In this paper, we study a natural modification of Szász–Mirakjan operators. It is shown by discussing many important established results for Szász–Mirakjan operators. The results do hold for this modification as well, be they local in nature or global, be they qualitative or quantitative. It is also shown that this generalization is meaningful by means of examples and graphical representations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider a parallel fiber-reinforced periodic elastic composite that present an imperfect contact of spring type between the fiber and the matrix. We use the elliptic integral of Cauchy type for solving the plane strain local problems that arise from the asymptotic homogenization method. Several general conditions are assumed, which include that the fibers are disposed of arbitrary manner in the local cell, that all fibers present contact perfect with different constants of imperfection, and that their cross section is a smooth closed arbitrary curve. We find that there are infinity solutions for these problems, and we find relations between these solutions and effective coefficients of the composite. Finally, we obtain analytic formulae for the circular fiber case and show some numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper investigates the nonlinear boundary value problem, resulting from the exact reduction of the Navier–Stokes equations for unsteady laminar boundary layer flow caused by a stretching surface in a quiescent viscous incompressible fluid. We prove existence of solutions for all values of the relevant parameters and provide unique results in the case of a monotonic solution. The results are obtained using a topological shooting argument, which varies a parameter related to the axial shear stress. To solve this equation, a numerical method is proposed based on a rational Chebyshev functions spectral method. Using the operational matrices of derivative, we reduced the problem to a set of algebraic equations. We also compare this work with some other numerical results and present a solution that proves to be highly accurate. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper investigates the orbital stability of solitary waves for the coupled Klein–Gordon–Zakharov (KGZ) equations

where *α* ≠ 0. Firstly, we rewrite the coupled KGZ equations to obtain its Hamiltonian form. And then, we present a pair of sech-type solutions of the coupled KGZ equations. Because the abstract orbital stability theory presented by Grillakis, Shatah, and Strauss (1987) cannot be applied directly, we can extend the abstract stability theory and use the detailed spectral analysis to obtain the stability of the solitary waves for the coupled KGZ equations. In our work, *α* = 1,*β* = 0 are advisable. Hence, we can also obtain the orbital stability of solitary waves for the classical KGZ equations which was studied by Chen. Copyright © 2016 John Wiley & Sons, Ltd.

The main aim of the paper is to research dynamic properties of a mechanical system consisting of a ball jumping between a movable baseplate and a fixed upper stop. The model is constructed with one degree of freedom in the mechanical oscillating part. The ball movement is generated by the gravity force and non-harmonic oscillation of the baseplate in the vertical direction. The impact forces acting between the ball and plate and the stop are described by the nonlinear Hertz contact law. The ball motion is then governed by a set of two nonlinear ordinary differential equations. To perform their solving, the Runge–Kutta method of the fourth order with adaptable time step was applied. As the main result, it is shown that the systems exhibit regular, irregular, and chaotic pattern for different choices of parameters using the newly introduced 0–1 test for chaos, detecting bifurcation diagram, and researching Fourier spectra. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we are interested in a model derived from the 1-D Keller-Segel model on the half line *x* > as follows:

where *l* is a constant. Under the conserved boundary condition, we study the asymptotic behavior of solutions. We prove that the problem is always globally and classically solvable when the initial data is small, and moreover, we obtain the decay rates of solutions. The paper mainly deals with the case of *l* > 0. In this case, the solution to the problem tends to a conserved stationary solution in an exponential decay rate, which is a very different result from the case of *l* < 0. Copyright © 2016 John Wiley & Sons, Ltd.

The scalar spherical wave functions (SWFs) are solutions to the scalar Helmholtz equation obtained by the method of separation of variables in spherical polar coordinates. These functions are complete and orthogonal over a sphere, and they can, therefore, be used as a set of basis functions in solving boundary value problems by spherical wave expansions. In this work, we show that there exists a theory of functions with quaternionic values and of three real variables, which is determined by the Moisil–Theodorescu-type operator with quaternionic variable coefficients, and which is intimately related to the radial, angular and azimuthal wave equations. As a result, we explain the connections between the null solutions of these equations, on one hand, and the quaternionic hyperholomorphic and anti-hyperholomorphic functions, on the other. We further introduce the quaternionic spherical wave functions (QSWFs), which refine and extend the SWFs. Each function is a linear combination of SWFs and products of -hyperholomorphic functions by regular spherical Bessel functions. We prove that the QSWFs are orthogonal in the unit ball with respect to a particular bilinear form. Also, we perform a detailed analysis of the related properties of QSWFs. We conclude the paper establishing analogues of the basic integral formulae of complex analysis such as Borel–Pompeiu's and Cauchy's, for this version of quaternionic function theory. As an application, we present some plot simulations that illustrate the results of this work. Copyright © 2016 John Wiley & Sons, Ltd.

