the problem's far field pattern. As an indication of its accuracy and effectiveness, the proposed algebraic procedure is shown to recover already known low-frequency coefficients and also to deduce higher order of approximations. The proposed track of calculations leads to a closed-form expression of any such coefficient and has been recently applied on impenetrable spherical scatterers as well, offering equally accurate results. The effectiveness of the proposed procedure indicates an underlying general efficient method applicable to a wider class of starshaped scatterers. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper investigates an inverse problem for parabolic equations backward in time, which is solved by total-variation-like (TV-like, in abbreviation) regularization method with cost function ∥*u*_{x}∥^{2}. The existence, uniqueness and stability estimate for the regularization problem are deduced in the linear case. For numerical illustration, the variational adjoint method, which presents a simple method to derive the gradient of the optimization functional, is introduced to reconstruct the unknown initial condition for both linear and nonlinear parabolic equations. The conjugate gradient method is used to iteratively search for the optimal approximation. Numerical results validate the feasibility and effectiveness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

Water bodies located nearby cities are much prone to pollution, especially in the developing countries, where effluents treatment facilities are generally lacking. The main reason for this phenomenon is the increasing population in the cities, and the large number of industries located near them. This leads to generation of huge amounts of domestic and industrial sewage that is discharged into the water bodies, increasing their organic pollutant load and resulting in the depletion of dissolved oxygen. In this paper, we propose a mathematical model for this situation, focusing especially on the resulting quality of the water, determined by the level of dissolved oxygen. The model also accounts for resources needed for the population survival and for the industrial operations. In addition, we describe also the decomposition of organic pollutants by bacteria in the aquatic medium. Feasibility conditions and stability criteria of the system's equilibria are determined analytically. The results show that human population and industries are relevant influential factors responsible for the increase in organic pollutants and the decrease in dissolved oxygen in the water body, in the sense that they may exert a destabilizing effect on the system. The numerical simulations confirm the analytical results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We investigate the behavior of the solution of a mixed problem for the Poisson equation in a domain with two moderately close holes. If *ϱ*_{1} and *ϱ*_{2} are two positive parameters, we define a perforated domain Ω(*ϱ*_{1},*ϱ*_{2}) by making two small perforations in an open set: the size of the perforations is *ϱ*_{1}*ϱ*_{2}, while the distance of the cavities is proportional to *ϱ*_{1}. Then, if *r*_{∗} is small enough, we analyze the behavior of the solution for (*ϱ*_{1},*ϱ*_{2}) close to the degenerate pair (0,*r*_{∗}). Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents a numerical method for shape optimization of a body immersed in an incompressible viscous flow governed by Stokes–Oseen equations. The purpose of this work is to optimize the shape that minimizes a given cost functional. Based on the continuous adjoint method, the shape gradient of the cost functional is derived by involving a Lagrangian functional with the function space parametrization technique. Then, a gradient-type algorithm is applied to the shape optimization problem. The numerical examples indicate the proposed algorithm is feasible and effective in low Reynolds number flow. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In the present paper, our aim is to establish several formulas involving integral transforms, fractional derivatives, and a certain family of extended generalized hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results. A probability density function involving the extended generalized hypergeometric function is introduced, and its properties are studied. The corresponding properties of some of the classical probability distributions and their associated probability density functions are easily derivable as special cases of our general results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with a generalized Arzela–Ascoli's lemma, which has been extensively applied in almost periodic problems by the continuation theorem of degree theory. We give a counter example to show that this lemma is incorrect, and there is a gap in the proof of some existing literature, where the addressed generalized Arzela–Ascoli's lemma was used. Moreover, we make some final comments and introduce an open problem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We establish the existence of local in time semi-strong solutions and global in time strong solutions for the system of equations describing flows of viscous and incompressible asymmetric fluids with variable density in general three-dimensional domains with boundary uniformly of class *C*^{3}. Under suitable assumptions, uniqueness of local semi-strong solutions is also proved. Copyright © 2016 John Wiley & Sons, Ltd.

