We consider the one and one-half dimensional multi-species relativistic Vlasov–Maxwell system with non-decaying (in space) initial data. We prove its well-posedness and nonrelativistic limit as the speed of light . These results mainly rely on a delicate analysis of energy structure and application of estimates along the characteristic lines. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper deals with the non-uniform dependence and persistence properties for a coupled Camassa–Holm equations. Using the method of approximate solutions in conjunction with well-posedness estimate, it is proved that the solution map of the Cauchy problem for this coupled Camassa–Holm equation is not uniformly continuous in Sobolev spaces *H*^{s} with *s* > 3/2. On the other hand, the persistence properties in weighted *L*^{p} spaces for the solution of this coupled Camassa–Holm system are considered. Copyright © 2016 John Wiley & Sons, Ltd.

The solution form of the system of nonlinear difference equations

where the coefficients *a*,*b*,*α*,*β* and the initial values *x*_{ − i},*y*_{ − i},*i*∈{0,1,…,*k*} are non-zero real numbers, is obtained. Furthermore, the behavior of solutions of the aforementioned system when *p* = 1 is examined. Copyright © 2016 John Wiley & Sons, Ltd.

This work proposes a general class of estimators for the population total of a sensitive variable using auxiliary information. Under a general randomized response model, the optimal estimator in this class is derived. Design-based properties of proposed estimators are obtained. A simulation study reflects the potential gains from the use of the proposed estimators instead of the customary estimators. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider a linear parabolic problem in a thick junction domain which is the union of a fixed domain and a collection of periodic branched trees of height of order 1 and small width connected on a part of the boundary. We consider a three-branched structure, but the analysis can be extended to n-branched structures. We use unfolding operator to study the asymptotic behavior of the solution of the problem. In the limit problem, we get a multi-sheeted function in which each sheet is the limit of restriction of the solution to various branches of the domain. Homogenization of an optimal control problem posed on the above setting is also investigated. One of the novelty of the paper is the characterization of the optimal control via the appropriately defined unfolding operators. Finally, we obtain the limit of the optimal control problem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we introduce a new generalization of the Helgason–Fourier transform using the angular Dirac operator on both the hyperboloid and unit ball models. The explicit integral kernels of even dimension are derived. Furthermore, we establish the formal generating function of the even dimensional kernels. In the computations, fractional integration plays a key unifying role. Copyright © 2016 John Wiley & Sons, Ltd.

]]>An isogeometric discontinuous Galerkin method for Euler equations is proposed. It integrates the idea of isogeometric analysis with the discontinuous Galerkin framework by constructing each element through the knots insertion and degree elevation techniques in non-uniform rational B-splines. This leads to the solution inherently shares the same function space as the non-uniform rational B-splines representation, and results in that the curved boundaries as well as the interfaces between neighboring elements are naturally and exactly resolved. Additionally, the computational cost is reduced in contrast to that of structured grid generation. Numerical tests demonstrate that the presented method can be high order of accuracy and flexible in handling curved geometry. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study existence of invariant curves of an iterative equation

which is from dissipative standard map. By constructing an invertible analytic solution *g*(*x*) of an auxiliary equation of the form

invertible analytic solutions of the form *g*(*λ**g*^{ − 1}(*x*)) for the original iterative functional equation are obtained. Besides the hyperbolic case 0 < |*λ*| < 1, we focus on those *λ* on the unit circle *S*^{1}, that is, |*λ*| = 1. We discuss not only those *λ* at resonance, that is, at a root of the unity, but also those *λ* near resonance under the Brjuno condition. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a generalized Kirchhoff equation in a bounded domain under the effect of a sublinear nonlinearity. Under suitable assumptions on the data of the problem, we show that, with a simple change of variable, the equation can be reduced to a classical semilinear equation and then studied with standard tools. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper deals with the following chemotaxis system:

under homogeneous Neumann boundary conditions in a bounded domain
with smooth boundary. Here, *δ* and *χ* are some positive constants and *f* is a smooth function that satisfies

with some constants *a*⩾0,*b* > 0, and *γ* > 1. We prove that the classical solutions to the preceding system are global and bounded provided that

Copyright © 2016 John Wiley & Sons, Ltd.