]]>An asymptotic expansion of the contrasting structure-like solution of the generalized Kolmogorov–Petrovskii–Piskunov equation is presented. A generalized maximum principle for the pseudoparabolic equations is developed. This, together with the generalized differential inequalities method, allows to prove the consistence and convergence of the asymptotic series method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The quaternion Fourier transform (QFT) is one of the key tools in studying color image processing. Indeed, a deep understanding of the QFT has created the color images to be transformed as whole, rather than as color separated component. In addition, understanding the QFT paves the way for understanding other integral transform, such as the quaternion fractional Fourier transform, quaternion linear canonical transform, and quaternion Wigner–Ville distribution. The aim of this paper is twofold: first to provide some of the theoretical background regarding the quaternion bound variation function. We then apply it to derive the quaternion Fourier and linear canonical inversion formulas. Secondly, to provide some in tuition for how the quaternion Fourier and linear canonical inversion theorems work on the absolutely integrable function space. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we derive five new versions of Landau-type theorems for biharmonic mappings of unit disk. Moreover, we prove that all five results are sharp when the bounds are equal to 1. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we focus on a diffuse interface model named by Hele–Shaw–Cahn–Hilliard system, which describes a two-phase Hele–Shaw flow with matched densities and arbitrary viscosity contrast in a bounded domain. The diffuse interface thickness is measured by *ϵ*, and the mobility coefficient (the diffusional Peclet number) is *ϵ*^{α}. We will prove rigorously that the global weak solutions of the Hele–Shaw–Cahn–Hilliard system converge to a varifold solution of the sharp interface model as *ϵ*0 in the case of 0≤*α* < 1. Copyright © 2016 John Wiley & Sons, Ltd.

In the present work, the steady-state crack propagation in a chain of oscillators with non-local interactions is considered. The interactions are modelled as linear springs, while the crack is presented by the absence of extra springs. The problem is reduced to the Wiener-Hopf type, and solution is presented in terms of the inverse Fourier transform. It is shown that the non-local interactions may change the structure of the solution well known from the classical local interactions formulation. In particular, it may change the range of the region of stable crack motion. The conclusions of the analysis are supported by numerical results. Namely, the observed phenomenon is partially clarified by evaluation of the structure profiles on the crack line ahead. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We present an algorithm for the identification of the relaxation kernel in the theory of diffusion systems with memory (or of viscoelasticity), which is linear, in the sense that we propose a linear Volterra integral equation of convolution type whose solution is the relaxation kernel. The algorithm is based on the observation of the flux through a part of the boundary of a body.

The identification of the relaxation kernel is ill posed, as we should expect from an inverse problem. In fact, we shall see that it is mildly ill posed, precisely as the deconvolution problem which has to be solved in the algorithm we propose. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we consider a nonlinear coupled wave equations with initial-boundary value conditions and nonlinear damping and source terms. Under suitable assumptions on the damping terms and source terms and initial data in the stable set, we obtain that the decay estimates of the energy function is exponential or polynomial by using Nakao's method. By using the energy method, we obtain the blow-up result of solution with some positive or nonpositive initial energy. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we prove the local-in-time existence and a blow-up criterion of solutions in the Besov spaces for the Euler-*α* equations of inviscid incompressible fluid flows in
. We also establish the convergence rate of the solutions of the Euler-*α* equations to the corresponding solutions of the Euler equations as the regularization parameter *α* approaches 0 in
. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a uniform elliptic nonlocal operator

- (1)

which is a weighted form of fractional Laplacian. We firstly establish three maximum principles for antisymmetric functions with respect to the nonlocal operator. Then, we obtain symmetry, monotonicity, and nonexistence of solutions to some semilinear equations involving the operator
on bounded domain,
and
, by applying direct moving plane methods. Finally, we show the relations between the classical operator − Δ and the nonlocal operator in ((1)) as *α*2. Copyright © 2016 John Wiley & Sons, Ltd.

This paper discusses uncertainty principles of images defined on the square, or, equivalently, uncertainty principles of signals on the 2-torus. Means and variances of time and frequency for signals on the 2-torus are defined. A set of phase and amplitude derivatives are introduced. Based on the derivatives, we obtain three comparable lower bounds of the product of variances of time and frequency, of which the largest lower bound corresponds to the strongest uncertainty principles known for periodic signals. Examples, including simulations, are provided to illustrate the obtained results. To the authors' knowledge, it is in the present paper, and for the first time, that uncertainty principles on the torus are studied. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In the current paper, we consider a stochastic parabolic equation that actually serves as a mathematical model describing the operation of an electrostatic actuated micro-electro-mechanical system. We first present the derivation of the mathematical model. Then after establishing the local well posedeness of the problem, we investigate under which circumstances a *finite-time quenching* for this stochastic partial differential equation, corresponding to the mechanical phenomenon of *touching down*, occurs. For that purpose, the Kaplan's eigenfunction method adapted in the context of stochastic partial differential equations is employed. Copyright © 2016 John Wiley & Sons, Ltd.

A new variant of the Adaptive Cross Approximation (ACA) for approximation of dense block matrices is presented. This algorithm can be applied to matrices arising from the Boundary Element Methods (BEM) for elliptic or Maxwell systems of partial differential equations. The usual interpolation property of the ACA is generalised for the matrix valued case. Some numerical examples demonstrate the efficiency of the new method. The main example will be the electromagnetic scattering problem, that is, the exterior boundary value problem for the Maxwell system. Here, we will show that the matrix valued ACA method works well for high order BEM, and the corresponding high rate of convergence is preserved. Another example shows the efficiency of the new method in comparison with the standard technique, whilst approximating the smoothed version of the matrix valued fundamental solution of the time harmonic Maxwell system. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study the exact controllability of *q* uncoupled damped string equations by means of the same control function. This property is called simultaneous controllability. An observability inequality is proved, which implies the simultaneous controllability of the system. Our results generalize the previous results on the linear wave without the dampings. Copyright © 2016 John Wiley & Sons, Ltd.