We prove the existence of pullback and uniform attractors for the process associated to a non-autonomous SIR model, with several types of non-autonomous features. The Hausdorff dimension of the pullback attractor is also estimated. We illustrate some examples of pullback attractors by numerical simulations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Galfenol offers potential opportunities to build composite materials with higher magnetoelectric (ME) coupling coefficients. This work is mainly focused on exploring the ME coupling in composites made with Galfenol. A revision of effective properties estimation is based on the prediction obtained through an implementation of the asymptotic homogenization method for magneto-electro-elastic composites considering two types of composites: laminated and fiber reinforced. As constituents, Barium Titanate is considered as the piezoelectric phase and Galfenol as the magnetostrictive one. For the sake of comparison, Terfenol-D and Cobalt Ferrite are also considered. The herein obtained ME coefficients are higher than most ones reported in the literature when comparison is possible. In the literature, two types of ME coefficients can be found: one is the voltage ME, reported in V/cmOe, and the other one is the coupling ME reported in Ns/VC. Herein, we focused on the connection between these coefficients in order to facilitate the application of the coupling ME calculated by asymptotic homogenization method on the experimental measured voltage ME coefficient. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The initial-boundary value problems encountered in the regime of the dynamic theory of gradient elasticity are characterized by several crucial and intrinsic issues. One of the most important characteristics is that the involving differential equation is of fourth order and mainly that the second order time derivative – appearing in it – is not given explicitly but in the contrary is incorporated implicitly in mixed-type differential terms (via spatial-time derivatives). The aim of the present work is the investigation of the emerged initial-boundary value problems of gradient elasticity in one spatial dimension and the establishment of the suitable functional setting assuring existence and uniqueness of weak (or strong) solutions. Furthermore, the spectral analysis of the gradient elastic operator is investigated and compared with the well-known results of classical elasticity. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study the convergence to equilibria, as time tends to infinity, of trajectories of dissipative wave systems with time-dependent velocity feedbacks and subject to nonlinear potential energies. Estimates for the speed of convergence are obtained in terms of the damping coefficient and the Łojasiewicz–Simon exponent. We allow for both restoring and amplifying effects of exterior forces, which makes our results possess wide applicability. As an example of application, we show that the trajectories of a sine-Gordon system, with nonautonomous damping, approach equilibria at least polynomially. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Over the last years, considerable attention has been paid to the role of the prolate spheroidal wave functions (PSWFs) introduced in the early sixties by D. Slepian and H.O. Pollak to many practical signal and image processing problems. The PSWFs and their applications to wave phenomena modeling, fluid dynamics, and filter design played a key role in this development. In this paper, we introduce the prolate spheroidal quaternion wave functions (PSQWFs), which refine and extend the PSWFs. The PSQWFs are ideally suited to study certain questions regarding the relationship between quaternionic functions and their Fourier transforms. We show that the PSQWFs are orthogonal and complete over two different intervals: the space of square integrable functions over a finite interval and the three-dimensional Paley–Wiener space of bandlimited functions. No other system of classical generalized orthogonal functions is known to possess this unique property. We illustrate how to apply the PSQWFs for the quaternionic Fourier transform to analyze Slepian's energy concentration problem. We address all of the aforementioned and explore some basic facts of the arising quaternionic function theory. We conclude the paper by computing the PSQWFs restricted in frequency to the unit sphere. The representation of these functions in terms of generalized spherical harmonics is explicitly given, from which several fundamental properties can be derived. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we develop the energy argument in homogeneous Besov space framework to study the large time behavior of global-in-time strong solutions to the Cauchy problem of the three-dimensional incompressible nematic liquid crystal flows with low regularity assumptions on initial data. More precisely, if the small initial data
with 1 < *p* < *∞* and further assume that
with 1 < *q*≤*p* and
, then the global-in-time strong solution (*u*,*d*) to the nematic liquid crystal flows admits the following temporal decay rate:

Here, is a constant unit vector. The highlight of our argument is to show that the -norms (with ) of solution are preserved along time evolution. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We pose a Bayesian formulation of the inverse problem associated to recovering both the support and the refractive index of a convex obstacle given measurements of near-field scattered waves. Aiming at sampling efficiently from the arising posterior distribution using Markov Chain Monte Carlo, we construct a sampler (probability transition kernel) that is invariant under affine transformations of space. A point cloud method is used to approximate the scatterer support. We show that affine invariant sampling can successfully address the presence of multiple scales in inverse scattering in inhomogeneous media. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with the following Schrödinger elliptic system

where the potential *V* is periodic and 0 lies in a gap of the spectrum of −Δ+*V*, *f*(*x*,*t*) and *g*(*x*,*t*) are superlinear in *t* at infinity. By using non-Nehari manifold method developed recently by Tang, we demonstrate the existence of the ground state solutions of Nehari-Pankov type for the above problem with periodic and asymptotically periodic nonlinearity. Copyright © 2016 John Wiley & Sons, Ltd.

Sufficient conditions are obtained for the nonexistence of solutions to the nonlinear higher order pseudo-parabolic equation

where
is the Kohn-Laplace operator on the (2*N* + 1)-dimensional Heisenberg group
, *m*≥1,*p* > 1. Then, this result is extended to the case of a 2 × 2-system of the same type. Our technique of proof is based on judicious choices of the test functions in the weak formulation of the sought solutions. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we are concerned with the system of the non-isentropic compressible Navier–Stokes equations coupled with the Maxwell equations through the Lorentz force in three space dimensions. The global existence of solutions near constant steady states is established, and the time-decay rates of perturbed solutions are obtained. The proof for existence is due to the classical energy method, and the investigation of large-time behavior is based on the linearized analysis of the non-isentropic Navier–Stokes–Poisson equations and the electromagnetic part for the linearized isentropic Navier–Stokes–Maxwell equations. In the meantime, the time-decay rates obtained by Zhang, Li, and Zhu [*J. Differential Equations, 250(2011), 866-891*] for the linearized non-isentropic Navier–Stokes–Poisson equations are improved. Copyright © 2016 John Wiley & Sons, Ltd.