We consider a fourth-order variational model, to solve the image inpainting problem, with the emphasis on the recovery of low-dimensional sets (edges and corners) and the curvature of the edges. The model permits also to perform simultaneously the restoration (filtering) of the initial image where this one is available. The multiscale character of the model follows from an adaptive selection of the diffusion parameters that allows us to optimize the regularization effects in the neighborhoods of the small features that we aim to preserve. In addition, because the model is based on the high-order derivatives, it favors naturally the accurate capture of the curvature of the edges, hence to balance the tasks of obtaining long curved edges and the obtention of short edges, tip points, and corners. We analyze the method in the framework of the calculus of variations and the Γ-convergence to show that it leads to a convergent algorithm. In particular, we obtain a simple discrete numerical method based on a standard mixed-finite elements with well-established approximation properties. We compare the method to the Cahn–Hilliard model for the inpainting, and we present several numerical examples to show its performances. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the initial boundary value problem for generalized logarithmic improved Boussinesq equation. By using the Galerkin method, logarithmic Sobolev inequality, logarithmic Gronwall inequality, and compactness theorem, we show the existence of global weak solution to the problem. By potential well theory, we show the norm of the solution will grow up as an exponential function as time goes to infinity under some suitable conditions. Furthermore, for the generalized logarithmic improved Boussinesq equation with damped term, we obtain the decay estimate of the energy. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we discuss two inverse problems for differential pencils with boundary conditions dependent on the spectral parameter. We will prove the Hochstadt–Lieberman type theorem of – except for arbitrary one eigenvalue and the Borg type theorem of – except for at most arbitrary two eigenvalues, respectively. The new results are generalizations of the related results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the Maxwell equations for a composite material consisting of two phases and enjoying a periodical structure in the presence of a time-harmonic current source. We perform the two-scale homogenization taking into account both the interfacial layer thickness and the complex conductivity of the interfacial layer. We introduce a variational formulation of the differential system equipped with boundary and interfacial conditions. We show the unique solvability of the variational problem. Then, we analyze the low frequency case, high and very high frequency cases, with different strength of the interfacial currents. We find the macroscopic equations and determine the effective constant matrices such as the magnetic permeability, dielectric permittivity, and electric conductivity. The effective matrices depend strongly on the frequency of the current source; the dielectric permittivity and electric conductivity also depend on the strength of the interfacial currents. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper deals with an inverse problem of determining the diffusion coefficient, spacewise dependent source term, and the initial value simultaneously for a one-dimensional heat equation based on the boundary control, boundary measurement, and temperature distribution at a given single instant in time. By a Dirichlet series representation for the boundary observation, the identification of the diffusion coefficient and initial value can be transformed into a spectral estimation problem of an exponential series with measurement error, which is solved by the matrix pencil method. For the identification of the source term, a finite difference approximation method in conjunction with the truncated singular value decomposition is adopted, where the regularization parameter is determined by the generalized cross-validation criterion. Numerical simulations are performed to verify the result of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Ductal carcinoma *in situ* – a special cancer – is confined within the breast ductal only. We derive the mathematical ductal carcinoma *in situ* model in a form of a nonlinear parabolic equation with initial, boundary, and free boundary conditions. Existence, uniqueness, and stability of problem are proved. Algorithm and illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2016 John Wiley & Sons, Ltd.

The aim of this paper is twofold. We first establish the Besov spaces on metric spaces endowed with a doubling measure, via the remarkable orthonormal wavelet basis constructed recently by T. Hytönen and O. Tapiola, and characterize the dual spaces of these Besov spaces. Second, we prove the *T*1 type theorem for the boundedness of Calderón–Zygmund operators on these Besov spaces. Finally, we introduce a new class of Lipschitz spaces and characterize these spaces via the Littlewood–Paley theory. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we investigate the influence of boundary dissipation on the decay property of solutions for a transmission problem of Kirchhoff-type wave equations with a memory condition on one part of the boundary. Without the condition *u*_{0} = 0 on Γ_{0}, we establish a general decay of energy depending on the behavior of relaxation function by introducing suitable energy and Lyapunov functionals. This result allows a wider class of relaxation functions. Copyright © 2016 John Wiley & Sons, Ltd.