The shrinkage of fossil fuel resources motivates many countries to search alternative energy sources. *Jatropha curcas* is a small drought-resistant shrub from whose seeds a high grade fuel biodiesel can be produced. It is cultivated in many tropical countries including India. However, the plant is affected by the mosaic virus (*Begomovirus*) through infected white-flies (*Bemisia tabaci*) which causes mosaic disease. Disease control is an important factor to obtain healthy crop but in agricultural practice, farming awareness is equally important. Here, we propose a mathematical model for media campaigns for raising awareness among people to protect this plant in small plots and control disease. In order to archive high crop yield, we consider the awareness campaign to be arranged in impulsive way to make people aware from infected white-flies to protect Jatropha plants from mosaic virus. The study reveals that the spread of mosaic disease can be contained or even eradicated by the awareness campaigns. To attain an effective eradication, awareness campaign should be implemented at sufficiently short time intervals. Copyright © 2016 John Wiley & Sons, Ltd.

Our aim in this paper is to study the well-posedness of a singular reaction-diffusion equation which is related with brain lactate kinetics, when spatial diffusion is taken into account. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we are concerned with the backward problem of reconstructing the initial condition of a time-fractional diffusion equation from interior measurements. We establish uniqueness results and provide stability analysis. Our method is based on the eigenfunction expansion of the forward solution and the Tikhonov regularization to tackle the ill-posedness issue of the underlying inverse problem. Some numerical examples are included to illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we establish a blow-up criterion for the three-dimentional viscous, compressible magnetohydrodynamic flows. It is shown that for the Cauchy problem and the initial-boundary-value problem with initial density allowed to vanish, the strong or smooth solution for the three-dimentional magnetohydrodynamic flows exists globally if the density, temperature, and magnetic field is bounded from above. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, the stochastic stability under small Gauss type random excitation is investigated theoretically and numerically. When *p* is larger than 0, the *p*-moment stability theorem of stochastic models is proved by Lyapunov method, Ito isometry formula, matrix theory and so on. Then the application of *p*-moment such as *k*-order moment of the origin and *k*-order moment of the center is introduced and analyzed. Finally, *p*-moment stability of the power system is verified through the simulation example of a one machine and infinite bus system. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider the strongly nonlinear Nernst–Planck equations coupled with the quasi-linear Poisson equation under inhomogeneous, moreover, nonlinear boundary conditions. This system describes joint multi-component electrokinetics in a pore phase. The system is supplemented by the force balance and by the volume and positivity constraints. We establish well-posedness of the problem in the variational setting. Namely, we prove the existence theorem supported by the energy and the entropy a-priori estimates, and we provide the Lyapunov stability of the solution as well as its uniqueness in special cases. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The main work is related to show the existence and uniqueness of solution for the fractional impulsive differential equation of order *α*∈(1,2) with an integral boundary condition and finite delay. Using the application of the Banach and Sadovaskii fixed-point theorems, we obtain the main results. An example is presented at the end to verify the results of the paper. Copyright © 2016 John Wiley & Sons, Ltd.

In this study, we solve an inverse nodal problem for *p*-Laplacian Dirac system with boundary conditions depending on spectral parameter. Asymptotic formulas of eigenvalues, nodal points and nodal lengths are obtained by using modified Prüfer substitution. The key step is to apply modified Prüfer substitution to derive a detailed asymptotic estimate for eigenvalues. Furthermore, we have shown that the functions *r(x)* and *q(x)* in Dirac system can be established uniquely by using nodal parameters with the method used by Wang et al. Obtained results are more general than the classical Dirac system. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we propose a new method called the fractional natural decomposition method (FNDM). We give the proof of new theorems of the FNDM, and we extend the natural transform method to fractional derivatives. We apply the FNDM to construct analytical and approximate solutions of the nonlinear time-fractional Harry Dym equation and the nonlinear time-fractional Fisher's equation. The fractional derivatives are described in the Caputo sense. The effectiveness of the FNDM is numerically confirmed. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We establish a Bochner type characterization for Stepanov almost periodic functions, and we prove a new result about the integration of almost periodic functions. This result is then used together with a reduction principle to investigate the nature of bounded solutions of some almost periodic partial neutral functional differential equations. More specifically, we prove that all bounded solutions on are almost periodic. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The initial-boundary value problems for a Sobolev equation with exponential nonlinearities, classical, and nonclassical boundary conditions are considered. For this model, which describes processes in crystalline semiconductors, the blow-up phenomena are studied. The sufficient blow-up conditions and the blow-up time are analyzed by the method of the test functions. This analytical *a priori* information is used in the numerical experiments, which are able to determine the process of the solution's blow-up more accurately. The model derivation and some questions of local solvability and uniqueness are also discussed. Copyright © 2016 John Wiley & Sons, Ltd.

