In this paper, we investigate a Vector-Borne disease model with nonlinear incidence rate and 2 delays: One is the incubation period in the vectors and the other is the incubation period in the host. Under the biologically motivated assumptions, we show that the global dynamics are completely determined by the basic reproduction number *R*_{0}. The disease-free equilibrium is globally asymptotically stable if *R*_{0}≤1; when *R*_{0}>1, the system is uniformly persistent, and there exists a unique endemic equilibrium that is globally asymptotically. Numerical simulations are conducted to illustrate the theoretical results.

For a fissured medium with uncertainty in the knowledge of fractures' geometry, a conservative tangential flow field is constructed, which is consistent with the physics of stationary fluid flow in porous media and an interpolated geometry of the cracks. The flow field permits computing preferential fluid flow directions of the medium, rates of mechanical energy dissipations, and a stochastic matrix modeling stream lines and fluid mass transportation, for the analysis of solute/contaminant mass advection-diffusion as well as drainage times.

]]>This paper is concerned with the following nonlinear fractional Schrödinger equation

where *ε*>0 is a small parameter, *V*(*x*) is a positive function, 0<*s*<1, and
. Under some suitable conditions, we prove that for any positive integer *k*, one can construct a nonradial sign-changing (nodal) solutions with exactly *k* maximum points and *k* minimum points near the local minimum point of *V*(*x*).

This paper studies the dynamics of the generalized Lengyel-Epstein reaction-diffusion model proposed in a recent study by Abdelmalek and Bendoukha. Two main results are shown in this paper. The first of which is sufficient conditions that guarantee the nonexistence of Turing patterns, ie, nonconstant solutions. Second, more relaxed conditions are derived for the stability of the system's unique steady-state solution.

]]>A class of inverse problems for restoring the right-hand side of a fractional heat equation with involution is considered. The results on existence and uniqueness of solutions of these problems are presented.

]]>A deterministic model for the transmission dynamics of measles in a population with fraction of vaccinated individuals is designed and rigorously analyzed. The model with standard incidence exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibrium whenever the associated reproduction number is less than unity. This phenomenon can be removed if either measles vaccine is assumed to be perfect or disease related mortality rates are negligible. In the latter case, the disease-free equilibrium is shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, the model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. This equilibrium is shown, using a nonlinear Lyapunov function of Goh-Volterra type, to be globally asymptotically stable for a special case.

]]>Malaria is one of the most common mosquito-borne diseases widespread in the tropical and subtropical regions. Few models coupling the within-host malaria dynamics with the between-host mosquito-human dynamics have been developed. In this paper, by adopting the nested approach, a malaria transmission model with immune response of the host is formulated. Applying age-structured partial differential equations for the between-host dynamics, we describe the asymptomatic and symptomatic infectious host population for malaria transmission. The basic reproduction numbers for the within-host model and for the coupled system are derived, respectively. The existence and stability of the equilibria of the coupled model are analyzed. We show numerically that the within-host model can exhibit complex dynamical behavior, possibly even chaos. In contrast, equilibria in the immuno-epidemiological model are globally stable and their stabilities are determined by the reproduction number. Increasing the activation rate of the within-host immune response “dampens” the sensitivity of the population level reproduction number and prevalence to the increase of the within-host reproduction of the pathogen. From public health perspective this means that treatment in a population with higher immunity has less impact on the population-level reproduction number and prevalence than in a population with less immunity.

]]>We provide a non-linear image transformation by means of conformal geometric algebra (CGA) elements and operations. We show the correspondence to the fisheye correction algorithms; more precisely, we prove the proportionality of classical models with CGA algorithm and provide exact formula in terms of CGA. Consequently, we show that the geometric construction allows to determine the inverse model quite universally.

]]>In this paper, we consider the integration of systems of second-order linear inhomogeneous initial value problems with constant coefficients. Hybrid Numerov methods are used that are constructed in the sense of Runge-Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. We present the order conditions taking advantage of the special structure of the problem at hand. These equations are solved using differential evolution technique, and we present a method with algebraic order eighth at a cost of only 5 function evaluations per step. Numerical results over some linear problems, especially arising from the semidiscretization of the wave equation indicate the superiority of the new method.

]]>In this paper, we investigate the dynamics of a multigroup disease propagation model with distributed delays and nonlinear incidence rates, which accounts for the relapse of recovered individuals. The main concern is the stability of the equilibria, sufficient conditions for global stability being obtained by applying Lyapunov-LaSalle invariance principle and using Lyapunov functionals, which are constructed using their single-group counterparts. The situation in which the deterministic model is subject to perturbations of white noise type is also investigated from a stability viewpoint.