We analyze a highly nonlinear system of partial differential equations related to a model solidification and/or melting of thermoviscoelastic isochoric materials with the possibility of motion of the material during the process. This system consists of an internal energy balance equation governing the evolution of temperature, coupled with an evolution equation for a phase field whose values describe the state of material and a balance equation for the linear moments governing the material displacements. For this system, under suitable dissipation conditions, we prove global existence and uniqueness of weak solutions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the existence of homoclinic solutions for a class of fourth-order nonautonomous differential equations

where *w* is a constant,
and
. By using variational methods and the mountain pass theorem, some new results on the existence of homoclinic solutions are obtained under some suitable assumptions. The interesting is that *a*(*x*) and *f*(*x*,*u*) are nonperiodic in *x*,*a* does not fulfil the coercive condition, and *f* does not satisfy the well-known (*A**R*)-condition. Furthermore, the main result partly answers the open problem proposed by Zhang and Yuan in the paper titled with Homoclinic solutions for a nonperiodic fourth-order differential equations without coercive conditions (see Appl. Math. Comput. 2015; 250:280–286). Copyright © 2016 John Wiley & Sons, Ltd.

We consider the Lamé system for an elastic medium consisting of an inclusion embedded in a homogeneous background medium. Based on the field expansion method and layer potential techniques, we rigorously derived the asymptotic expansion of the perturbed displacement field because of small perturbations in the interface of the inclusion. We extend these techniques to determine a relationship between traction-displacement measurements and the shape of the object and derive an asymptotic expansion for the perturbation in the elastic moment tensors because of the presence of small changes in the interface of the inclusion. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a class of fourth-order Sturm-Liouville problems with transmission conditions is considered. The eigenvalues depend not only continuously but also smoothly on the problem. An expression for the derivative of the eigenvalues with respect to a given parameter: an endpoint, a boundary condition, a transmission condition, a coefficient, or the weight function, is found. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The process of transporting nanoparticles at the blood vessels level stumbles upon various physical and physiological obstacles; therefore, a Mathematical modeling will provide a valuable means through which to understand better this complexity. In this paper, we consider the motion of nanoparticles in capillaries having cylindrical shapes (i.e., tubes of finite size). Under the assumption that these particles have spherical shapes, the motion of these particles reduces to the motion of their centers. Under these conditions, we derive the mathematical model, to describe the motion of these centers, from the equilibrium of the gravitational force, the hemodynamic force and the van der Waals interaction forces. We distinguish between the interaction between the particles and the interaction between each particle and the walls of the tube. Assuming that the minimum distance between the particles is large compared with the maximum radius *R* of the particles and hence neglecting the interactions between the particles, we derive simpler models for each particle taking into account the particles-to-wall interactions. At an error of order *O*(*R*) or *O*(*R*^{3})(depending if the particles are 'near' or 'very near' to the walls), we show that the horizontal component of each particle's displacement is solution of a nonlinear integral equation that we can solve via the fixed point theory. The vertical components of the displacement are computable in a straightforward manner as soon as the horizontal components are estimated. Finally, we support this theory with several numerical tests. Copyright © 2016 John Wiley & Sons, Ltd.