The paper presents a novel view on the scattering of a flexural wave in a Kirchhoff plate by a semi-infinite discrete system. Blocking and channelling of flexural waves are of special interest. A quasi-periodic two-source Green's function is used in the analysis of the waveguide modes. An additional ‘effective waveguide’ approximation has been constructed. Comparisons are presented for these two methods in addition to an analytical solution for a finite truncated system. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we prove existence and regularity results for weak solutions in the framework of anisotropic Sobolev spaces for a class of nonlinear anisotropic elliptic equations in the whole with variable exponents and locally integrable data. Our approach is based on the anisotropic Sobolev inequality, a smoothness, and compactness results. The functional setting involves Lebesgue–Sobolev spaces with variable exponents. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we first utilize the vanishing diffusivity method to prove the existence of global quasi-strong solutions and get some higher order estimates, and then prove the global well-posedness of the two-dimensional Boussinesq system with variable viscosity for *H*^{3} initial data. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we study the two-mode Korteweg–de Vries (TKdV) equation, which describes the propagation of two different waves modes simultaneously. We show that the TKdV equation gives multiple soliton solutions for specific values of the nonlinearity and dispersion parameters involved in the equation. We also derive other distinct exact solutions for general values of these parameters. We apply the simplified Hirota's method to study the specific of the parameters, which gives multiple soliton solutions. We also use the tanh/coth method and the tan/cot method to obtain other set of solutions with distinct physical structures. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is focused on higher order differentiation of Fourier series of functions. By means of Stokes's transformation, the recursion relations between the Fourier coefficients in Fourier series of different order (partial) derivatives of the functions as well as the general formulas for Fourier series of higher order (partial) derivatives of the functions are acquired. And then, the sufficient conditions for term-by-term differentiation of Fourier series of the functions are presented. These findings are subsequently used to reinvestigate the Fourier series methods for linear elasto-dynamical systems. The results given in this paper on the constituent elements, together with their combinatorial modes and numbering, of the sets of coefficients concerning 2*r*th order linear differential equation with constant coefficients are found to be different from the results deduced by Chaudhuri back in 2002. And it is also shown that the displacement solution proposed by Li in 2009 is valid only when the second order mixed partial derivative of the displacement vanishes at all of the four corners of the rectangular plate. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we investigate a Mimura–Tsujikawa model with nonlocal delay effect under the homogeneous Neumann boundary condition. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability, and Hopf bifurcation of nontrivial steady-state solutions bifurcating from the nonzero steady-state solution. Moreover, we illustrate our general results by applications to models with a one-dimensional spatial domain. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with a class of neutral-type BAM neural networks with distributed leakage delays. By applying the exponential dichotomy of linear differential equations, Lyapunov functional method and contraction mapping principle, we establish some sufficient conditions which ensure the existence and exponential stability of almost periodic solutions for such BAM neural networks. An example is given to illustrate the effectiveness of the theoretical findings. The results obtained in this article are completely new and complement the previously known studies. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the attraction–repulsion chemotaxis system with rotational flux terms

where
is a bounded domain with smooth boundary. Here, *S*_{1} and *S*_{2} are given parameter functions on [0,*∞*)^{2}×Ω with values in
. It is shown that for any choice of suitably regular initial data (*u*_{0},*v*_{0},*w*_{0}) fulfilling a smallness condition on the norm of *v*_{0},*w*_{0} in *L*^{∞}(Ω), the corresponding initial-boundary value problem possesses a global bounded classical solution. Copyright © 2016 John Wiley & Sons, Ltd.

Tsunamis are rare events compared with other extreme natural hazards, but the growth of population along coastlines has increased their potential impact. Tsunamis are most often generated by earthquake-induced dislocations of the seafloor, which displace large water masses. They can be simulated effectively as long waves whose propagation is modeled by the nonlinear shallow water equations. In this note, we present a brief assessment of earthquake-generated tsunami hazards for the city of Heraklion, Crete. We employ numerical hydrodynamic simulations, including inundation computations with the model MOST, and use high-resolution bathymetry and topography data for the area of interest. MOST implements a splitting method in space to reduce the system of shallow water equations in two successive systems, one for each spatial variable, and it uses a dispersive, Godunov-type finite difference method to solve the equations in characteristic form. We perform probabilistic analysis to assess the effects of the earthquake epicenter location on the tsunami, for time windows of 100, 500, and 1000years. The tsunami hazard is assessed through computed values of the maximum inundation range and maximum flow depth. Finally, we present a brief vulnerability analysis for the city of Heraklion, Crete. The data needed to identify tsunami-vulnerable areas are obtained by combining remote sensing techniques and geographic information system technology with surveyed observations and estimates of population data. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Completing previous work, a new class of interior solutions for compact static fluid spheres exhibiting pressure anisotropy, admitting conformal motion, and having 7, 8, 9, and 10 spacetime dimensions, respectively, is presented. Einstein's field equations without cosmological constant are solved for a particular energy density distribution function, assuming non-commutative geometry of spacetime. The behavior of the physical quantities obtained does not exclude the possible existence of ultra-compact, though rather exotic, stars in higher spacetime dimensions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a multistage susceptible-infectious-recovered model with distributed delays and nonlinear incidence rate is investigated, which extends the model considered by Guo *et al*. *[H. Guo, M. Y. Li and Z. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM J. Appl. Math., 72 (2012), 261–279]*. Under some appropriate and realistic conditions, the global dynamics is completely determined by the basic reproduction number *R*_{0}. If *R*_{0}≤1, then the infection-free equilibrium is globally asymptotically stable and the disease dies out in all stages. If *R*_{0}>1, then a unique endemic equilibrium exists, and it is globally asymptotically stable, and hence the disease persists in all stages. The results are proved by utilizing the theory of non-negative matrices, Lyapunov functionals, and the graph-theoretical approach. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study the following generalized quasilinear Schrödinger equation