The present paper is concerned with the approximation properties of discrete version of Picard operators. We first give exact equalities for the moments of the operators. In calculations of these moments, Eulerian numbers play a crucial role. We discuss convergence of these operators in weighted spaces and give Voronovskaya-type asymptotic formula. The weighted approximation of the operators in quantitative mean in terms of different modulus of continuities is also considered. We emphasize that the rate of convergence of the operators is better than the one obtained in . Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider a coupled system of mixed hyperbolic–parabolic type, which describes the Biot consolidation model in poro-elasticity. We establish a local Carleman estimate for Biot consolidation system. Using this estimate, we prove the uniqueness and a Hölder stability in determining on the one hand a physical parameter arising in connection with secondary consolidation effects *λ*^{∗} and on the other hand the two spatially varying densities by a single measurement of solution over *ω* × (0,*T*), where *T* > 0 is a sufficiently large time and a suitable subdomain *ω* satisfying *∂**ω*⊃*∂*Ω. Copyright © 2016 John Wiley & Sons, Ltd.

A vector-valued signal in *N* dimensions is a signal whose value at any time instant is an *N*-dimensional vector, that is, an element of
. The sum of an arbitrary number of such signals *of the same frequency* is shown to trace an ellipse in *N*-dimensional space, that is, to be *confined to a plane*. The parameters of the ellipse (major and minor axes, represented by *N*-dimensional vectors; and phase) are obtained algebraically in terms of the directions of oscillation of the constituent signals, and their phases. It is shown that the major axis of the ellipse can always be determined algebraically. That is, a vector, whose value can be computed algebraically (without decisions or comparisons of magnitude) from parameters of the constituent signals, always represents the major axis of the ellipse. The ramifications of this result for the processing and Fourier analysis of signals with vector values or samples are discussed, with reference to the definition of Fourier transforms, particularly discrete Fourier transforms, such as have been defined in several hypercomplex algebras, including Clifford algebras. The treatment in the paper, however, is entirely based on signals with values in
. Although the paper is written in terms of vector *signals* (which are taken to include images and volumetric images), the analysis clearly also applies to a superposition of simple harmonic motions in *N* dimensions. Copyright © 2016 John Wiley & Sons, Ltd.

Hepatitis B virus (HBV) and its vaccination strategy may affect human immunodeficiency virus (HIV) transmission dynamics because both viruses have synergistic effects. To quantitatively assess the potential impact of HBV and its vaccination strategy on HIV transmission dynamics at the population level, in this paper, we formulate a deterministic compartmental model that describes the joint dynamics of HBV and HIV. We first derive the explicit expressions for the basic reproduction numbers of HIV and HBV and analyze the dynamics of HIV and HBV subsystems, respectively. Then a systematic qualitative analysis of the full system is also provided, which includes the local and global behavior. By using a set of reasonable parameter values, the full system is numerically investigated to assess the potential impact of HBV and its vaccination strategy on HIV transmission. The direct and indirect population level impact of HBV on HIV is demonstrated by calculating the fraction of HIV infections attributable to HBV and the difference between HIV prevalence in the presence and absence of HBV, respectively. The findings imply that the increase of HBV vaccination rate may unusually accelerate HIV epidemics indirectly, although the direct effect of HBV on HIV transmission decreases as HBV vaccination rate increases. Finally, the potential impact of HIV on HBV transmission dynamics is investigated by way of parenthesis. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the quasi-static evolution of thermo-visco-elastic material. The main goal of this paper is to present how taking into account the additional effects may improve the result of solutions' existence. We added a micropolarity effect to thermo-visco-elastic model regarding Norton-Hoff-type constitutive function. This additional phenomenon improves the regularity of solution. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the zero viscosity and capillarity limit problem for the one-dimensional compressible isentropic Navier–Stokes–Korteweg equations when the corresponding Euler equations have rarefaction wave solutions. In the case that either the effects of initial layer are ignored or the rarefaction waves are smooth, we prove that the solutions of the Navier–Stokes–Korteweg equation with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity and capillarity number vanish, and we also obtain a rate of convergence, which is valid uniformly for all time. These results are showed by a scaling argument and elementary energy analysis. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We prove an existence theorem for an abstract operator equation associated with a quasi-subdifferential operator and then apply it to concrete elliptic variational and quasi-variational inequalities. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we show the existence and uniqueness results for periodic solutions of Weyl fractional order integral systems. A numerical example is given to illustrate our theoretical results. Our results show that periodic orbits can be obtained by putting the periodic conditions to some certain fractional order integral systems. Copyright © 2016 John Wiley & Sons, Ltd.