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term. Based on a low-frequency and high-frequency decomposition, Green's function method and the classical energy method, we not only obtain *L*^{2} time-decay estimates but also establish the global existence of solutions to Cauchy problem when the initial data *u*_{0}(*x*) satisfies the smallness condition on
, but not on
. Furthermore, by taking a time-frequency decomposition, we obtain the optimal decay estimates of solutions. Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents a computational technique for the solution of the nonlinear mixed Volterra–Fredholm integral equations of the second kind. Using the properties of three-dimensional modification of hat functions, these are types of equations to a nonlinear system of algebraic equations. Also, convergence results and error analysis are discussed. The efficiency and accuracy of the proposed method is illustrated by numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, new Lyapunov-type inequalities are obtained for the case when one is dealing with a class of fractional two-point boundary value problems. As an application of this result, we obtain a lower bound for the eigenvalues of corresponding equations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the long-time dynamics of solutions to a nonlinear nonautonomous extensible plate equation with a strong damping. Under some suitable assumptions on the initial data, the nonlinear term and external force, we establish the existence of global solutions that generate a family of processes for the problem and obtain uniform attractors corresponding to strong and weak symbol spaces in a bounded domain . Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the one-dimensional compressible Navier–Stokes equations with periodic boundary conditions, with initial conditions in a small neighborhood of a state of uniform density and uniform nonzero velocity. We prove that, with a control given only by a body force localized in a subinterval, we can steer the system to uniform density and velocity. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Recently, numerous inventory models were developed for ameliorating items (say, fish, ducklings, chicken, etc.) considering the constant demand rate. However, such types of problems are not useful in the real market. The demand rate of ameliorating items is fluctuates in their life-period. The consumption and demand of ameliorating items are not generally steady. In a few seasons, the demand rate increases; ordinarily, it is static, and sometimes, it declines. With the outcome that their demand rate can be properly portrayed by a trapezoidal-type. In the proposed model, an inventory model for ameliorating/deteriorating items are considered with inflationary condition and time discounting rate. Additionally, having shortages that is completely backlogged. The demand rate is taken as the continuous trapezoidal-type function of time. The amelioration and deterioration rate are considered as Weibull distribution. To obtain the minimum cost, mathematical formulation of the proposed model with solution procedure is talked about. Numerical cases are given to be checked the optimal solution. Additionally, we have talked about the convexity of the proposed model through graphically. Conclusion with future worked are clarified appropriately. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper addresses a study of the controllability for a class of heat equations with memory in one spacial dimension. Unlike the classical heat equation, a heat equation with memory in general is not null controllable. There always exists a set of initial values such that the property of the null controllability fails. Also, one does not know whether there are nontrivial initial values, which can be driven to zero with a boundary control. In this paper, we give a characterization of the set of such nontrivial initial values. On the other hand, if a moving control is imposed on this system with memory, we prove the null controllability of it in a suitable state space for any initial value. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The Dirac-type time-frequency distribution (TFD), regarded as *ideal* TFD, has long been desired. It, until the present time, cannot be implemented, due to the fact that there has been no appropriate representation of signals leading to such TFD. Instead, people have been developing other types of TFD, including the Wigner and the windowed Fourier transform types. This paper promotes a practical passage leading to a Dirac-type TFD. Based on the proposed function decomposition method, viz., adaptive Fourier decomposition, we establish a rigorous and practical Dirac-type TFD theory. We do follow the route of analytic signal representation of signals founded and developed by Garbo, Ville, Cohen, Boashash, Picinbono, and others. The difference, however, is that our treatment is theoretically throughout and rigorous. To well illustrate the new theory and the related TFD, we include several examples and experiments of which some are in comparison with the most commonly used TFDs. Copyright © 2016 John Wiley & Sons, Ltd.

We present a new Lyapunov function for laminar flow, in the *x*-direction, between two parallel planes in the presence of a coplanar magnetic field for three-dimensional perturbations with stress-free boundary planes that provides conditional nonlinear stability for all Reynolds numbers(*R*_{e}) and magnetic Reynolds numbers(*R*_{m}) below *π*^{2}/2*M*. Compared with previous results on the nonlinear stability of this problem, the radius of stability ball and the energy decay rate obtained in this paper are independent of the magnetic field. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is devoted to the analysis of complex dynamics of a generalized Lorenz–Stenflo hyperchaotic system. First, on the local dynamics, the bifurcation of periodic solutions at the zero-zero-Hopf equilibrium (that is, an isolated equilibrium with double zero eigenvalues and a pair of purely imaginary eigenvalues) of this hyperchaotic system is investigated, and the sufficient conditions, which insure that two periodic solutions will bifurcate from the bifurcation point, are obtained. Furthermore, on the global dynamics, the explicit ultimate bound sets of this hyperchaotic system are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper describes the procedure of extracting information about the dynamics of highway traffic speed. The wavelet shrinkage is used to diminish the effect of the noise. Afterwards, the dynamical properties of the system are estimated through the 0–1 test for chaos, Lyapunov exponents and the notion of Shannon entropy. The results indicate the strong chaotic dynamics in the traffic speed data. In addition to that, the predictability of the system is related to the values of the maximal Lyapunov exponent and Shannon entropy. The higher those values are, the worse the predictability of the system is. Furthermore, it is shown that Shannon entropy can be used to detect changes in dynamics on different time scales. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the following second-order dynamical system:

where *c*⩾0 is a constant,
and
. When *g* admits a singularity at zero of repulsive type without the restriction of strong force condition, we apply the coincidence degree theory to prove that the system admits nonplanar collisionless rotating periodic solutions taking the form *u*(*t* + *T*) = *Q**u*(*t*),
with *T* > 0 and *Q* an orthogonal matrix under the assumption of Landesman–Lazer type. Copyright © 2016 John Wiley & Sons, Ltd.

