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.

]]>A novel second-order two-scale (SOTS) analysis method is developed for predicting the transient heat conduction performance of porous materials with periodic configurations in curvilinear coordinates. Under proper coordinate transformations, some non-periodic porous structures in Cartesian coordinates can be transformed into periodic structures in general curvilinear coordinates, which is our particular interest in this study. The SOTS asymptotic expansion formulas for computing the temperature field of transient heat conduction problem in curvilinear coordinates are constructed, some coordinate transformations are discussed, and the related SOTS formulas are given. The feature of this asymptotic model is that each of the cell functions defined in the periodic cell domain is associated with the macroscopic coordinates and the homogenized material coefficients varies continuously in the macroscopic domain behaving like the functional gradient material. Finally, the corresponding SOTS finite element algorithms are brought forward, and some numerical examples are given in detail. The numerical results demonstrate that the SOTS method proposed in this paper is valid to predict transient heat conduction performance of porous materials with periodicity in curvilinear coordinates. By proper coordinate transformations, the SOTS asymptotic analysis method can be extended to more general non-periodic porous structures in Cartesian coordinates. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We propose a new finite volume scheme for 2D anisotropic diffusion problems on general unstructured meshes. The main feature lies in the introduction of two auxiliary unknowns on each cell edge, and then the scheme has both cell-centered primary unknowns and cell edge-based auxiliary unknowns. The auxiliary unknowns are interpolated by the multipoint flux approximation technique, which reduces the scheme to a completely cell-centered one. The derivation of the scheme satisfies the linearity-preserving criterion that requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is then called as a cell edge-based linearity-preserving scheme. The optimal convergence rates are numerically obtained on unstructured grids in case that the diffusion tensor is taken to be anisotropic and/or discontinuous. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The purpose of this paper is to introduce a family of *q*-Szász–Mirakjan–Kantorovich type positive linear operators that are generated by Dunkl's generalization of the exponential function. We present approximation properties with the help of well-known Korovkin's theorem and determine the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's *K*-functional, and the second-order modulus of continuity. Furthermore, we obtain the approximation results for bivariate *q*-Szász–Mirakjan–Kantorovich type operators that are also generated by the aforementioned Dunkl generalization of the exponential function. Copyright © 2017 John Wiley & Sons, Ltd.

We discuss bound and anti-bound states for 2×2 matrix Schrödinger operator. We analyze the Fredholm determinant for Hamiltonians that can be represented in a multi-channel framework. Our analysis covers the whole and the half-line problems. We obtain some results on counting anti-bound states between successive bound states. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, by applying nonstandard finite difference scheme, we propose a discrete multigroup Susceptible-Infective-Removed (SIR) model with nonlinear incidence rate. Using Lyapunov functions, it is shown that the global dynamics of this model are completely determined by the basic reproduction number . If , then the disease-free equilibrium is globally asymptotically stable; if , then there exists a unique endemic equilibrium and it is globally asymptotically stable. Example and numerical simulations are presented to illustrate the results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we consider global nonexistence of a solution for coupled quasilinear system with damping and source under Dirichlet boundary condition. We obtain a global nonexistence result of solution by using the perturbed energy method, where the initial energy is assumed to be positive. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We prove that the existence of peakon as weak traveling wave solution and as global weak solution for the nonlinear surface wind waves equation, so as to correct the assertion that there exists no peakon solution for such an equation in the literature. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the mean squared derivative cost functions that arise in various applications such as in motor control, biometrics and optimal transport theory. We provide qualitative properties, explicit analytical formulas and computational algorithms for the cost functions. We also perform numerical simulations to illustrate the analytical results. In addition, as a by-product of our analysis, we obtain an explicit formula for the inverse of a Wronskian matrix that is of independent interest in linear algebra and differential equations theory. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this article, we establish sufficient conditions for the regularity of solutions of 3D MHD equations in the framework of the anisotropic Lebesgue spaces. In particular, we obtain the anisotropic regularity criterion via partial derivatives, and it is a generalization of the some previous results. Besides, the anisotropic integrability regularity criteria in terms of the magnetic field and the third component of the velocity field are also investigated. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In the paper, we study some ‘a priori’ properties of mild solutions to a single reaction–diffusion equation with discontinuous nonlinear reaction term on the two-dimensional sphere close to its poles. This equation is the counterpart of the well-studied bistable reaction–diffusion equation on the Euclidean plane. The investigation of this equation on the sphere is mainly motivated by the phenomenon of the fertilization of oocytes or recent studies of wave propagation in a model of immune cells activation, in which the cell is modeled by a ball. Because of the discontinuous nature of reaction kinetics, the standard theory cannot guarantee the solution existence and its smoothness properties. Moreover, the singular nature of the diffusion operator near the north/south poles makes the analysis more involved. Unlike the case in the Euclidean plane, the (axially symmetric) Green's function for the heat operator on the sphere can only be represented by an infinite series of the Legendre polynomials. Our approach is to consider a formal series in Legendre polynomials obtained by assuming that the mild solution exists. We show that the solution to the equation subject to the Neumann boundary condition is *C*^{1} smooth in the spatial variable up to the north/south poles and Hölder continuous with respect to the time variable. Our results provide also a sort of ‘a priori’ estimates, which can be used in the existence proofs of mild solutions, for example, by means of the iterative methods. Copyright © 2017 John Wiley & Sons, Ltd.