]]>In this paper, we investigate a backward problem for a space-fractional partial differential equation. The main purpose is to propose a modified regularization method for the inverse problem. The existence and the uniqueness for the modified regularized solution are proved. To derive the gradient of the optimization functional, the variational adjoint method is introduced, and hence, the unknown initial value is reconstructed. Finally, numerical examples are provided to show the effectiveness of the proposed algorithm. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study a reaction-diffusion equation where nonlinear memory and concentration effects are considered at the same time. We are specially concerned with the local solvability of this problem for singular initial data in Lebesgue spaces. We also analyze the vanishing concentration problem and prove a blow-up alternative. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we present the approximate solution of damped Boussinesq equation using extended Raviart–Thomas mixed finite element method. In this method, the numerical solution of this equation is obtained using triangular meshes. Also, for discretization in time direction, we use an implicit finite difference scheme. In addition, error estimation and stability analysis of both methods are shown. Finally, some numerical examples are considered to confirm the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we consider the almost periodic dynamics of an impulsive multispecies Lotka–Volterra competition system with time delays on time scales. By establishing some comparison theorems of dynamic equations with impulses and delays on time scales, a permanence result for the model is obtained. Furthermore, based on the permanence result, by studying the Lyapunov stability theory of impulsive dynamic equations on time scales, we establish the existence and uniformly asymptotic stability of a unique positive almost periodic solution of the system. Finally, we give an example to show the feasibility of our main results, and our example also shows that the continuous time system and its corresponding discrete time system have the same dynamics. Our results of this paper are completely new even if for both the case of the time scale and the case of the time scale . Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we use the domain decomposition method to prove well-posedness and smoothness results in anisotropic weighted Sobolev spaces for a multidimensional high-order parabolic equation set in conical time-dependent domains of . Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study a model of hematopoiesis with time-varying delays and discontinuous harvesting, which is described by a nonsmooth dynamical system. Based on a newly developed method, nonsmooth analysis, and the generalized Lyapunov method, some new delay-dependent criteria are established to ensure the existence and global exponential stability of positive periodic solutions. Moreover, an example with numerical simulations is presented to demonstrate the effectiveness of theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Quaternion-valued signals along with quaternion Fourier transforms (QFT) provide an effective framework for vector-valued signal and image processing. However, the sampling theory of quaternion-valued signals has not been well developed. In this paper, we present the generalized sampling expansions associated with QFT by using the generalized translation and convolution. We show that a *σ*-bandlimited quaternion-valued signal in QFT sense can be reconstructed from the samples of output signals of *M* linear systems based on QFT. Quaternion linear canonical transform is a generalization of QFT with six parameters. Using the relationship between QFT, we derive the sampling formula for *σ*-bandlimited quaternion-valued signal in quaternion linear canonical transform sense. Examples are given to illustrate our results. Copyright © 2017 John Wiley & Sons, Ltd.

We consider general virus dynamics model with virus-to-target and infected-to-target infections. The model is incorporated by intracellular discrete or distributed time delays. We assume that the virus-target and infected-target incidences, the production, and clearance rates of all compartments are modeled by general nonlinear functions that satisfy a set of reasonable conditions. The non-negativity and boundedness of the solutions are studied. The existence and stability of the equilibria are determined by a threshold parameter. We use suitable Lyapunov functionals and apply LaSalle's invariance principle to prove the global asymptotic stability of the all equilibria of the model. We confirm the theoretical results by numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.

]]>One of the fundamental problems in communications is finding the energy distribution of signals in time and frequency domains. It should therefore be of great interest to find the quaternionic signal whose time-frequency energy distribution is most concentrated in a given time-frequency domain. The present paper finds a new kind of quaternionic signals whose energy concentration is maximal in both time and frequency under the quaternionic Fourier transform. The new signals are a generalization of the classical prolate spheroidal wave functions to a quaternionic space, which are called the quaternionic prolate spheroidal wave functions. The purpose of this paper is to present the definition and fundamental properties of the quaternionic prolate spheroidal wave functions and to show that they can reach the extreme case within the energy concentration problem both from the theoretical and experimental description. The superiority of the proposed results can be widely applied to the application of 4D valued problems. In particular, these functions are shown as an effective method for bandlimited quaternionic signals relying on the extrapolation problem. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider the Calderón problem in an infinite cylindrical domain, whose cross section is a bounded domain of the plane. We prove log–log stability in the determination of the isotropic periodic conductivity coefficient from partial Dirichlet data and partial Neumann boundary observations of the solution. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Calgero–Bogoyavlenskii–Schiff (CBS) equation is analytically solved through two successive reductions into an ordinary differential equation (ODE) through a set of optimal Lie vectors. During the second reduction step, CBS equation is reduced using hidden vectors. The resulting ODE is then analytically solved through the singular manifold method in three steps; First, a Bäcklund truncated series is obtained. Second, this series is inserted into the ODE, and finally, a seminal analysis leads to a Schwarzian differential equation in the eigenfunction *φ*(*η*). Solving this differential equation leads to new analytical solutions. Then, through two backward substitution steps, the original dependent variable is recovered. The obtained results are plotted for several Lie hidden vectors and compared with previous work on CBS equation using Lie transformations. Copyright © 2017 John Wiley & Sons, Ltd.

The theory of real quaternion differential equations has recently received more attention, but significant challenges remain the non-commutativity structure. They have numerous applications throughout engineering and physics. In the present investigation, the Laplace transform approach to solve the linear quaternion differential equations is achieved. Specifically, the process of solving a quaternion different equation is transformed to an algebraic quaternion problem. The Laplace transform makes solving linear ODEs and the related initial value problems much easier. It has two major advantages over the methods discussed in literature. The corresponding initial value problems can be solved without first determining a general solution. More importantly, a particularly powerful feature of this method is the use of the Heaviside functions. It is helpful in solving problems, which is represented by complicated quaternion periodic functions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper studies the time-averaged energy dissipation rate ⟨*ε*_{SMD}(*u*)⟩ for the combination of the Smagorinsky model and damping function. The Smagorinsky model is well known to over-damp. One common correction is to include damping functions that reduce the effects of model viscosity near walls. Mathematical analysis is given here that allows evaluation of ⟨*ε*_{SMD}(*u*)⟩ for any damping function. Moreover, the analysis motivates a modified van Driest damping. It is proven that the combination of the Smagorinsky with this modified damping function does not over-dissipate and is also consistent with Kolmogorov phenomenology. Copyright © 2017 John Wiley & Sons, Ltd.