The present article deals with the growth of biofilms produced by bacteria within a saturated porous medium. Starting from the pore-scale, the process is essentially described by attachment/detachment of mobile microorganisms to a solid surface and their ability to build biomass. The increase in biomass on the surface of the solid matrix changes the porosity and impedes flow through the pores. Using formal periodic homogenization, we derive an averaged model describing the process via Darcy's law and upscaled transport equations with effective coefficients provided by the evolving microstructure at the pore-scale. Assuming, that the underlying pore geometry may be described by a single parameter, for example, porosity, the level set equation locating the biofilm-liquid interface transforms into an ordinary differential equation (ODE) for the parameter. For such a setting, we state significant analytical and algebraic properties of these effective parameters. A further objective of this article is the analytical investigation of the resulting coupled PDE–ODE model. In a weak sense, unique solvability either global in time or at least up to a possible clogging phenomenon is shown. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we are introducing pertinent Euler–Lagrange–Jensen type *k*-quintic functional equations and investigate the ‘Ulam stability’ of these new *k*-quintic functional mappings *f*:*X**Y*, where *X* is a real normed linear space and *Y* a real complete normed linear space. We also solve the Ulam stability problem for Euler–Lagrange–Jensen alternative *k*-quintic mappings. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, Sturmian comparison theory is developed for the pair of second-order differential equations; first of which is the nonlinear differential equations of the form

- (1)

and the second is the half-linear differential equations

- (2)

where Φ_{α}(*s*) = |*s*|^{α − 1}*s* and *α*_{1} > ⋯ > *α*_{m} > *β* > *α*_{m + 1} > ⋯ > *α*_{n} > 0. Under the assumption that the solution of (2) has two consecutive zeros, we obtain Sturm–Picone type and Leighton type comparison theorems for (1) by employing the new nonlinear version of Picone formula that we derive. Wirtinger type inequalities and several oscillation criteria are also attained for (1). Examples are given to illustrate the relevance of the results. Copyright © 2016 John Wiley & Sons, Ltd.

Let
be a metric measure space of homogeneous type and *L* be a one-to-one operator of type *ω* on
for *ω*∈[0, *π*/2). In this article, under the assumptions that *L* has a bounded *H*_{∞}-functional calculus on
and satisfies (*p*_{L}, *q*_{L}) off-diagonal estimates on balls, where *p*_{L}∈[1, 2) and *q*_{L}∈(2, *∞*], the authors establish a characterization of the Sobolev space
, defined via *L*^{α/2}, of order *α*∈(0, 2] for *p*∈(*p*_{L}, *q*_{L}) by means of a quadratic function *S*_{α, L}. As an application, the authors show that for the degenerate elliptic operator *L*_{w}: =− *w*^{ − 1}div(*A*∇) and the Schrödinger type operator
with *a*∈(0, *∞*) on the weighted Euclidean space
with *A* being real symmetric, if *n*⩾3,
with *q*∈[1, 2],
, *p*∈(1, *∞*) and
with
, then, for all
,
, where the implicit equivalent positive constants are independent of *f*,
denotes the class of Muckenhoupt weights,
the reverse Hölder class, and *D*(*L*_{w}) and
the domains of *L*_{w} and
, respectively. Copyright © 2016 John Wiley & Sons, Ltd.

This paper studies the chemotaxis-haptotaxis system with nonlinear diffusion

subject to the homogeneous Neumann boundary conditions and suitable initial conditions, where *χ*, *ξ* and *μ* are positive constants, and
(*n*⩾2) is a bounded and smooth domain. Here, we assume that *D*(*u*)⩾*c*_{D}*u*^{m − 1} for all *u* > 0 with some *c*_{D} > 0 and *m*⩾1. For the case of non-degenerate diffusion, if *μ* > *μ*^{∗}, where

it is proved that the system possesses global classical solutions which are uniformly-in-time bounded. In the case of degenerate diffusion, we show that the system admits a global bounded weak solution under the same assumptions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Based on the weak form quadrature element method, a perturbation approach is developed. Waves propagating in periodic beams on a nonlinear elastic foundation are studied by using the new proposed method. The feasibility and accuracy of the proposed method are verified by comparing the present results with those available in literatures in linear cases. Detailed modal analysis of the linear cases is conducted in order to obtain the dispersion relations of the nonlinear cases. The theoretical results show that the dispersion relations of the nonlinear cases are amplitude dependent. Furthermore, the effects of geometric parameters and degree of nonlinearity on the amplitude-dependent dispersion relations are discussed in detail. This work provides a new method for analyzing the dispersion relations of nonlinear periodic structures and gives some useful guidelines for designing periodic beams or pipelines with nonlinear structure–foundation interaction. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The evaluation of the diagonal of matrix functions arises in many applications and an efficient approximation of it, without estimating the whole matrix *f*(*A*), would be useful. In the present paper, we compare and analyze the performance of three numerical methods adjusted to attain the estimation of the diagonal of matrix functions *f*(*A*), where
is a symmetric matrix and *f* a suitable function. The applied numerical methods are based on extrapolation and Gaussian quadrature rules. Various numerical results illustrating the effectiveness of these methods and insightful remarks about their complexity and accuracy are demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.