]]>A predator–prey model with disease amongst the prey and ratio-dependent functional response for both infected and susceptible prey is proposed and its features analysed. This work is based on previous mathematical models to analyse the important ecosystem of the Salton Sea in Southern California and New Mexico where birds (particularly pelicans) prey on fish (particularly tilapia). The dynamics of the system around each of the ecologically meaningful equilibria are presented. Natural disease control is considered before studying the impact of the disease in the absence of predators and the interaction of predators and healthy prey and the disease effects on predators in the absence of healthy prey. Our theoretical results are confirmed by numerical simulation. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We compute a local linearization for the nonlinear, *inverse* problem of identifying the stored energy function of a hyperelastic material from the full knowledge of the displacement field. The displacement field is described as a solution of the nonlinear, dynamic, elastic wave equation, where the first Piola–Kirchhoff stress tensor is given as the gradient of the stored energy function. We assume that we have a dictionary at hand such that the energy function is given as a conic combination of the dictionary's elements. In that sense, the mathematical model of the *direct problem* is the nonlinear operator that maps the vector of expansion coefficients to the solution of the hyperelastic wave equation. In this article, we summarize some continuity results for this operator and deduce its Fréchet derivative as well as the adjoint of this derivative. Because the stored energy function encodes mechanical properties of the underlying, hyperelastic material, the considered inverse problem is of highest interest for structural health monitoring systems where defects are detected from boundary measurements of the displacement field. For solving the inverse problem iteratively by the Landweber method or Newton-type methods, the knowledge of the Fréchet derivative and its adjoint is of utmost importance. Copyright © 2016 John Wiley & Sons, Ltd.

The main objective of this paper was to study the global stability of the positive solutions and the periodic character of the difference equation

where the parameters *a*, *b*, *c*, *d*, and *e* are positive real numbers and the initial conditions *x*_{−t},*x*_{−t + 1},...,*x*_{−1}, *x*_{0} are positive real numbers where *t* = *m**a**x*{*l*,*k*,*s*}. Some numerical examples will be given to explicate our results. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we implement the natural decomposition method (NDM) to solve nonlinear partial differential equations. We apply the NDM to obtain exact solutions for three applications of nonlinear partial differential equations. The new method is a combination of the natural transform method and the Adomian decomposition method. We prove some of the properties that are related to the natural transform method. The results are compared with existing solutions obtained by other methods, and one can conclude that the NDM is easy to use and efficient. Copyright © 2016 John Wiley & Sons, Ltd.

]]>By using the Kolmogorov–Arnold–Moser theory, we investigate the stability of the equilibrium solution of the difference equation

where *A*,*B*,*D* > 0,*u*_{−1},*u*_{0}>0. We also use the symmetries to find effectively the periodic solutions with feasible periods. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we formulate a (*n* + 3)-dimensional nonlinear virus dynamics model that considers *n*-stages of the infected cells and *n* + 1 distributed time delays. The model incorporates humoral immune response and general nonlinear forms for the incidence rate of infection, the generation and removal rates of the cells and viruses. Under a set of conditions on the general functions, the basic infection reproduction number
and the humoral immune response activation number
are derived. Utilizing Lyapunov functionals and LaSalle's invariance principle, the global asymptotic stability of all steady states of the model are proven. Numerical simulations are carried out to confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is concerned with the existence of positive solutions to a class of nonlocal boundary value problem of the *p*-Kirchhoff type
where
is a bounded smooth domain and *M*,*f*, and *g* are continuous functions. The existence of a positive solution is stated through an iterative method based on mountain pass theorem. Copyright © 2016 John Wiley & Sons, Ltd.