The Riesz probability distribution on any symmetric cone and, in particular, on the cone of positive definite symmetric matrices represents an important generalization of the Wishart and of the matrix gamma distributions containing them as particular examples. The present paper is a continuation of the investigation of the properties of this probability distribution. We first establish a property of invariance of this probability distributions by a subgroup of the orthogonal group. We then show that the Pierce components of a Riesz random variable are independent, and we determine their probability distributions. Some moments and some useful expectations related to the Riesz probability distribution are also calculated. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The present contribution focuses on the estimation of the geometric acceleration and of the geometric jolt (namely, the derivative of the acceleration) of a multidimensional, structured gyroscopic signal. A gyroscopic signal encodes the instantaneous orientation of a rigid body during a full three-dimensional rotation that is regarded as a trajectory in the curved space SO(3) of the special orthogonal matrices. The geometric acceleration and jolt associated to a gyroscopic signal are evaluated through the rules of calculus prescribed by differential geometry. Such an endeavor is motivated by recent studies on the smoothness of human body movement in biomechanical engineering, sports science, and rehabilitation neuroengineering. Two indexes of smoothness are compared, namely, a normalized proper geometric acceleration index and a normalized proper geometric jolt index. Our investigation concludes that, in the considered experiments with measured signals, for relatively low values of the acceleration and of the jolt indexes, such indexes are strongly positively correlated. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper presents a method for computing numerical solutions of two-dimensional Stratonovich Volterra integral equations using one-dimensional modification of hat functions and two-dimensional modification of hat functions. The problem is transformed to a linear system of algebraic equations using the operational matrix associated with one-dimensional modification of hat functions and two-dimensional modification of hat functions. The error analysis of the method is given. The method is computationally attractive, and applications are demonstrated by a numerical example. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier–Stokes equations with a slip boundary condition. We establish the local well-posedness of the strong solutions for initial boundary value problems for such systems. Furthermore, the vanishing viscosity limit process is established, and a strong rate of convergence is obtained as the boundary of the domain is flat. In addition, it is needed to add some additional condition for density to match well the boundary condition. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This work provides sufficient conditions for the existence of homoclinic solutions of fourth-order nonlinear ordinary differential equations. Using Green's functions, we formulate a new modified integral equation that is equivalent to the original nonlinear equation. In an adequate function space, the corresponding nonlinear integral operator is compact, and it is proved an existence result by Schauder's fixed point theorem. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper is addressed to a study of the persistent regional null controllability problems for one-dimensional linear degenerate wave equations through a distributed controller. Different from non-degenerate wave equations, the classical null controllability results do not hold for some degenerate wave equations. Thus, persistent regional null controllability is introduced, which means finding a control such that the corresponding state of the degenerate wave equation may vanish in a suitable subset of the space domain in a period of time. In order to solve this problem, we need to establish the regional null controllability for degenerate wave equations. This problem is reduced to a suitable observability problem of a linear degenerate wave equation. The key point is to choose a suitable multiplier in order to establish this observability inequality. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In the current study, we consider the approximate solutions of fractional-order PDEs with initial-boundary conditions based on the shifted Chebyshev polynomials. The proposed method is combined with the operational matrix of fractional-order differentiation described in the Caputo's sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations with constant coefficients by dispersing unknown variables. The validity and effectiveness of the approach are demonstrated via some numerical examples. Lastly, the error analysis of the proposed method has been investigated. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We study Hankel transform of the sequences (*u*,*l*,*d*),*t*, and the classical Motzkin numbers. Using the method based on orthogonal polynomials, we give closed-form evaluations of the Hankel transform of the aforementioned sequences, sums of two consecutive, and shifted sequences. We also show that these sequences satisfy some interesting convolutional properties. Finally, we partially consider the Hankel transform evaluation of the sums of two consecutive shifted (*u*,*l*,*d*)-Motzkin numbers. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we have studied a prey–predator model living in a habitat that divided into two regions: an unreserved region and a reserved (refuge) region. The migration between these two regions is allowed. The interaction between unreserved prey and predator is Crowley–Martin-type functional response. The local and global stability of the system is discussed. Further, the system is extended to incorporate the effect of time delay. Then the dynamical behavior of the system is analyzed, taking delay as a bifurcation parameter. The direction of Hopf bifurcation and the stability of the bifurcated periodic solution are determined with the help of normal form theory and centre manifold theorem. We have also discussed the influence of prey refuge on densities of prey and predator species. The analytical results are supplemented with numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, a novel simulation methodology based on the reproducing kernels is proposed for solving the fractional order integro-differential transport model for a nuclear reactor. The analysis carried out in this paper thus forms a crucial step in the process of development of fractional calculus as well as nuclear science models. The fractional derivative is described in the Captuo Riemann–Liouville sense. Results are presented graphically and in tabulated forms to study the efficiency and accuracy of method. The present scheme is very simple, effective, and appropriate for obtaining numerical simulation of nuclear science models. Copyright © 2017 John Wiley & Sons, Ltd.

]]>It has been reported that training deep neural networks is more difficult than training shallow neural networks. Hinton *et al*. proposed deep belief networks with a learning algorithm that trains one layer at a time. A much better generalization can be achieved when pre-training each layer with an unsupervised learning algorithm. Since then, deep neural networks have been extensively studied. On the other hand, it has been revealed that singular points affect the training dynamics of the learning models such as neural networks and cause a standstill of training. Naturally, training deep neural networks suffer singular points. As described in this paper, we present a deep neural network model that has fewer singular points than the usual one. First, we demonstrate that some singular points in the deep real-valued neural network, which is equivalent to a deep complex-valued neural network, have been resolved as its inherent property. Such deep neural networks are less likely to become trapped in local minima or plateaus caused by critical points. Results of experiments on the two spirals problem, which has an extreme nonlinearity, support our theory. Copyright © 2017 John Wiley & Sons, Ltd.