where *N*≥3,
is a *C*^{1} even function, *g*(0) = 1 and *g*^{′}(*s*) > 0 for all *s* > 0. Under some suitable conditions, we prove that the equation has a ground state solution and infinitely many pairs ±*u* of geometrically distinct solutions. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, the global solvability to the mixed problem involving the wave equation with memory term and acoustic boundary conditions for non-locally reacting boundary is considered. Moreover, the general decay of the energy functionality is established by the techniques of Messaoudi. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The main purpose of this article is to investigate the optimal wholesaler's replenishment decisions for deterioration items under two levels of the trade credit policy and two storage facilities in order to reflect the supply chain management situation within the economic order quantity framework. In this study, each of the following assumptions have been made: (1) The own warehouse with limited capacity always is not sufficient to store the order quantity, so that a rented warehouse is needed to store the excess units over the capacity of the own warehouse; (2) The wholesaler always obtains the partial trade credit, which is independent of the order quantity offered by the supplier, but the wholesaler offers the full trade credit to the retailer; (3) The wholesaler must take a loan to pay his or her supplier the partial payment immediately when the order is received and then pay off the loan with the entire revenue. Under these three conditions, the wholesaler can obtain the least costs. Furthermore, this study models the wholesaler's optimal replenishment decisions under the aforementioned conditions in the supply chain management. Two theorems are developed to efficiently determine the optimal replenishment decisions for the wholesaler. Finally, numerical examples are given to illustrate the theorems that are proven in this study, and the sensitivity analysis with respect to the major parameters in this study is performed. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the model equations for the Timoshenko beam as a first-order system in the framework of evolutionary equations. The focus is on boundary damping, which is implemented as a dynamic boundary condition. A change of material laws allows the inclusion of a large class of cases of boundary damping. By choosing a particular material law, it is shown that the first-order approach to Sturm–Liouville problems with boundary damping is also covered. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We will prove that for piecewise C^{2}-concave domains in
Korn's first inequality holds for vector fields satisfying homogeneous normal or tangential boundary conditions with explicit Korn constant
. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study the following quasilinear chemotaxis–haptotaxis system

- (⋆)

in a bounded smooth domain
under zero-flux boundary conditions, where the nonlinearities *D*,*S*_{1}, and *S*_{2} are supposed to generalize the prototypes

with
, and *f*∈*C*^{1}([0,+*∞*) × [0,+*∞*)) satisfies

with *r* > 0 and *b* > 0. If the nonnegative initial data *u*_{0}(*x*)∈*W*^{1,∞}(Ω),*v*_{0}(*x*)∈*W*^{1,∞}(Ω), and
for some *α*∈(0,1), it is proved that

- For
*n*= 1, if and then (⋆) has a unique nonnegative classical solution, which is globally bounded. - For
*n*= 2, if and then (⋆) has a unique nonnegative classical solution, which is globally bounded. - For
*n*≥3, if and then (⋆) has a unique nonnegative classical solution, which is globally bounded.

Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this work, we consider the question of controllability of a class of integrodifferential equations on Hilbert space with measures as controls. We assume that the linear part has a resolvent operator in the sense given by R. Grimmer. We generalize the original work of N. Ahmed on vector measures, and we use it to develop necessary and sufficient conditions for weak and the exact controllability of the integrodifferential equation. Using the latter, we prove that exact controllability of the integrodifferential equation implies exact controllability of a perturbed integrodifferential equation. Controllability problem for the perturbed system is formulated fixed point problem in the space of vector measures. Our results cover impulsive controls as well as regular controls. Copyright © 2016 John Wiley & Sons, Ltd.

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

]]>We investigate sufficient conditions for existence of multiple solutions to a coupled system of fractional-order differential equations with three-point boundary conditions. By coupling the method of upper and lower solutions together with the method of monotone iterative technique, we develop conditions for iterative solutions. Based on these conditions, we study maximal and minimal solutions to the problem under consideration. We also study error estimates and provide an example. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This article contains an existence result for a class of quasiconvex stored energy functions satisfying the material non-interpenetrability condition, which primarily obstructs applying classical techniques from the vectorial calculus of variations to nonlinear elasticity. The fundamental concept of reversibility serves as the starting point for a theory of nonlinear elasticity featuring the basic duality inherent to the Eulerian and Lagrangian points of view. Motivated by this concept, split-quasiconvex stored energy functions are shown to exhibit properties, which are very alluding from a mathematical point of view. For instance, any function with finite energy is automatically a Sobolev homeomorphism; existence of minimizers can be readily established and first variation formulae hold. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, along the idea of Souplet and Zhang, we deduce a local elliptic-type gradient estimates for positive solutions of the nonlinear parabolic equation:

on
for *α* ≥ 1 and *α* ≤ 0. As applications, related Liouville-type theorem is exported. Our results are complement of known results. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we study two operators that arise in electromagnetic scattering in chiral media. We first consider electromagnetic scattering by a chiral dielectric with a perfectly conducting core. We define a chiral Calderon-type surface operator in order to solve the direct electromagnetic scattering problem. For this operator, we state coercivity and prove compactness properties. In order to prove existence and uniqueness of the problem, we define some other operators that are also related to the chiral Calderon-type operator, and we state some of their properties that they and their linear combinations satisfy. Then we sketch how to use these operators in order to prove the existence of the solution of the direct scattering problem. Furthermore, we focus on the electromagnetic scattering problem by a perfect conductor in a chiral environment. For this problem, we study the chiral far-field operator that is defined on a unit sphere and contains the far-field data, and we state and prove some of its properties that are preliminaries properties for solving the inverse scattering problem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the elliptic boundary blow-up problem

where Ω is a bounded smooth domain of
are positive continuous functions supported in disjoint subdomains Ω_{+},Ω_{−} of Ω, respectively, *a*_{+} vanishes on the boundary of
satisfies *p*(*x*)≥1 in Ω,*p*(*x*) > 1 on *∂*Ω and
, and *ε* is a parameter. We show that there exists *ε*^{∗}>0 such that no positive solutions exist when *ε* > *ε*^{∗}, while a minimal positive solution *u*_{ε} exists for every *ε*∈(0,*ε*^{∗}). Under the additional hypotheses that
is a smooth *N* − 1-dimensional manifold and that *a*_{+},*a*_{−} have a convenient decay near Γ, we show that a second positive solution *v*_{ε} exists for every *ε*∈(0,*ε*^{∗}) if
, where *N*^{∗}=(*N* + 2)/(*N* − 2) if *N* > 2 and
if *N* = 2. Our results extend that of Jorge Garcá-Melián in 2011, where the case that *p* > 1 is a constant and *a*_{+}>0 on *∂*Ω is considered. Copyright © 2016 John Wiley & Sons, Ltd.

Consider the following fractional Kirchhoff equations involving critical exponent:

where (−Δ)^{α} is the fractional Laplacian operator with *α*∈(0,1),
,
, *λ*_{2}>0 and
is the critical Sobolev exponent, *V*(*x*) and *k*(*x*) are functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space, the minimax arguments, Pohozaev identity, and suitable truncation techniques, we obtain the existence of a nontrivial weak solution for the previously mentioned equations without assuming the Ambrosetti–Rabinowitz condition on the subcritical nonlinearity *f*. Copyright © 2016 John Wiley & Sons, Ltd.

The aim of this paper is to propose mixed two-grid finite difference methods to obtain the numerical solution of the one-dimensional and two-dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large-sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two-grid method, where the two-grid method is used for solving the large-sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of *h* and Δ*t*. The numerical examples show the efficiency of this algorithm for solving the one-dimensional and two-dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a class of nonlinear fractional differential equations on the infinite interval

with the integral boundary conditions

By using Krasnoselskii fixed point theorem, the existence results of positive solutions for the boundary value problem in three cases and , are obtained, respectively. We also give out two corollaries as applications of the existence theorems and some examples to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we investigate a stochastic non-autonomous SIRS (susceptible-infected-recovered-susceptible) model. The extinction and the prevalence of the disease are discussed, and so, the threshold is given. Especially, we show there is a positive nontrivial periodic solution. At last, some examples and simulations are provided to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>No abstract is available for this article.

]]>In this paper, a class of neutral-type neural networks with delays in the leakage term on time scales are considered. By using the Banach fixed point theorem and the theory of calculus on time scales, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solutions for this class of neural networks. The results of this paper are new and complementary to the previously known results. Finally, an example is presented to illustrate the effectiveness of our results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The objective of this paper is to investigate the almost periodic dynamics for a class of delayed predator–prey model with mutual interference and Beddington–DeAngelis type functional response, in which the harvesting policies are modeled by discontinuous functions. Based on the theory of functional differential inclusions theory and set-valued analysis, the solution in sense of Filippov of system with the discontinuous harvesting policies is given, and the local and global existence of positive the solution in sense of Filippov of the system is studied. By employing generalized differential inequalities, some useful Lemmas are obtained. After that, sufficient conditions which guarantee the permanence of the system are obtained in view of the constructed Lemmas. By constructing some suitable generalized Lyapunov functional, a series of useful criteria on existence, uniqueness, and global attractivity of the almost positive periodic solution to the system are derived in view of functional differential inclusions theory and nonsmooth analysis theory. Some suitable examples together with their numeric simulations are given to substantiate the theoretical results and to illustrate various dynamical behaviors of the system. Copyright © 2016 John Wiley & Sons, Ltd.