This paper considers the two-dimensional Riemann problem for a system of conservation laws that models the polymer flooding in an oil reservoir. The initial data are two different constant states separated by a smooth curve. By virtue of a nonlinear coordinate transformation, this problem is converted into another simple one. We then analyze rigorously the expressions of elementary waves. Based on these preparations, we obtain respectively four kinds of non-selfsimilar global solutions and their corresponding criteria. It is shown that the intermediate state between two elementary waves is no longer a constant state and that the expression of the rarefaction wave is obtained by constructing an inverse function. These are distinctive features of the non-selfsimilar global solutions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study the dynamics of a Nicholson's blowflies equation with state-dependent delay. For the constant delay, it is known that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and global existence of periodic solutions has been established. Here, we consider the state-dependent delay instead of the constant delay and generalize the results on the existence of slowly oscillating periodic solutions under a set of mild conditions on the parameters and the delay function. In particular, when the positive equilibrium gets unstable, a global unstable manifold connects the positive equilibrium to a slowly oscillating periodic orbit. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Cancer stem cells are responsible for tumor survival and resurgence and are thus essential in developing novel therapeutic strategies against cancer. Mathematical models can help understand cancer stem and differentiated cell interaction in tumor growth, thus having the potential to help in designing experiments to develop novel therapeutic strategies against cancer. In this paper, by using theory of functional and ordinary differential equations, we study the existence and stability of nonlinear growth kinetics of breast cancer stem cells. First, we provide a sufficient condition for the existence and uniqueness of the solution for nonlinear growth kinetics of breast cancer stem cells. Then we study the uniform asymptotic stability of the zero solution. By using linearization techniques, we also provide a criteria for uniform asymptotic stability of a nontrivial steady-state solution with and without time delays. We present a theorem from complex analysis that gives certain conditions that allow for this criteria to be satisfied. Next, we apply these theorems to a special case of the system of functional differential equations that has been used to model nonlinear growth kinetics of breast cancer stem cells. The theoretical results are further justified by numerical testing examples. Consistent with the theories, our numerical examples show that the time delays can disrupt the stability. All the results can be easily extended to study more general cell lineage models. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We investigate the qualitative behavior of a host-parasitoid model with a strong Allee effect on the host. More precisely, we discuss the boundedness, existence and uniqueness of positive equilibrium, local asymptotic stability of positive equilibrium and existence of Neimark–Sacker bifurcation for the given system by using bifurcation theory. In order to control Neimark–Sacker bifurcation, we apply pole-placement technique that is a modification of OGY method. Moreover, the hybrid control methodology is implemented in order to control Neimark–Sacker bifurcation. Numerical simulations are provided to illustrate theoretical discussion. Copyright © 2017 John Wiley & Sons, Ltd.