We prove optimal lower bounds for the growth of the potential energy over balls of minimizers to the vectorial Allen–Cahn energy in two spatial dimensions, as the radius tends to infinity. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with the existence and uniqueness of time periodic solutions in the whole-space
for a heat equation with nonlinear term. The nonlinear term we considered is of this type, |*u*|^{q − 1}*u* + *f*(*x*,*t*), with
, *N* > 2. We show that there exists a unique time periodic solution when the source term *f* is small. In fact,
is a critical exponent; when
, there is no time periodic solution. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a plate equation with infinite memory in the presence of delay and source term. Under suitable conditions on the delay and source term, we establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity. Our result allows a wider class of relaxation functions and improves earlier results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Based on a general isospectral problem of fractional order, a fractional bilinear form variational identity, the new integrable coupling of fractional L-hierarchy and the Hamiltonian structures of the integrable coupling of fractional L-hierarchy are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, three conservative finite volume element schemes are proposed and compared for the modif ied Korteweg–de Vries equation, especially with regard to their accuracy and conservative properties. The schemes are constructed basing on the discrete variational derivative method and the finite volume element method to inherit the properties of the original equation. The theoretical analysis show that three schemes are conservative under suitable boundary conditions as well as unconditionally linear stability. Numerical experiments are given to confirm the theoretical results and the capacity of the proposed methods for capturing the solitary wave phenomena. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study the properties of the Lagrange multiplier for an Allen–Cahn equation with a double obstacle potential. Here, the dynamic boundary condition, including the Laplace–Beltrami operator on the boundary, is investigated. We then establish the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier of our problem. We present remarks on a trace problem as well as on the Neumann boundary condition. Moreover, we describe a numerical experiment for a problem with Neumann boundary condition using the Lagrange multiplier. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We point out some mistakes in a known paper. Some existence results for solutions of two classes of boundary value problems for nonlinear impulsive fractional differential equations are established. Our analysis relies on the well-known Schauder fixed point theorem. Examples are given to illustrate the main results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The aim of this paper is to propose optimal strategies for dengue reduction and prevention in Cali, Colombia. For this purpose, we consider two variants of a simple dengue transmission model, epidemic and endemic, each of which is amended with two control variables. These variables express feasible control actions to be taken by an external decision-maker. First control variable stands for the insecticide spraying and thus targets to suppress the vector population. The second one expresses the protective measures (such as use of repellents, mosquito nets, and insecticide-treated clothes) that are destined to reduce the number of contacts (bites) between female mosquitoes (principal dengue transmitters) and human individuals. We use the Pontryagin's maximum principle in order to derive the optimal strategies for dengue control and then perform the cost-effectiveness analysis of these strategies in order to choose the most sustainable one in terms of cost–benefit relationship. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we establish some new sufficient conditions on the existence of homoclinic solution for a class of second-order impulsive Hamiltonian systems. By using the mountain pass theorem, we demonstrate that the limit of a 2*k**T*-periodic approximation solution is a homoclinic solution of our problem. We also present some examples to illustrate the applications of our main results. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, a new impulsive Lasota–Wazewska model with patch structure and forced perturbed terms is proposed and analyzed on almost periodic time scales. For this, we introduce the concept of matrix measure on time scales and obtain some of its properties. Then, sufficient conditions are established which ensure the existence and exponential stability of positive almost periodic solutions of the proposed biological model. Our results are new even when the time scale is almost periodic, in particular, for periodic time scales on or . An example is given to illustrate the theory. Finally, we present some phenomena which are triggered by almost periodic time scales. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper deals with constructing generalized ‘fractional’ power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non-linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The Melan beam equation modeling suspension bridges is considered. A slightly modified equation is derived by applying variational principles and by minimizing the total energy of the bridge. The equation is nonlinear and nonlocal, while the beam is hinged at the endpoints. We show that the problem always admits at least one solution whereas the uniqueness remains open although some numerical results suggest that it should hold. We also emphasize the qualitative difference with some simplified models. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In a recent paper, the notion of *quantum perceptron* has been introduced in connection with projection operators. Here, we extend this idea, using these kind of operators to produce a *clustering machine*, that is, a framework that generates different clusters from a set of input data. Also, we consider what happens when the orthonormal bases first used in the definition of the projectors are replaced by frames and how these can be useful when trying to connect some noised signal to a given cluster. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider the one-dimensional Schrödinger operator on bounded time scales. We construct a space of boundary values of the minimal operator and describe all maximal dissipative, maximal accretive, self-adjoint, and other extensions of the dissipative Schrödinger operators in terms of boundary conditions. In particular, using Lidskii's theorem, we prove a theorem on completeness of the system of root vectors of the dissipative Schrödinger operators on bounded time scales. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The present contribution is concerned with an analytical presentation of the low-frequency electromagnetic fields, which are scattered off a highly conductive ring torus that is embedded within an otherwise lossless ambient and interacting with a time-harmonic magnetic dipole of arbitrary orientation, located nearby in the three-dimensional space. Therein, the particular 3-D scattering boundary value problem is modeled with respect to the solid impenetrable torus-shaped body, where the toroidal geometry fits perfectly. The incident, the scattered, and the total non-axisymmetric magnetic and electric fields are expanded in terms of positive integral powers of the real-valued wave number of the exterior medium at the low-frequency regime, whereas the static Rayleigh approximation and the first three dynamic terms provide the most significant part of the solution, because all the additional terms are small contributors and, hence, they are neglected. Consequently, the typical Maxwell-type physical problem is transformed into intertwined either Laplace's or Poisson's potential-type boundary value problems with the proper conditions, attached to the metallic surface of the torus. The fields of interest assume representations via infinite series expansions in terms of standard toroidal eigenfunctions, obtaining in that way analytical closed-form solutions in a compact fashion. Although this mathematical procedure leads to infinite linear systems for every single case, these can be readily and approximately solved at a certain level of desired accuracy through standard cut-off techniques. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study a multiphasic incompressible fluid model, called the Kazhikhov–Smagulov model, with a particular viscous stress tensor, introduced by Bresch and co-authors, and a specific diffusive interface term introduced for the first time by Korteweg in 1901. We prove that this model is globally well posed in a 3D bounded domain. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The current article devoted on the new method for finding the exact solutions of some time-fractional Korteweg–de Vries (KdV) type equations appearing in shallow water waves. We employ the new method here for time-fractional equations viz. time-fractional KdV-Burgers and KdV-mKdV equations for finding the exact solutions. We use here the fractional complex transform accompanied by properties of local fractional calculus for reduction of fractional partial differential equations to ordinary differential equations. The obtained results are demonstrated by graphs for the new solutions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a class of cellular neural networks with neutral proportional delays and time-varying leakage delays is considered. Some results on the finite-time stability for the equations are obtained by using the differential inequality technique. In addition, an example with numerical simulations is given to illustrate our results, and the generalized exponential synchronization is also established. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we introduced a summation-integral type modification of Szász–Mirakjan operators. Calculation of moments, density in some space, a direct result and a Voronvskaja-type result, are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we establish finite-region stability (FRS) and finite-region boundedness analysis methods to investigate the transient behavior of discrete two-dimensional Roesser models. First, by building special recursive formulas, a sufficient FRS condition is built via solvable linear matrix inequalities constraints. Next, by designing state feedback controllers, the finite-region stabilization issue is analyzed for the corresponding two-dimensional closed-loop system. Similar to FRS analysis, the finite-region boundedness problem is addressed for Roesser models with exogenous disturbances and corresponding criteria, and linear matrix inequalities conditions are reported. To conclude the paper, we provide numerical examples to confirm the validity of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study a cantilevered Euler–Bernoulli beam fixed to a base in a translational motion at one end and to a tip mass at its free end. The beam is subject to undesirable vibrations, and it is made of a viscoelastic material that permits a certain weak damping. By applying a control force at the base, we shall attenuate these vibrations in a fast manner. In fact, we establish the exponential stability of the system. Our method is based on the multiplier technique. Copyright © 2016 John Wiley & Sons, Ltd.