The equations describing the steady flow of Cosserat–Bingham fluids are considered, and existence of weak solution is proved for the three-dimensional boundary-value problem with the use of the Lipschitz truncation argument. In contrast to the classical Bingham fluid, the micropolar Bingham fluid supports local micro-rotations and two types of plug zones. Our approach is based on an approximation of the constitutive relation by a generalized Newtonian constitutive relation and a subsequent limiting process. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Evolution of human language and learning processes have their foundation built on grammar that sets rules for construction of sentences and words. These forms of replicator–mutator (game dynamical with learning) dynamics remain however complex and sometime unpredictable because they involve children with some predispositions. In this paper, a system modeling evolutionary language and learning dynamics is investigated using the Crank–Nicholson numerical method together with the new differentiation with non-singular kernel. Stability and convergence are comprehensively proven for the system. In order to seize the effects of the non-singular kernel, an application to game dynamical with learning dynamics for a population with five languages is given together with numerical simulations. It happens that such dynamics, as functions of the learning accuracy *μ*, can exhibit unusual bifurcations and limit cycles followed by chaotic behaviors. This points out the existence of fickle and unpredictable variations of languages as time goes on, certainly due to the presence of learning errors. More interestingly, this chaos is shown to be dependent on the order of the non-singular kernel derivative and speeds up as this derivative order decreases. Hence, can people use that order to control chaotic behaviors observed in game dynamical systems with learning! Copyright © 2016 John Wiley & Sons, Ltd.

This article investigates the solvability and optimal controls of systems monitored by fractional delay evolution inclusions with Clarke subdifferential type. By applying a fixed-point theorem of condensing multivalued maps and some properties of Clarke subdifferential, an existence theorem concerned with the mild solution for the system is proved under suitable assumptions. Moreover, an existence result of optimal control pair that governed by the presented system is also obtained under some mild conditions. Finally, an example is given to illustrate our main results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this study, the generation of smooth trajectories of the end effector of a rotating extensible manipulator arm is considered. Possible trajectories are modelled using Cartesian and polar piecewise cubic interpolants expressed as polynomial Hermite-type functions. The use of polar piecewise cubic interpolants devises continuous first-order and – in some cases – second-order derivatives and allows easy calculation of kinematics variables such as velocity and acceleration. Moreover, the manipulator equations of motion can be easily handled, and the constrained trajectory of the non-active end of the manipulator derived directly from the position of the end-effector. To verify the proposed approach, numerical simulations are conducted for two different configurations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we propose an eco-epidemiological predator–prey model, modeling the spread of infectious keratoconjunctivitis among domestic and wild ungulates, during the summer season, when they intermigle in high mountain pastures. The disease can be treated in the domestic animals, but for the wild herbivores, it leads to blindness, with consequent death. The model shows that the disease can lead infected herbivores or their predators to extinction, even if it does not affect the latter. Boundedness of solutions and equilibria feasibility are obtained. Stability around the different equilibrium points is analyzed through eigenvalues and the Routh–Hurwitz criterion. Simulations are carried out to support the theoretical results. Sensitivity with respect of some parameters is investigated. The prey vaccination as control measure is introduced and simulated, although at present, the vaccine is not yet available, but just being developed. It would then possibly eradicate the infection in the domestic animals, which are considered a disease reservoir. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we propose a space-time spectral method for solving a class of time fractional convection diffusion equations. Because both fractional derivative and spectral method have global characteristics in bounded domains, we propose a space-time spectral-Galerkin method. The convergence result of the method is proved by providing a priori error estimate. Numerical results further confirm the expected convergence rate and illustrate the versatility of our method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The aim of this work is to study *μ*-pseudo almost automorphic solutions of abstract fractional integro-differential neutral equations with an infinite delay. Thanks to some restricted hypothesis on the delayed data in the phase space, we ensure the existence of the ergodic component of the desired solution. Copyright © 2016 John Wiley & Sons, Ltd.