]]>No abstract is available for this article.

]]>In this article, we discuss the modelling of elastic and electromagnetic wave propagation through one-dimensional and two-dimensional structured piezoelectric solids. Dispersion and the effect of piezoelectricity on the group velocity and positions of stop bands are studied in detail. We will also analyse the reflection and transmission associated with the problem of scattering of an elastic wave by a heterogeneous piezoelectric stack. Special attention is given to the occurrence of transmission resonances in finite stacks and their dependence on a piezoelectric effect. A two-dimensional doubly periodic piezoelectric checkerboard structure is subsequently introduced, for which the dispersion surfaces for Bloch waves have been constructed and analysed, with the emphasis on the dynamic anisotropy and special features of standing waves within the piezoelectric structure. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The behaviour of the anti-plane local fields of a two-phase piezo-composite made of a fibre embedded in an infinite matrix, and in perfect contact with it, is studied. Both constituents belong to the 622 crystal symmetry class. A solution for the case of a far-field in-plane electrical load and a far-field anti-plane mechanical load is presented. The limit cases are analysed, whereas the results validation is made through comparisons with semi-analytical results derived from the application of complex variable methods to the solution of the local problems that arise from the application of the asymptotic homogenization method to the related problem of periodic distributions of fibres with small cross-section area. These results could be useful in bone mechanics applications. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this work, three-phase piezoelectric fibrous composites distributed in a parallelepiped cell is studied. The statement of the mathematical problems and the formulation of the local problems by means of the asymptotic homogenization method are presented. Closed-form formulae are obtained for the effective properties of the composites for different configurations of the cells. The present method for thick and thin mesophases can provide a point of reference for comparisons with other numerical and approximate methods. Copyright © 2016 John Wiley & Sons, Ltd.

]]>An analysis of the static behavior using the self-consistent effective medium method for a composite with randomly aligned spheroidal inclusions embedded in a matrix is studied. The constituents of which may have piezoelectric properties. Based on the self-consistent method proposed by F.J. Sabina and V. Levin an analysis of the static overall properties is developed for piezoelectric composites. The electroelastic Green's functions by Willis's approach is developed and the spheroidal shape inclusion is studied with the help of it. In that sense, the influence of various geometrical forms of the inclusions is analyzed. Comparisons with other self-consistent methods, finite element calculations, two scale asymptotic homogenization method and with experimental data show good results. Copyright © 2016 John Wiley & Sons, Ltd.

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

]]>Determination of effective physical parameters of gas-filled porous media is of practical interest to a wide spectrum of applications from construction industry to petrophysics. The classical equations of thermal conductivity are not applicable for sufficiently small pores, the characteristic size of which is comparable with the mean free path of gas molecules. In this case, for description of the gas behavior, it is necessary to use the methods of physical kinetics and rarefied gas dynamics. In this work, we have considered so-called slip flow regime, 0 < Kn < 1, where Kn =*λ*/*R* is the Knudsen number, *λ* is the mean free path, and *R* is the characteristic pore size. In this regime, we can use the classical equations of mechanics of continuum media with modified boundary conditions that take into account the temperature and energy flux jumps on the pore boundaries. We have solved the so-called one-particle problem of thermal polarization of gas-filled inclusion. The obtained solution was then used in some self-consistent homogenization methods to calculate the effective conductivity coefficient. Our calculations have shown that the effective conductivity depends substantially on the pore size (the Knudsen number). The obtained results demonstrate the feasibility of pore sizes determination by using measurements of effective thermal conductivity. Copyright © 2015 John Wiley & Sons, Ltd.