]]>No abstract is available for this article.

]]>In this research article, a complete analysis of symmetries and conservation laws for the charged squashed Kaluza–Klein black hole space-time in a Riemannian space is discussed. First, a comprehensive group analysis of the underlying space-time metric using Lie point symmetries is presented, and then the *n*-dimensional optimal system of this space-time metric, for *n* = 1,…,4, are computed. It is shown that there is no any *n*-dimensional optimal system of Lie symmetry subalgebra associated to the system of geodesic for *n*≥5. Then the point symmetries of the one-parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian that means Noether symmetries are found, and then the conservation laws associated to the system of geodesic equations are calculated via Noether's theorem. Copyright © 2015 John Wiley & Sons, Ltd.

We consider a problem in the inelastic deformation theory with a quasistatic deformation process of the gradient-monotone type. We assume that the body has contact with a rigid foundation: the body moves on the foundation with friction. The frictional contact is modelled by a velocity-dependent dissipation functional. This makes an evolution problem with two nonlinear monotone operators. We consider the gradient-monotone inelastic constitutive function with a rapid growth at infinity. This leads us to a nonreflexive Orlicz space as an operational base. The frictional dissipation potential brings about a minimalization problem in this nonreflexive Orlicz space. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this present paper, the existence of pullback attractors for the 2D Navier–Stokes equation with weak damping, distributed delay, and continuous delay has been considered, by virtue of classical Galerkin's method, we derived the existence and uniqueness of global weak and strong solutions. Using the Aubin–Lions lemma and some energy estimate in the Banach space with delay, we obtained the uniform bounded and existence of uniform pullback absorbing ball for the solution semi-processes; we concluded the pullback attractors via verifying the pullback asymptotical compactness by the generalized Arzelà–Ascoli theorem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In the manuscript, a pseudospectral method is developed for approximate and efficient solution of nonlinear singular Lane–Emden–Fowler initial and boundary value problems arising in astrophysics. In the proposed method, the Gauss pseudospectral method is utilized to reduce the problem to the solution of a system of algebraic equations. Furthermore, the Gauss pseudospectral method is developed for finding the first zero of the solution of this equation that gives the radius of the star, in which the numerous properties of the star such as mass, central pressure, and binding energy can be computed through their relations to this solution. The main advantage of the proposed method is that good results are obtained even by using a small number of discretization points and the rate of convergence is high. The accuracy and performance of the proposed method are examined by means of some numerical experiments. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, a model of hematopoiesis with an oscillating circulation loss rate is investigated. By applying the exponential dichotomy theory, contraction mapping fixed-point theorem, and differential inequality techniques, a set of sufficient conditions are obtained for the existence and exponential stability of positive pseudo almost periodic solutions of the model. Some numerical simulations are carried out to support the theoretical findings. Our results improve and generalize those of the previous studies. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we study the nonlinear Schrödinger equation on Zoll manifolds with odd order nonlinearities. We will obtain the local well-poesdness in the critical space
. This extends the recent results in the literature to the Zoll manifolds of dimension *d*≥2 with general odd order nonlinearities and also partially improves the previous results in the subcritical spaces of Yang to the critical cases. Copyright © 2015 John Wiley & Sons, Ltd.