We are concerned with the identification and reconstruction of the coefficients of a linear parabolic system from finite time observations of the solution on the boundary. We present two procedures depending on whether the spectrum of the system is simple or multiple. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Protein structure calculations using nuclear magnetic resonance (NMR) experiments are one of the most important applications of distance geometry. The chemistry of proteins and the NMR data allow us to define an atomic order, such that the distances related to the pairs of atoms {*i*−3,*i*},{*i*−2,*i*},{*i*−1,*i*} are available, and solve the problem iteratively using a combinatorial method, called branch-and-prune. The main step of BP algorithm is to intersect three spheres centered at the positions for atoms *i*−3,*i*−2,*i*, with radius given by the atomic distances *d*_{i−3,i},*d*_{i−2,i},*d*_{i−1,i}, respectively, to obtain the position for atom *i*. Because of uncertainty in NMR data, some of the distances *d*_{i−3,i} may not be precise or even not be available. Using conformal Clifford algebra, in addition to take care of NMR uncertainties, which implies that we have to calculate sphere intersections considering that their centers and radius may not be fixed anymore, we consider a more flexible atomic order, where distances *d*_{i−3,i} are replaced by *d*_{j,i}, where *j*⩽*i*−3. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, an iteration process is considered to solve linear ill-posed problems. Based on the randomness of the involved variables, this kind of problems is regarded as simulation problems of the posterior distribution of the unknown variable given the noise data. We construct a new ensemble Kalman filter-based method to seek the posterior target distribution. Despite the ensemble Kalman filter method having widespread applications, there has been little analysis of its theoretical properties, especially in the field of inverse problems. This paper analyzes the propagation of the error with the iteration step for the proposed algorithm. The theoretical analysis shows that the proposed algorithm is convergence. We compare the numerical effect with the Bayesian inversion approach by two numerical examples: backward heat conduction problem and the first kind of integral equation. The numerical tests show that the proposed algorithm is effective and competitive with the Bayesian method. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Both numerical and asymptotic analyses are performed to study the similarity solutions of three-dimensional boundary-layer viscous stagnation point flow in the presence of a uniform magnetic field. The three-dimensional boundary-layer is analyzed in a non-axisymmetric stagnation point flow, in which the flow is developed because of influence of both applied magnetic field and external mainstream flow. Two approaches for the governing equations are employed: the Keller-box numerical simulations solving full nonlinear coupled system and a corresponding linearized system that is obtained under a far-field behavior and in the limit of large shear-to-strain-rate parameter (*λ*). From these two approaches, the flow phenomena reveals a rich structure of new family of solutions for various values of the magnetic number and *λ*. The various results for the wall stresses and the displacement thicknesses are presented along with some velocity profiles in both directions. The analysis discovered that the flow separation occurs in the secondary flow direction in the absence of magnetic field, and the flow separation disappears when the applied magnetic field is increased. The flow field is divided into a near-field (due to viscous forces) and far-field (due to mainstream flows), and the velocity profiles form because of an interaction between two regions. The magnetic field plays an important role in reducing the thickness of the boundary-layer. A physical explanation for all observed phenomena is discussed. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we establish global existence of strong solutions to the 3D incompressible two-fluid MHD equations with small initial data. In addition, the explicit convergence rate of strong solutions from the two-fluid MHD equations to the Hall-MHD equations is obtained as . Copyright © 2017 John Wiley & Sons, Ltd.

]]>We investigate the correctness of the initial boundary value problem of longitudinal impact on a piecewise-homogeneous semi-infinite bar consisting of a semi-infinite elastic part and finite length visco-elastic part whose hereditary properties are described by linear integral relations with an arbitrary difference kernel. Introducing nonstationary regularization in boundary conditions and in the contact conditions, the well-posedness of the considered problem is proved. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider a semi-discrete in time Crank–Nicolson scheme to discretize a weakly damped forced nonlinear fractional Schrödinger equation *u*_{t}−*i*(−Δ)^{α}*u*+*i*|*u*|^{2}*u*+*γ**u*=*f* for
considered in the the whole space
. We prove that such semi-discrete equation provides a discrete infinite-dimensional dynamical system in
that possesses a global attractor in
. We show also that if the external force is in a suitable weighted Lebesgue space, then this global attractor has a finite fractal dimension. Copyright © 2017 John Wiley & Sons, Ltd.

In this work successive differentiation method is applied to solve highly nonlinear partial differential equations (PDEs) such as Benjamin–Bona–Mahony equation, Burger's equation, Fornberg–Whitham equation, and Gardner equation. To show the efficacy of this new technique, figures have been incorporated to compare exact solution and results of this method. Wave variable is used to convert the highly nonlinear PDE into ordinary differential equation with order reduction. Then successive differentiation method is utilized to obtain the numerical solution of considered PDEs in this paper. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper is devoted to the existence of positive solutions for a fourth-order impulsive boundary value problem with integral boundary conditions on time scales. Existence results of at least two and three positive solutions are established via the double fixed point theorem and six functionals fixed point theorem, respectively. Also, an example is given to illustrate the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