]]>A mathematical model to understand the dynamics of malaria–visceral leishmaniasis co-infection is proposed and analyzed. Results show that both diseases can be eliminated if *R*_{0}, the basic reproduction number of the co-infection, is less than unity, and the system undergoes a backward bifurcation where an endemic equilibrium co-exists with the disease-free equilibrium when one of *R*_{m} or *R*_{l}, the basic reproduction numbers of malaria-only and visceral leishmaniasis-only, is precisely less than unity. Results also show that in the case of maximum protection against visceral leishmaniasis (VL), the disease-free equilibrium is globally asymptotically stable if malaria patients are protected from VL infection; similarly, in the case of maximum protection against malaria, the disease-free equilibrium is globally asymptotically stable if VL and post-kala-azar dermal leishmaniasis patients and the recovered humans after VL are protected from malaria infection. Numerical results show that if *R*_{m} and *R*_{l} are greater than unity, then we have co-existence of both disease at an endemic equilibrium, and malaria incidence is higher than visceral leishmaniasis incidence at steady state. Copyright © 2016 John Wiley & Sons, Ltd.

An optimal boundary control problem in a domain with oscillating boundary has been investigated in this paper. The controls are acting periodically on the oscillating boundary. The controls are applied with suitable scaling parameters. One of the major contribution is the representation of the optimal control using the unfolding operator. We then study the limiting analysis (homogenization) and obtain two limit problems according to the scaling parameters. Another notable observation is that the limit optimal control problem has three controls, namely, a distributed control, a boundary control, and an interface control. Copyright © 2016 John Wiley & Sons, Ltd.

]]>There is an increasing reliance on mathematical modelling to assist in the design of piezoelectric ultrasonic transducers since this provides a cost-effective and quick way to arrive at a first prototype. Given a desired operating envelope for the sensor, the inverse problem of obtaining the associated design parameters within the model can be considered. It is therefore of practical interest to examine the well-posedness of such models. There is a need to extend the use of such sensors into high-temperature environments, and so this paper shows, for a broad class of models, the well-posedness of the magneto-electro-thermo-elastic problem. Because of its widespread use in the literature, we also show the well-posedness of the quasi-electrostatic case. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Our aim in this paper is to study generalizations of the nonconserved and conserved Caginalp phase-field systems based on the Maxwell–Cattaneo law with two temperatures for heat conduction. In particular, we obtain well-posedness results and study the dissipativity of the associated solution operators. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we first address the space-time decay properties for higher-order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a parabolic interpolation inequality, bootstrap argument, and some weighted estimates. Secondly, we present a new solution integration formula for the Boussinesq system, which will be employed to establish the existence of strong solutions for small initial data in some scaling invariant function spaces. The smallness conditions are somehow weaker than those presented by Brandolese and Schonbek. We further investigate the asymptotic profiles and decay properties of these strong solutions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Antiplane stress state of a piecewise-homogeneous elastic body with a semi-infinite crack along the interface is considered. The longitudinal displacements along one of the crack edges on a finite interval, adjacent to the crack tip, are known. Shear stresses are applied to the body along the crack edges and at infinity. The problem reduces to a Riemann–Hilbert boundary-value matrix problem with a piecewise-constant coefficient for a complex potential in the class of symmetric functions. The complex potential is found explicitly using a Gaussian hypergeometric function. The stress state of the body close to the singular points is investigated. The stress intensity factors are determined. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Collective behavior of a group of individuals is studied. Each individual adopts one of two alternative decisions on the basis of a neural network bistable dynamical system. The parameters of this system are regulated by collective behavior of the group with the purpose to control the number of individuals with certain decision. It is shown how behavior of the group depends on the distribution of initial states of individuals before they begin the process of decision making. If this distribution is narrow, then it can be impossible to achieve a stable coexistence of two decisions, and oscillations in the number of individuals with given decisions are observed. Various implications of this theory are discussed. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study the Cauchy problem for a class of strongly damped multidimensional generalized Boussinesq equations *u*_{tt}−Δ*u*−Δ*u*_{tt}+Δ^{2}*u*+Δ^{2}*u*_{tt}−*k*Δ*u*_{t}=Δ*f*(*u*), where *k* is a positive constant. Under some assumptions and by using potential well method, we prove the existence and nonexistence of global weak solution without solution without establishing the local existence theory. Copyright © 2016 John Wiley & Sons, Ltd.