The present paper is devoted to the analysis of a nonlinear system modeling unsteady flows of an incompressible non-Newtonian fluid mixed with a reactant. We are interested on generalized second grade fluids, which are chemically reacting and whose viscosity depends both on the shear-rate and the concentration. We prove existence and uniqueness of strong–weak solution for a flow filling in the plane and subject to space periodic boundary conditions. This result is established under the fulfillment of some assumptions on the viscosity stress tensor and the flux vector of the diffusion–convection equation reflecting the chemical reaction. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This research gives a complete Lie group classification of the one-dimensional nonlinear delay Klein–Gordon equation. First, the determining equations are derived and their complete solutions are found. Then the complete group classification and representations of all invariant solutions are obtained. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We consider here the existence of rari-constant anisotropic layers and show that actually there are two distinct classes of such materials, mutually exclusive. Also, we show that the correct condition for establishing that a material is of the rari-constant type is that the number of independent linear tensor invariants of the elastic tensors must reduce to one. We characterize these materials and show that they can be designed by using some basic rules of homogenization. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we are concerned with asymptotic stability of a class of Bresse-type system with three boundary dissipations. The beam has a rigid body attached to its free end. We show that exponential stabilization can be achieved by applying force and moment feedback boundary controls on the shear, longitudinal, and transverse displacement velocities at the point of contact between the mass and the beam. Our method is based on the operator semigroup technique, the multiplier technique, and the contradiction argument of the frequency domain method. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, a delayed Susceptible-Exposed-Infectious-Susceptible (SEIS) infectious disease model with logistic growth and saturation incidence is investigated, where the time delay describes the latent period of the disease. By analyzing corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is discussed. The existence of Hopf bifurcations at the endemic equilibrium is established. By using the persistence theory for infinite dimensional dynamic systems, it is proved that if the basic reproduction number is greater than unity, the system is permanent. By means of suitable Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are obtained for the global stability of the disease-free equilibrium and the endemic equilibrium, respectively. Numerical simulations are carried out to illustrate the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Computer simulations of the injection molding process of fiber-reinforced plastics critically depend on the accuracy of the constitutive models. Of prime importance for the process simulation is the precise knowledge of the viscosity. Industrial applications generally feature both high shear rates and high fiber volume fractions. Thus, both the shear-thinning behavior of the melt and the strong anisotropic effects induced by the fibers play a dominant role. Unfortunately, the viscosity cannot be determined experimentally in its full anisotropy, and analytical models cease to be accurate for the high fiber volume fractions in question. Computing the effective viscosity by a simplified homogenization approach serves as a possible remedy. This paper is devoted to the analysis of a cell problem determining the effective viscosity. We provide primal as well as dual formulations and prove corresponding existence and uniqueness theorems for Newtonian and Carreau fluids in suitable Sobolev spaces. In the Newtonian regime, the primal formulation leads to a saddle point problem, whereas a dual formulation can be obtained in terms of a coercive and symmetric bilinear form. This observation has deep implications for numerical formulations. As a by-product, we obtain the invertibility of the effective viscosity, considered as a function, mapping the macroscopic shear rate to the macroscopic shear stress. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the existence of global attractor and exponential attractor for some dynamical system generated by nonlinear parabolic equations in bounded domains with the dimension *N*≤4 which describe double-diffusive convection phenomena in a porous medium. We deal with both of homogeneous Dirichlet and Neumann boundary condition cases. Especially, when Neumann condition is imposed, we need some assumptions and restrictions for the external forces and the average of initial data, since the mass conservation law holds. Copyright © 2015 John Wiley & Sons, Ltd.

This paper presents a computational technique based on the pseudo-spectral method for the solution of distributed optimal control problem for the viscous Burgers equation. By using pseudo-spectral method, the problem is converted to a classical optimal control problem governed by a system of ordinary differential equations, which can be solved by well-developed direct or indirect methods. For solving the resulting optimal control problem, we present an indirect method by deriving and numerically solving the first-order optimality conditions. Numerical tests involving both unconstrained and constrained control problems are considered. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper is concerned with local and global existence of solutions to the parabolic-elliptic chemotaxis system
. Marinoschi (J. Math. Anal. Appl. 2013; 402:415–439) established an abstract approach using nonlinear *m*-accretive operators to giving existence of local solutions to this system when 0 < *D*_{0}≤*D*′(*r*)≤*D*_{∞}<*∞* and (*r*_{1},*r*_{2})↦*K*(*r*_{1},*r*_{2})*r*_{1} is Lipschitz continuous on
, provided that the initial data is assumed to be small. The smallness assumption on the initial data was recently removed (J. Math. Anal. Appl. 2014; 419:756–774). However the case of non-Lipschitz and degenerate diffusion, such as *D*(*r*) = *r*^{m}(*m* > 1), is left incomplete. This paper presents the local and global solvability of the system with non-Lipschitz and degenerate diffusion by applying (J. Math. Anal. Appl. 2013; 402:415–439) and (J. Math. Anal. Appl. 2014; 419:756–774) to an approximate system. In particular, the result in the present paper does not require any properties of boundedness, smoothness and radial symmetry of initial data. This makes it difficult to deal with nonlinearity. Copyright © 2015 John Wiley & Sons, Ltd.