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

Global exponential stability for coupled neutral stochastic delayed systems on networks (CNSDSNs) is investigated in this paper. By means of combining the Razumikhin method with graph theory, some sufficient conditions that can be verified easily are derived to ensure the global exponential stability for CNSDSNs. Finally, a specific model of CNSDSNs is discussed, and numerical test manifests the effectiveness of the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The main purpose of this research is to evaluate the impact of Darcy–Forchheimer flow in an incompressible third-grade liquid through Cattaneo–Christov heat flux approach. The Cattaneo–Christov heat flux theory is adopted to govern the mathematical expression of energy, which involves the heat flux relaxation time chracteristics. Time-dependent thermal conductivity is accounted. The steady problem is reduced to ordinary differential equations via suitable transformation. Numerical solutions for the resulting flow expressions have been computed with the help of Euler's explicit technique. Impact of influential variables on the velocity, temperature and skin-friction coefficient have been demonstrated and discussed through graphs. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The key purpose of the present work is to constitute a numerical scheme based on *q*-homotopy analysis transform method to examine the fractional model of regularized long-wave equation. The regularized long-wave equation explains the shallow water waves and ion acoustic waves in plasma. The proposed technique is a mixture of *q*-homotopy analysis method, Laplace transform, and homotopy polynomials. The convergence analysis of the suggested scheme is verified. The scheme provides
and *n*-curves, which show that the range convergence of series solution is not a local point effects and elucidate that it is superior to homotopy analysis method and other analytical approaches. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, a time-fractional diffusion equation with singular source term is considered. The Caputo fractional derivative with order 0<*α*⩽1 is applied to the temporal variable. Under specific initial and boundary conditions, we find that the time-fractional diffusion equation presents quenching solution that is not globally well-defined as time goes to infinity. The quenching time is estimated by using the eigenfunction of linear fractional diffusion equation. Moreover, by implementing a finite difference scheme, we give some numerical simulations to demonstrate the theoretical analysis. Copyright © 2017 John Wiley & Sons, Ltd.

In earlier literature, a version of a classical three-species food chain model, with modified Holling type IV functional response, is proposed. Results on the global boundedness of solutions to the model system under certain parametric restrictions are derived, and chaotic dynamics is shown. We prove that in fact the model possesses explosive instability, and solutions can explode/blow up in finite time, for certain initial conditions, even under the parametric restrictions of the literature. Furthermore, we derive the Hopf bifurcation criterion, route to chaos, and Turing bifurcation in case of the spatially explicit model. Lastly, we propose, analyze, and simulate a version of the model, incorporating gestation effect, via an appropriate time delay. The delayed model is shown to possess globally bounded solutions, for any initial condition. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider a semilinear wave equation with nonlinear damping in the whole space . Local-in-time existence and uniqueness results are obtained in the class of Bessel-potential spaces . Copyright © 2017 John Wiley & Sons, Ltd.

]]>Ren and Zeng (2013) introduced a new kind of *q*-Bernstein–Schurer operators and studied some approximation properties. Acu *et al*. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using *q*-Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's *K*-functional. Next, we introduce the bivariate case of *q*-Bernstein–Schurer–Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's *K*-functional. Finally, we define the generalized Boolean sum operators of the *q*-Bernstein–Schurer–Kantorovich type and investigate the approximation of the Bögel continuous and Bögel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright © 2017 John Wiley & Sons, Ltd.

This paper presents an operatorial model based on fermionic operators for the description of the dynamics of political parties affected by turncoat-like behaviors. By observing the political landscape in place in Italy over the last years, appropriate macro-groups have been identified on the basis of the behavior of politicians in terms of disloyal attitude as well as openness towards accepting chameleons from other parties. Once introduced, a time-dependent number-like operator for each physical observable relevant for the description of the political environment, the analysis of the party system dynamics is carried out by combining the action of a quadratic Hamiltonian operator with certain rules acting periodically on the system in such a way that the parameters entering the model are repeatedly changed so as to express a sort of dependence of them upon the variations of the mean values of the observables. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this article, we discuss the integral representation of quaternionic harmonic functions in the half space with the general boundary condition. Next, we derive a lower bound from an upper one for quaternionic harmonic functions. These results generalize some of the classic results from the case of plane to the case of noncommutative quaterninionic half space. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, multi-switching combination–combination synchronization scheme has been investigated between a class of four non-identical fractional-order chaotic systems. The fractional-order Lorenz and Chen's systems are taken as drive systems. The combination–combination of multi drive systems is then synchronized with the combination of fractional-order Lü and Rössler chaotic systems. In multi-switching combination–combination synchronization, the state variables of two drive systems synchronize with different state variables of two response systems simultaneously. Based on the stability of fractional-order chaotic systems, the multi-switching combination–combination synchronization of four fractional-order non-identical systems has been investigated. For the synchronization of four non-identical fractional-order chaotic systems, suitable controllers have been designed. Theoretical analysis and numerical results are presented to demonstrate the validity and feasibility of the applied method. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this work, we integrate both density-dependent diffusion process and Beddington–DeAngelis functional response into virus infection models to consider their combined effects on viral infection and its control. We perform global analysis by constructing Lyapunov functions and prove that the system is well posed. We investigated the viral dynamics for scenarios of single-strain and multi-strain viruses and find that, for the multi-strain model, if the basic reproduction number for all viral strains is greater than 1, then each strain persists in the host. Our investigation indicates that treating a patient using only a single type of therapy may cause competitive exclusion, which is disadvantageous to the patient's health. For patients infected with several viral strains, the combination of several therapies is a better choice. Copyright © 2017 John Wiley & Sons, Ltd.