We propose and investigate a delayed model that studies the relationship between HIV and the immune system during the natural course of infection and in the context of antiviral treatment regimes. Sufficient criteria for local asymptotic stability of the infected and viral free equilibria are given. An optimal control problem with time delays both in state variables (incubation delay) and control (pharmacological delay) is then formulated and analyzed, where the objective consists to find the optimal treatment strategy that maximizes the number of uninfected *C**D*4^{ + } T cells as well as cytotoxic T lymphocyte immune response cells, keeping the drug therapy as low as possible. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we investigate a boundary problem with non-local conditions for mixed parabolic–hyperbolic-type equation with three lines of type changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part, we use solution of the first boundary problem with appropriate Green's function, and in hyperbolic parts, we use corresponding solutions of the Cauchy problem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider a non-stationary Stokes system in a thin porous medium Ω_{ϵ} of thickness *ϵ* which is perforated by periodically solid cylinders of size *a*_{ϵ}. We are interested here to give the limit behavior when *ϵ* goes to zero. To do so, we apply an adaptation of the unfolding method. Time-dependent Darcy's laws are rigorously derived from this model depending on the comparison between *a*_{ϵ} and *ϵ*. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is concerned with the solvability of a fully nonlinear third-order *m*-point boundary value problem at resonance posed on the half line. The nonlinearity which depends on the first and the second derivatives satisfies a sublinear-like growth condition. Our main existence result is based on Mawhin's coincidence degree theory. An illustrative example of application is included. Copyright © 2016 John Wiley & Sons, Ltd.

Many researchers have studied simple low order ODE model problems for fluid flows in order to gain new insight into the dynamics of complex fluid flows. We investigate the existence of a global attractor for a low order ODE system that has served as a model problem for transition to turbulence in viscous incompressible fluid flows. The ODE system has a linear term and an energy-conserving, non-quadratic nonlinearity. Standard energy estimates show that solutions remain bounded and converge to a global attractor when the linear term is negative definite, that is, the linear term is energy decreasing; however, numerical results indicate the same result is true in some cases when the linear term does not satisfy this condition. We give a new condition guaranteeing solutions remain bounded and converge to a global attractor even when the linear term is not energy decreasing. We illustrate the new condition with examples. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Graph theory is a fundamental tool in the study of economic issues, and input–output tables are one of the main examples. We use the interpretation of the labour market through networks to obtain a better understanding on its overall functioning. One benefit of the network perspective is that a large body of mathematics exists to help analyze many forms of networks models. If an economic system has obtained a suitable model, then it becomes possible to utilize relevant mathematical tools, such as graph theory, to better understand the way the labour market works. This interpretation allows us to employ the concepts of coverage, invariance, orbit and the structural functions supply–demand and competition and interpret them from the point of view of circular flow. In this paper, we aim to interpret the labour market through networks that are represented by graphs and where characteristic concepts of chaos theory such as cover, invariance and orbits interact with the concept circular flow. Finally, an example of this approach to labour markets is described, and some conclusions are drawn. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with the reachable set estimation problem of singular systems with time-varying delay and bounded disturbance inputs. Based on a novel Lyapunov–Krasovskii functional that contains four triple integral terms, reciprocally convex approach and free-weighting matrix method, two sufficient conditions are derived in terms of linear matrix inequalities to guarantee that the reachable set of singular systems with time-varying delay is bounded by the intersection of ellipsoid. Finally, two numerical examples are given to demonstrate the effectiveness and superiority of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Recent studies have shown that the initiation of human cancer is due to the malfunction of some genes (such as E2F, CycE, CycD, Cdc25A, P27^{Kip1}, and Rb) at the R-checkpoint during the G_{1}-to-S transition of the cell cycle. Identifying and modeling the dynamics of these genes provide new insight into the initiation and progression of many types of cancers. In this study, a cancer subnetwork that has a mutual activation between phosphatase Cdc25A and the CycE/Cdk2 complex and a mutual inhibition between the Cdk inhibitor P27^{Kip1} and the CycE/Cdk2 complex is identified. A new mathematical model for the dynamics of this cancer subnetwork is developed. Positive steady states are determined and rigorously analyzed. We have found a condition for the existence of positive steady states from the activation, inhibition, and degradation parameter values of the dynamical system. We also found a robust condition that needs to be satisfied for the steady states to be asymptotically stable. We determine the parameter value(s) under which the system exhibits a saddle–node bifurcation. We also identify the condition for which the system exhibits damped oscillation solutions. We further explore the possibility of Hopf and homoclinic bifurcations from the saddle–focus steady state of the system. Our analytic and numerical results confirm experimental results in the literature, thus validating our model. Copyright © 2016 John Wiley & Sons, Ltd.