An equilibrium problem for an elastic body is considered. It is assumed that the body has a thin elastic inclusion and a thin rigid inclusion. We analyze a junction problem assuming that the inclusions have a joint point. Different equivalent problem formulations are discussed, and existence of solutions is proved. A set of junction conditions is found. We investigate a convergence to infinity and to zero of a rigidity parameter of the elastic inclusion. A delamination of the elastic inclusion is also investigated. In this case, inequality-type boundary conditions are imposed at the crack faces to prevent a mutual penetration between crack faces. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In the present paper, we prove quantitative *q*-Voronovskaya type theorems for *q*-Baskakov operators in terms of weighted modulus of continuity. We also present a new form of Voronovskaya theorem, that is, *q*-Grüss-Voronovskaya type theorem for *q*-Baskakov operators in quantitative mean. Hence, we describe the rate of convergence and upper bound for the error of approximation, simultaneously. Our results are valid for the subspace of continuous functions although classical ones is valid for differentiable functions. Copyright © 2015 John Wiley & Sons, Ltd.

We introduce and study certain distributions generalizing the operation of curvilinear integration for the case where the path of integration is not rectifiable. Then we apply that distributions for solving of boundary value problems of Riemann—Hilbert type in domains with non-rectifiable boundaries. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non-singular one by new fractional-order Legendre functions. The fractional-order Legendre functions are generated by change of variable on well-known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we study the existence of radially symmetric convex solutions for Dirichlet problems of Monge-Ampère equations. By applying a well-known fixed point theorem in cones, we shall establish several new criteria for the existence of nontrivial radially symmetric convex solutions for the systems of Monge-Ampère equations with or without an eigenvalue parameter. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the dynamical properties for a model of delay differential equations, which describes a virus-immune interaction *in vivo*. By analyzing corresponding characteristic equations, the local stability of the equilibria for infection-free, antibody-free, and antibody response and the existence of Hopf bifurcation with antibody response delay as a bifurcation parameter at the antibody-activated infection equilibrium are established, respectively. Global stability of the equilibria for infection-free, antibody-free, and antibody response, respectively, also are established by applying the Lyapunov functionals method. The numerical simulations are performed in order to illustrate the dynamical behavior of the model. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we analyze the behavior of a family of solutions of a nonlinear elliptic equation with nonlinear boundary conditions, when the boundary of the domain presents a highly oscillatory behavior, which is uniformly Lipschitz and nonlinear terms, are concentrated in a region, which neighbors the boundary of domain. We prove that this family of solutions converges to the solutions of a limit problem in *H*^{1}an elliptic equation with nonlinear boundary conditions which captures the oscillatory behavior of the boundary and whose nonlinear terms are transformed into a flux condition on the boundary. Indeed, we show the upper semicontinuity of this family of solutions.Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we study the following biharmonic equation

where
, *K*(1) > 0,*K*′(1) > 0, *B*_{1}(0) is the unit ball in
(*N*≥6). We show that the aforementioned problem has infinitely many peak solutions, whose energy can be made arbitrarily large. Copyright © 2016 John Wiley & Sons, Ltd.

Based on Takenaka–Malmquist (TM) system, a new nonparametric estimator for probability density function is proposed. The TM estimation method is completely different from the existent density estimation methods in that the estimator depends on an approximate system with poles in a complex plane. Compared with the classic Fourier estimator, the TM estimator will offer more flexibility and adaptivity for real data due to the poles and nonlinearity of the phase of TM system. We compare the TM estimator with kernel, wavelet, and spline estimators by simulations. It shows that the introduced TM estimator is a more promising method than the existing and commonly used methods. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper presents a superconvergence result of the stationary natural convection equations. The superconvergence result is obtained by applying a coarse mesh projection for finite element approximation. This projection method is actually a postprocessing procedure that constructs a new approximation based on a high order polynomial on coarse mesh, under some regularity assumption. Finally, numerical experiment is presented for testing, which confirms the theoretic analysis. Copyright © 2015 John Wiley & Sons, Ltd.

]]>A dimension splitting method (DSM) with Crank–Nicolson time discrete strategy for a three-dimensional heat equation is proposed. The basic idea is to simulate the three-Dimensional problem by numerically solving a series of two-dimensional problems in parallel fashion. Convergence and error estimation for the DSM scheme are derived in the paper. Numerical experiments demonstrate the feasibility and efficiency of the DSM scheme. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the integrability and equivalence relationships of six coupled Korteweg–de Vries equations. It is shown that the six coupled Korteweg–de Vries equations are identical under certain invertible transformations. We reconsider the matrix representations of the prolongation algebra for the Painlevé integrable coupled Korteweg–de Vries equation in [Appl. Math. Lett. 23 (2010) 665-669] and propose a new Lax pair of this equation that can be used to construct exact solutions with vanishing boundary conditions. It is also pointed out that all the six coupled Korteweg–de Vries equations have fourth-order Lax pairs instead of the fifth-order ones. Moreover, the Painlevé integrability of the six coupled Korteweg–de Vries equations are examined. It is proved that the six coupled Korteweg–de Vries equations are all Painlevé integrable and have the same resonant points, which further determines the equivalence among them. Finally, the auto-Bäcklund transformation and exact solutions of one of the six coupled Korteweg–de Vries equations are proposed explicitly. Copyright © 2016 John Wiley & Sons, Ltd.

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