]]>A five-dimensional ordinary differential equation model describing the transmission of *Toxoplamosis gondii* disease between human and cat populations is studied in this paper. Self-diffusion modeling the spatial dynamics of the *T. gondii* disease is incorporated in the ordinary differential equation model. The normalized version of both models where the unknown functions are the proportions of the susceptible, infected, and controlled individuals in the total population are analyzed. The main results presented herein are that the ODE model undergoes a trans-critical bifurcation, the system has no periodic orbits inside the positive octant, and the endemic equilibrium is globally asymptotically stable when we restrict the model to inside of the first octant. Furthermore, a local linear stability analysis for the spatially homogeneous equilibrium points of the reaction diffusion model is carried out, and the global stability of both the disease-free and endemic equilibria are established for the reaction–diffusion system when restricted to inside of the first octant. Finally, numerical simulations are provided to support our theoretical results and to predict some scenarios about the spread of the disease. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we establish a new blowup criterions for the strong solution to the Dirichlet problem of the three-dimensional compressible MHD system with vacuum. Specifically, we obtain the blowup criterion in terms of the concentration of density in *B**M**O* norm or the concentration of the integrability of the magnetic field at the first singular time. The BMO-type estimate for the Lam
system and a variant of the Brezis-Waigner's inequality play a critical role in the proof. Copyright © 2017 John Wiley & Sons, Ltd.

This paper is devoted to establish Bielecki–Ulam–Hyers–Rassias stability, generalized Bielecki–Ulam–Hyers–Rassias stability, and Bielecki–Ulam–Hyers stability on a compact interval [0,*T*], for a class of higher-order nonlinear differential equations with fractional integrable impulses. The phrase ‘fractional integrable’ brings one to fractional calculus. Hence, applying usual methods for analysis offers many difficulties in proving the results of existence and uniqueness of solution and stability theorems. Picard operator is applied in showing existence and uniqueness of solution. Stability results are obtained by using the tools of fractional calculus and Hölder's inequality of integration. Along with tools of fractional calculus, Bielecki's normed Banach spaces are considered, which made the results more interesting. Copyright © 2017 John Wiley & Sons, Ltd.

In this work, we study the approximation of traveling wave solutions propagated at minumum speeds *c*_{0}(*h*) of the delayed Nicholson's blowflies equation:

In order to do that, we construct a subsolution and a super solution to (∗). Also, through that construction, an alternative proof of the existence of traveling waves moving at minimum speed is given. Our basic hypothesis is that *p*/*δ*∈(1,*e*] and then, the monostability of the reaction term. Copyright © 2017 John Wiley & Sons, Ltd.

The Darboux transformation method with 4×4 spectral problem has more complexity than 2×2 and 3×3 spectral problems. In this paper, we start from a new discrete spectral problem with a 4×4 Lax pairs and construct a lattice hierarchy by properly choosing an auxiliary spectral problem, which can be reduced to a new discrete soliton hierarchy. For the obtained lattice integrable coupling equation, we establish a Darboux transformation and apply the gauge transformation to a specific equation and then the explicit solutions of the lattice integrable coupling equation are obtained. Copyright © 2017 John Wiley & Sons, Ltd.

]]>No abstract is available for this article.

]]>We consider a Euler–Bernoulli beam equation with a boundary control condition of fractional derivative type. We study stability of the system using the semigroup theory of linear operators and a result obtained by Borichev and Tomilov. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper concerns new continuum phenomenological model for epitaxial thin-film growth with three different forms of the Ehrlich–Schwoebel current. Two of these forms were first proposed by Politi and Villain 1996 and then studied by Evans, Thiel, and Bartelt 2006. The other one is completely new. Energy structure and properties of the new model are studied. Following the techniques used in Li and Liu 2003, we present rigorous analysis of the well-posedness, regularity, and time stability for the new model. We also studied both the global and the local behavior of the surface roughness in the growth process. By using a convex–concave time-splitting scheme, one can naturally build unconditionally stable semi-implicit numerical discretizations with linear implicit parts, which is much easier to implement than conventional models requiring nonlinear implicit parts. Despite this fundamental difference in the model, numerical experiments show that the nonlinear morphological instability of the new model agrees well with results of other models published before which indicates that the new model correctly captures the essential morphological states in the thin-film growth process. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, our main aim is to establish some new fractional integral inequalities involving Hadamard-type *k*-fractional integral operators recently given by Mubeen *et al.* Furthermore, the paper discusses some of their relevance with known results. Copyright © 2017 John Wiley & Sons, Ltd.

The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot both be sharply localized. It plays an important role in signal processing and physics. This paper generalizes the uncertainty principle for measurable sets from complex domain to hypercomplex domain using quaternion algebras, associated with the quaternion Fourier transform. The performance is then evaluated in signal recovery problems where there is an interplay of missing and time-limiting data. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper is in continuation of the work performed by Kajla *et al*. (Applied Mathematics and Computation 2016; 275 : 372–385.) wherein the authors introduced a bivariate extension of *q*-Bernstein–Schurer–Durrmeyer operators and studied the rate of convergence with the aid of the Lipschitz class function and the modulus of continuity. Here, we estimate the rate of convergence of these operators by means of Peetre's *K*-functional. Then, the associated generalized Boolean sum operator of the *q*-Bernstein–Schurer–Durrmeyer type is defined and discussed. The smoothness properties of these operators are improved with the help of mixed *K*-functional. Furthermore, we show the convergence of the bivariate Durrmeyer-type operators and the associated generalized Boolean sum operators to certain functions by illustrative graphics using Maple algorithm. Copyright © 2017 John Wiley & Sons, Ltd.