We first prove the uniqueness of weak solutions (*ψ*,*A*) to the 3-D Ginzburg–Landau model in superconductivity with zero magnetic diffusivity and the Coulomb gauge if
, which is a critical space for some positive constant *T*. We also prove the global existence of solutions when
and *A*_{0}∈*L*^{3}. Copyright © 2016 John Wiley & Sons, Ltd.

Our work is devoted to an inverse problem for three-dimensional parabolic partial differential equations. When the surface temperature data are given, the problem of reconstructing the heat flux and the source term is investigated. There are two main contributions of this paper. First, an adjoint problem approach is used for analysis of the Fréchet gradient of the cost functional. Second, an improved conjugate gradient method is proposed to solve this problem. Based on Lipschitz continuity of the gradient, the convergence analysis of the conjugate gradient algorithm is studied. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the problem of the periodic solutions bifurcated from a homoclinic orbit for a pair of coupled ordinary differential equations in
. Assume that the autonomous system has a degenerate homoclinic solution *γ* in
. A functional analytic approach is used to consider the existence of periodic solution for the autonomous system with periodic perturbations. By exponential dichotomies and the method of Lyapunov–Schmidt, the bifurcation function defined between two finite dimensional subspaces is obtained, where the zeros correspond to the existence of periodic solutions for the coupled ordinary differential equations near
. Copyright © 2016 John Wiley & Sons, Ltd.

No abstract is available for this article.

]]>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.

]]>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.

In this paper, the ultimate bound for a new chaotic system is derived based on stability theory of dynamical systems. The meaningful contribution of this article is that the results presented in this paper contain the existing results as special cases. Finally, numerical simulations are given to verify the effectiveness and correctness of the obtained results. 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.

]]>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 couple regularization techniques of nondifferentiable optimization with the *h*-version of the boundary element method (*h*-BEM) to solve nonsmooth variational problems arising in contact mechanics. As a model example, we consider the delamination problem. The variational formulation of this problem leads to a hemivariational inequality with a nonsmooth functional defined on the contact boundary. This problem is first regularized and then discretized by an *h*-BEM. We prove convergence of the *h*-BEM Galerkin solution of the regularized problem in the energy norm, provide an *a priori* error estimate and give a numerical examples. 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.

]]>We show that the four-dimensional Martínez Alonso–Shabat equation is nonlinearly self-adjoint with differential substitution and the required differential substitution is just the admitted adjoint symmetry and vice versa. By means of computer algebra system, we obtain a number of local and nonlocal symmetries admitted by the equations under study. Then such symmetries are used to construct conservation laws of the equation under study and its reductions. 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.

]]>Blow-up phenomena for a nonlinear divergence form parabolic equation with weighted inner absorption term are investigated under nonlinear boundary flux in a bounded star-shaped region. We assume some conditions on weight function and nonlinearities to guarantee that the solution exists globally or blows up at finite time. Moreover, by virtue of the modified differential inequality, upper and lower bounds for the blow-up time of the solution are derived in higher dimensional spaces. Three examples are presented to illustrate applications of our results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we establish new Hartman–Wintner-type inequalities for a class of nonlocal fractional boundary value problems. As an application, we obtain a lower bound for the eigenvalues of corresponding equations. 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.

]]>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.

]]>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.

This paper presents the existence of positive periodic solutions for first-order neutral differential equation with distributed deviating arguments. We apply Krasnoselskii's fixed point theorem to obtain our results. An example is given to support the theory. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study a semilinear Timoshenko system having two damping effects. The observation that two damping effects might lead to smaller decay rates for solutions in comparison with one damping effect is rigorously proved here in providing optimality results. Moreover, the global well-posedness for small data in a low regularity class is presented for a larger class of nonlinearities than previously known and proved by a simpler approach. 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.

]]>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.

]]>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.

]]>Our main object is to establish a regularity criterion
with *p*≥*q* > 1 for the incompressible magnetohydrodynamics equations with zero magnetic diffusivity. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we use Krasnoselskii's fixed point theorem to study the existence and uniqueness of periodic solutions of an iterative functional differential equation

Copyright © 2016 John Wiley & Sons, Ltd.

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

]]>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.

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.