We study global existence and blow up in finite time for a one-dimensional fast diffusion equation with memory boundary condition. The problem arises out of a corresponding model formulated from tumor-induced angiogenesis. We obtain necessary and sufficient conditions for global existence of solutions to the problem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We provide the proof that the space of time series data is a Kolmogorov space with *T*_{0}-separation axiom using the loop space of time series data. In our approach, we define a cyclic coordinate of intrinsic time scale of time series data after empirical mode decomposition. A spinor field of time series data comes from the rotation of data around price and time axis by defining a new extradimension to time series data. We show that there exist hidden eight dimensions in Kolmogorov space for time series data. Our concept is realized as the algorithm of empirical mode decomposition and intrinsic time scale decomposition, and it is subsequently used for preliminary analysis on the real time series data. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we give an alternative proof to the global existence result, which is originally owing to the pioneering work of Klainerman and Christodoulou, for the Cauchy problem of quasilinear wave equations with null condition in three space dimensions. The proof can display the following three features simultaneously: the Lorentz boost operator is not employed in the generalized energy estimates; the generalized energy of the solution will be always small, which was first observed by Alinhac; and the initial data are not assumed to have a compact support. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this work, we deal with two-point boundary value problems for nonlinear impulsive Hamiltonian systems with sub-linear or linear growth. A theorem based on the Schauder fixed point theorem is established, which gives a result that yields existence of solutions without implications that solutions must be unique. An upper bound for the solution is also established. Examples are given to illustrate the main result. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we investigate positive solutions for boundary value problem of a fractional thermostat model with a parameter. Under different conditions of the function *f*, existence and nonexistence results for positive solutions are derived in terms of different values of *λ*. The results are illustrated with an example. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we study the existence and uniqueness of mild solutions for stochastic partial integrodifferential equations under local non-Lipschitz conditions on the coefficients. Our analysis makes use of the theory of resolvent operators as developed by R. Grimmer as well as a stopping time technique. Our results complement and improve several earlier related works. An example is provided to illustrate the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with a class of non-autonomous neutral functional differential equations with multi-proportional delays. It is shown that all solutions of the addressed system are globally exponentially convergent by employing the differential inequality technique and a novel argument. The obtained results improve and supplement existing ones. We also use numerical simulations to demonstrate our theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the vanishing viscosity limit problem for the 3D incompressible magnetohydrodynamic (MHD) system in a general bounded smooth domain of **R**^{3} with the generalized Navier slip boundary conditions. We also obtain rates of convergence of the solution of viscous MHD to the corresponding ideal MHD. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is concerned with the regularity criterion for a class of axisymmetric solutions to 3D incompressible magnetohydrodynamic equations. More precisely, for the solutions that have the form of *u* = *u*_{r}*e*_{r}+*u*_{θ}*e*_{θ}+*u*_{z}*e*_{z} and *b* = *b*_{θ}*e*_{θ}, we prove that if |*r**u*(*x*,*t*)|≤*C* holds for −1≤*t* < 0, then (*u*,*b*) is regular at time zero. This result can be thought as a generalization of recent results in for the 3D incompressible Navier-Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, the nonlinear model of the mechanism with two degrees of freedom will be studied. An approximate analytical solution of the differential equation of motion in the series showed the presence of features in the aspiration of the mass of one of the bodies to zero. It also gives an algorithm for finding the points of degeneracy of communication between small perturbations of the function of the problem and the derivatives of these functions at a time. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions of differential algebraic systems for ordinary differential equations. The reproducing kernel Hilbert space is constructed in which the initial conditions of the systems are satisfied. While, two smooth kernel functions are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper investigates the smooth solution of 2D Chaplygin gas equations on an asymptotically flat Riemannian manifold. Under the assumption that the initial data are close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of smooth solutions to the Cauchy problem for two-dimensional flow of Chaplygin gases on curved space. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper examines the initial-value problem for the nonhomogeneous incompressible nematic liquid crystals system with vacuum. This paper establishes two main results. The first result is involved with the global strong solutions to the 2D liquid crystals system in a bounded smooth domain. Our second result is concerned with the small data global existence result about the 3D system in the whole space. In addition, the local existence and a blow-up criterion of strong solutions are also mentioned. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a mathematical model of cancer treatment, in the form of a system of ordinary differential equations, by chemotherapy and radiotherapy where there is metastasis from a primary to a secondary site has been proposed and analyzed. The interaction between immune cells and cancer cells has been examined, and the chemotherapy agent has been considered as a predator on both normal and cancer cells. The metastasis may be time delayed. For better investigation of the treatment process and based on physical investigation, the immanent effects of inputs on cancer dynamic have been investigated. It is supposed that the interaction between NK cells and tumor cells changes during the chemotherapy. This novel approach is useful not only to gain a broad understanding of the specific system dynamics but also to guide the development of combination therapies. The analysis is carried out both analytically (where possible) and numerically. By considering such immanent effects, the tumor-free equilibrium point will be stable at the end of treatment, and the tumor can not recur again, and the patient will totally recover. So, the present analysis suggests that a proper treatment method should change the dynamics of the cancer instead of only reducing the population of cancer cells. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The goal of the paper is to study structurally damped elastic waves in 2D. A suitable diagonalization procedure allows to derive WKB representations for solutions to the Cauchy problems. As consequences, we obtain results on energy decay with and without additional regularity for the data, Gevrey smoothing, and propagation of singularities for the visco-elastic damped elastic wave model. 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 employ the method of lower and upper solutions coupled with the monotone iterative technique to obtain existence and uniqueness of extremal mild solutions to neutral fractional differential equation with infinite delay in an ordered Banach space. The results are obtained by using the theory of fractional calculus, monotone iterative technique, measure of noncompactness, and semigroup theory. Finally, we discuss an example to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The first integrals and exact solutions of mathematical models of epidemiology: a susceptible-infected-recovered-infected (SIRI) model and a tuberculosis model with demographic growth are analyzed. These models are represented by systems of first-order nonlinear ordinary differential equations, and this system is replaced by one which contains a second-order ordinary differential equation. The partial Lagrangian approach is then utilized to derive the first integrals of these models. Several cases arise. Then, we utilize the derived first integrals to construct exact solutions for the models under investigation and determine new solutions. The dynamic properties of these models are studied too. Copyright © 2016 John Wiley & Sons, Ltd.

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