A fibrous elastic composite is considered with transversely isotropic constituents. Three types of fibers are studied: circular, square, and rhombic. Fibers are distributed with the same periodicity along the two perpendicular directions to the fiber orientation, that is, the periodic cell of the composite is square. The composite exhibits imperfect contact at the interface between the fiber and matrix. Effective properties of this composite are calculated by means of a semi-analytic method, that is, the differential equations that described the local problems obtained by asymptotic homogenization method are solved using the finite element method. The finite element formulation can be applied to any type of element; particularly, three approaches are used: quadrilateral element of 4 nodes, quadrilateral element of 8 nodes, and quadrilateral element of 12 nodes. Numerical computations are implemented, and different comparisons are presented. Copyright © 2016 John Wiley & Sons, Ltd.

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

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

]]>In the present paper, an asymptotic model is constructed for the short-time deformation of an articular cartilage layer modeled as transversely isotropic, transversely homogeneous biphasic material. It is assumed that the layer thickness is relatively small compared with the characteristic size of the normal surface load applied to the upper surface of the cartilage layer, while the bottom surface is assumed to be firmly attached to a rigid impermeable substrate. In view of applications to articular contact problems, it is assumed that the interstitial fluid is not allowed to escape through the articular surface. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Within elastic solids subjected to illumination from uncorrelated sources, as those that arise from multiple scattering, it has been established that the displacement field has intensities that are similar to diffusion-like field. It is found that in this case, the average correlation of motions in the frequency domain, between two points, is proportional to the imaginary part of Green's function for those two receivers. For a single station, the average auto-correlation equals the average power spectrum, and this gives the imaginary part of Green's function at the source. To gain insight on the properties of Green's functions, particularly regarding their connection with diffuse fields, we study some of their characteristics for simplified cases. Specifically, we deal with 2D and 3D acoustic layers with various boundary conditions. In practice, we assume these Green's functions are related with a diffuse field, and we explore the analytical consequences. The aim of this study is to gather insight to understand patterns found when studying real data or to have a guide to interpret their trends. Copyright © 2016 John Wiley & Sons, Ltd.

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

]]>A typical defect of numerical solutions obtained for isothermal heavily loaded elastohydrodynamically lubricated (EHL) contacts such as solution instability is considered. A detailed analysis and numerical examples of solutions for dry and lubricated contacts with instability are presented. The main roots of numerical instability are discussed, and a regularization process, that is, the process that makes the numerical solution stable, is revealed and analyzed. A regularization of EHL problems slightly modifies the problem equations to make their numerical solution stable and convergent. The level of this problem modification is controlled by just one parameter. The effect of the level of regularization on parameters of EHL contacts is considered. The idea of such a regularization comes from the solution of a thermal EHL problem with low generation of energy in the contact. Some examples of regularized numerical solutions for classic EHL problem and an EHL problem that takes into account tangential surface displacements caused by frictional stresses are presented. The approach showed its effectiveness and application simplicity for very high values of pressure viscosity coefficients. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider periodic waveguides in the shape of a inhomogeneous string or beam partially supported by a uniform elastic Winkler foundation. A multi-parametric analysis is developed to take into account the presence of internal cutoff frequencies and strong contrast of the problem parameters. This leads to asymptotic conditions supporting non-typical quasi-static uniform or, possibly, linear microscale displacement variations over the high-frequency domain. Macroscale governing equations are derived within the framework of the Floquet–Bloch theory as well as using a high-frequency-type homogenization procedure adjusted to a string with variable parameters. It is found that, for the string problem, the associated macroscale equation is the same as that applying to a string resting on a Winkler foundation. Remarkably, for the beam problem, the macroscale behavior is governed by the same equation as for a beam supported by a two-parameter Pasternak foundation. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study the problem of recovering a scatterer object boundary by measuring the acoustic far field using Bayesian inference. This is the inverse acoustic scattering problem, and Bayesian inference is used to quantify the uncertainty on the unknowns (e.g., boundary shape and position). Aiming at sampling efficiently from the arising posterior probability distribution, we introduce a probability transition kernel (sampler) that is invariant under affine transformations of space. The sampling is carried out over a cloud of control points used to interpolate candidate boundary solutions. We demonstrate the performance of our method through a classical problem. Copyright © 2016 John Wiley & Sons, Ltd.

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