Time-dependent PDEs with fractional Laplacian ( − Δ)^{α} play a fundamental role in many fields and approximating ( − Δ)^{α} usually leads to ODEs' system like **u**^{′}(*t*) + *A***u**(*t*) = **g**(*t*) with *A* = *Q*^{α}, where
is a sparse symmetric positive definite matrix and *α* > 0 denotes the fractional order. The parareal algorithm is an ideal solver for this kind of problems, which is iterative and is characterized by two propagators
and
. The propagators
and
are respectively associated with large step size Δ*T* and small step size Δ*t*, where Δ*T* = *J*Δ*t* and *J*⩾2 is an integer. If we fix the
-propagator to the Implicit-Euler method and choose for
some proper Runge–Kutta (RK) methods, such as the second-order and third-order singly diagonally implicit RK methods, previous studies show that the convergence factors of the corresponding parareal solvers can satisfy
and
, where *σ*(*A*) is the spectrum of the matrix *A*. In this paper, we show that by choosing these two RK methods as the
-propagator, the convergence factors can reach
, provided the one-stage complex Rosenbrock method is used as the
-propagator. If we choose for both
and
, the complex Rosenbrock method, we show that the convergence factor of the resulting parareal solver can also reach
. Numerical results are given to support our theoretical conclusions. Copyright © 2017 John Wiley & Sons, Ltd.

In this work, we study the integrability aspects of the Schamel–Korteweg–de Vries equation that play an important role in studying the effect of electron trapping on the nonlinear interaction of ion-acoustic waves by including a quasi-potential. Lie symmetry analysis together with the simplest equation method and Kudryashov method is used to obtain exact traveling wave solutions for this equation. In addition, conservation laws are constructed using two different techniques, namely, the multiplier method and the new conservation theorem. Using the conservation laws and symmetries of the underlying equation, double reduction and exact solution were also constructed. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper contains results on well-posedness, stability, and long-time behavior of solutions to a class of plate models subject to damping and source terms given by the product of two nonlinear components [EQUATION1] where Ω is a bounded open set of **R**^{n} with smooth boundary, *γ*,*ρ*⩾0 and
are nonlocal functions. The main result states that the dynamical system {*S*(*t*)}_{t⩾0} associated with this problem has a compact global attractor. In addition, in the limit case *γ* = 0, it is also shown that {*S*(*t*)}_{t⩾0} has a finite dimensional global attractor by using an approach on quasi-stability because of Chueshov–Lasiecka (2010). Copyright © 2017 John Wiley & Sons, Ltd.

We investigate a linear, fully coupled thermoelasticity problem for a highly heterogeneous, two-phase medium. The medium in question consists of a connected matrix with disconnected, initially periodically distributed inclusions separated by a sharp interface undergoing an *a priori* known interface movement because of phase transformations. After transforming the moving geometry to an *ϵ*-periodic, fixed reference domain, we establish the well-posedness of the model and derive a number of *ϵ*-independent *a priori* estimates. Via a two-scale convergence argument, we then show that the *ϵ*-dependent solutions converge to solutions of a corresponding upscaled model with distributed time-dependent microstructures. Copyright © 2017 John Wiley & Sons, Ltd.

For graph domains without cycles, we show how unknown coefficients and source terms for a parabolic equation can be recovered from the dynamical Neumann-to-Dirichlet map associated with the boundary vertices. Through use of a companion wave equation problem, the topology of the tree graph, degree of the vertices, and edge lengths can also be recovered. The motivation for this work comes from a neuronal cable equation defined on the neuron's dendritic tree, and the inverse problem concerns parameter identification of *k* unknown distributed conductance parameters. Copyright © 2017 John Wiley & Sons, Ltd.

Starting form basic principles, we obtain mathematical models that describe the traffic of material objects in a network represented by a graph. We analyze existence, uniqueness, and positivity of solutions for some implicit models. Also, some linear models and their equilibria are analyzed. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, the diffusion is introduced to an immunosuppressive infection model with delayed antiviral immune response. The direction and stability of Hopf bifurcation are effected by time delay, in the absence of which the positive equilibrium is locally asymptotically stable by means of analyzing eigenvalue spectrum; however, when the time delay increases beyond a threshold, the positive equilibrium loses its stability via the Hopf bifurcation. The stability and direction of the Hopf bifurcation is investigated with the norm form and the center manifold theory. The stability of the Hopf bifurcation leads to the emergence of spatial spiral patterns. Numerical calculations are performed to illustrate our theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper is concerned with the exponential stability for the discrete-time bidirectional associative memory neural networks with time-varying delays. Based on Lyapunov stability theory, some novel delay-dependent sufficient conditions are obtained to guarantee the globally exponential stability of the addressed neural networks. In order to obtain less conservative results, an improved Lyapunov–Krasovskii functional is constructed and the reciprocally convex approach and free-weighting matrix method are employed to give the upper bound of the difference of the Lyapunov–Krasovskii functional. Several numerical examples are provided to illustrate the effectiveness of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study the *n*th-order nonlinear dynamic equation with Laplacians and a deviating argument

on an above-unbounded time scale, where *n*⩾2,

New oscillation criteria are established for the cases when *n* is even and odd and when *α* > *γ*,*α* = *γ*, and *α* < *γ*, respectively, with *α* = *α*_{1}⋯*α*_{n − 1}. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider a Riesz–Feller space-fractional backward diffusion problem with a time-dependent coefficient

We show that this problem is ill-posed; therefore, we propose a convolution regularization method to solve it. New error estimates for the regularized solution are given under *a priori* and *a posteriori* parameter choice rules, respectively. Copyright © 2016 John Wiley & Sons, Ltd.

Based on the extended extragradient-like method and the linesearch technique, we propose three projection methods for finding a common solution of a finite family of equilibrium problems. The linesearch used in the proposed algorithms has allowed to reduce some conditions imposed on equilibrium bifunctions. The strongly convergent theorems are established without the Lipschitz-type condition of bifunctions. The paper also helps in the design and analysis of practical algorithms and gives us a generalization of some previously known problems. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The paper is devoted to study the stability of nonlinear fractional order difference systems by their linear approximation. Additionally, we show the relation between the stability of linear fractional order differential systems and their discretizations. Copyright © 2016 John Wiley & Sons, Ltd.

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

]]>In many seemingly diverse areas of applications, reduction, summation, and transformation formulas for various families of hypergeometric functions in one, two, and more variables are potentially useful, especially in situations when these hypergeometric functions are involved in solutions of mathematical, physical, and engineering problems that can be modeled by (for example) ordinary and partial differential equations. The main object of this article is to investigate a number of reductions and transformations for the Appell functions *F*_{1},*F*_{2},*F*_{3}, and *F*_{4} in two variables and the corresponding (substantially more general) double-series identities. In particular, we observe that a certain reduction formula for the Appell function *F*_{3} derived recently by Prajapati *et al.*, together with other related results, were obtained more than four decades earlier by Srivastava. We give a new simple derivation of the previously mentioned Srivastava's formula . We also present a brief account of several other related results that are closely associated with the Appell and other higher-order hypergeometric functions in two variables. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, stability and bifurcation of a two-dimensional ratio-dependence predator–prey model has been studied in the close first quadrant
. It is proved that the model undergoes a period-doubling bifurcation in a small neighborhood of a boundary equilibrium and moreover, Neimark–Sacker bifurcation occurs at a unique positive equilibrium. We study the Neimark–Sacker bifurcation at unique positive equilibrium by choosing *b* as a bifurcation parameter. Some numerical simulations are presented to illustrate theocratical results. Copyright © 2017 John Wiley & Sons, Ltd.

Starting from the representation of the (*n* − 1) + *n* − dimensional Lorentz pseudo-sphere on the projective space
, we propose a method to derive a class of solutions underlying to a Dirac–Kähler type equation on the lattice. We make use of the Cayley transform
to show that the resulting group representation arises from the same mathematical framework as the conformal group representation in terms of the *general linear group*
. That allows us to describe such class of solutions as a commutative *n* − ary product, involving the quasi-monomials
) with membership in the paravector space
. Copyright © 2016 John Wiley & Sons, Ltd.

We establish a two-wave mode equation for the integrable Kadomtsev–Petviashvili equation, which describes the propagation of two different wave modes in the same direction simultaneously. We determine the necessary conditions that make multiple soliton solutions exist for this new equation. The simplified Hirota's method will be used to conduct this work. We also use other techniques to obtain other set of periodic and singular solutions for the two-mode Kadomtsev-Petviashvili equation. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, the approximate analytical solutions of Lotka–Volterra model with fractional derivative have been obtained by using hybrid analytic approach. This approach is amalgamation of homotopy analysis method, Laplace transform, and homotopy polynomials. First, we present an alternative framework of the method that can be used simply and effectively to handle nonlinear problems arising in several physical phenomena. Then, existence and uniqueness of solutions for the fractional Lotka–Volterra equations are discussed. We also carry out a detailed analysis on the stability of equilibrium. Further, we have derived the approximate solutions of predator and prey populations for different particular cases by using initial values. The numerical simulations of the result are depicted through different graphical representations showing that this hybrid analytic method is reliable and powerful method to solve linear and nonlinear fractional models arising in science and engineering. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, blow-up property to a system of nonlinear stochastic PDEs driven by two-dimensional Brownian motions is investigated. The lower and upper bounds for blow-up times are obtained. When the system parameters satisfy certain conditions, the explicit solutions of a related system of random PDEs are deduced, which allows us to use Yor's formula to obtain the distribution functions of several blow-up times. Particularly, the impact of noises on the life span of solutions is studied as the system parameters satisfy different conditions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we apply the family of potential wells to the initial boundary value problem of semilinear hyperbolic equations on the cone Sobolev spaces. We not only give some results of global existence and nonexistence of solutions but also obtain the vacuum isolating of solutions. Finally, we show blow-up in finite time of solutions on a manifold with conical singularities. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this report, we give a semi-discrete defect correction finite element method for the unsteady incompressible magnetohydrodynamics equations. The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Firstly, the nonlinear magnetohydrodynamics equations is solved with an artificial viscosity term. Then, the numerical solutions are improved on the same grid by a linearized defect-correction technique. Then, we give the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. In order to show the effect of our method, some numerical results are shown. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we aim to investigate the exponential stability of general hybrid stochastic functional differential systems with delayed impulses. By using the average impulsive interval and the Lyapunov function method, we derive some sufficient conditions for exponential stability, which are less conservative than those existing results based on the supremum or infimum of impulsive interval and more convenient to be applied than those Razumikhin-type conditions in the literature. Meanwhile, we show that unstable hybrid stochastic delay differential systems, both linear and nonlinear, can be stabilized by suitably impulsive sequence. Finally, two examples are discussed to illustrate the effectiveness and advantages of the obtained results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this work, we analyze the existence, uniqueness, and asymptotic behavior of solution to the model of a thermoelastic mixture of type III. We establish sufficient conditions to guarantee the exponential decay of solutions. When the decay is not of exponential type, we prove that the solutions decay polynomially and we find the optimal polynomial decay rate. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the effect of delay on the asymptotic behavior of Nicholson's blowflies model with patch structure and multiple time-varying delays. By using the fluctuation lemma and some differential inequality technique, delay-dependent criteria are obtained for the global attractivity of the addressed system. Meanwhile, some numerical examples are given to illustrate the feasibility of the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

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