In this paper, we introduce new modifications of Szász–Mirakyan operators based on (*p*,*q*)-integers. We first give a recurrence relation for the moments of new operators and present explicit formula for the moments and central moments up to order 4. Some approximation properties of new operators are explored: the uniform convergence over bounded and unbounded intervals is established, direct approximation properties of the operators in terms of the moduli of smoothness is obtained and Voronovskaya theorem is presented. For the particular case *p* = 1, the previous results for *q*-Sz ász–Mirakyan operators are captured. Copyright © 2015 John Wiley & Sons, Ltd.

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

]]>In this paper, it is shown how to change the integration basis in some Gaussian (weighted) quadrature rules in order to obtain new quadrature models and improve classical results in the sequel. The main advantage of this approach is its simplicity, which can be implemented in any numerical integration package. Several remarkable numerical evidences are then given to show the advantage and efficiency of the proposed approach with respect to classical methods. Copyright © 2015 John Wiley & Sons, Ltd.

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

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

In epidemiology, an epidemic is defined as the spread of an infectious disease to a large number of people in a given population within a short period of time. In the marketing context, a message is viral when it is broadly sent and received by the target market through person-to-person transmission. This specific marketing communication strategy is commonly referred as viral marketing. Because of this similarity between an epidemic and the viral marketing process and because the understanding of the critical factors to this communications strategy effectiveness remain largely unknown, the mathematical models in epidemiology are presented in this marketing specific field. In this paper, an epidemiological model susceptible-infected-recovered to study the effects of a viral marketing strategy is presented. It is made a comparison between the disease parameters and the marketing application, and MATLAB simulations are performed. Finally, some conclusions are carried out and their marketing implications are exposed: interactions across the parameters suggest some recommendations to marketers, as the profitability of the investment or the need to improve the targeting criteria of the communications campaigns. Copyright © 2015 John Wiley & Sons, Ltd.

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

]]>In this paper, the fully coupled theory of elasticity for solids with double porosity is considered. The explicit solutions of the basic boundary value problems (BVPs) in the fully coupled linear equilibrium theory of elasticity for the space with double porosity and spherical cavity are constructed. The solutions of these BVPs are represented by means of absolutely and uniformly convergent series. Copyright © 2015 John Wiley & Sons, Ltd.

]]>A new ordinary differential inequality without global solutions is proposed. Comparison with similar differential inequalities in the well-known concavity method is performed. As an application, finite time blow up of the solutions to nonlinear Klein–Gordon equation and generalized Boussinesq equation is proven. The initial energy is arbitrary high positive. The structural conditions on the initial data generalize the assumptions used in the literature for the time being. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We have found the motivation for this paper in the research of a quantized closed Friedmann cosmological model. There, the second-order linear ordinary differential equation emerges as a wave equation for the physical state functions. Studying the polynomial solutions of this equation, we define a new functional product in the space of real polynomials. This product includes the indexed weight functions which depend on the degrees of participating polynomials. Although it does not have all of the properties of an inner product, a unique sequence of polynomials can be associated with it by an additional condition. In the special case presented here, we consider the Hermite-type weight functions and prove that the associated polynomial sequence can be expressed in the closed form via the Hermite polynomials. Also, we find their Rodrigues-type formula and a four-term recurrence relation. In contrast to the zeros of Hermite polynomials, which are symmetrically located with respect to the origin, the zeros of the new polynomial sequence are all positive. Copyright © 2015 John Wiley & Sons, Ltd.

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

]]>A brief review of asymptotic methods to deal with frictionless unilateral contact problems for an elastic layer of finite thickness is presented. Under the assumption that the contact radius is small with respect to the layer thickness, an effective asymptotic method is suggested for solving the unilateral contact problem with a priori unknown contact radius. A specific feature of the method is that the construction of an asymptotic approximation is reduced to a linear algebraic system with respect to integral characteristics (polymoments) of the contact pressure. As an example, the sixth-order asymptotic model has been written out. Copyright © 2015 John Wiley & Sons, Ltd.

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

In this paper, we investigate the uniform convergence of the Fourier series expansions in terms of eigenfunctions for the spectral problem

where *λ* is a spectral parameter, *q*(*x*) is a real-valued continuous function on the interval [0,1], and *a*_{1},*b*_{0},*b*_{1},*c*_{1},*d*_{0}, and *d*_{1} are real constants that satisfy the conditions

Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we study the following Schrödinger–Poisson system:

where *λ* > 0 is a parameter,
with 2≤*p*≤+*∞*, and the function *f*(*x*,*s*) may not be superlinear in *s* near zero and is asymptotically linear with respect to *s* at infinity. Under certain assumptions on *V*, *K*, and *f*, we give the existence and nonexistence results via variational methods. More precisely, when *p*∈[2,+*∞*), we obtain that system (SP) has a positive ground state solution for *λ* small; when *p* =+ *∞*, we prove that system (SP) has a positive solution for *λ* small and has no any nontrivial solution for *λ* large. Copyright © 2015 John Wiley & Sons, Ltd.

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

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

]]>The paper is devoted to the study of the asymptotic behavior of the solutions of a kinetic model describing chemotaxis phenomena. Our interest focuses on the case, where the diffusion part dominates the chemotaxis part in the limit. More in detail, we prove that the solution of kinetic model exists globally and converges to a solution of diffusive limit. Copyright © 2015 John Wiley & Sons, Ltd.

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

In this paper, we model the growths of populations by means of local fractional calculus. We formulate the local fractional rate equation and the local fractional logistic equation. The exact solutions of local fractional rate equation and local fractional logistic equation with the Mittag-Leffler function defined on Cantor sets are presented. The obtained results illustrate the accuracy and efficiency for modeling the complexity of linear and nonlinear population dynamics (PD). Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we obtain the existence of infinitely solutions for a class of nonlocal elliptic systems of (*p*_{1}(*x*),⋯,*p*_{n}(*x*))-Kirchhoff type. Our main results are new. Our approach are based on general variational principle because of B. Ricceri and the theory of the variable exponent Sobolev spaces. Copyright © 2015 John Wiley & Sons, Ltd.

This paper is concerned with global existence and asymptotic behavior of *H*^{1} solutions to the Cauchy problem of one-dimensional full non-Newtonian fluids with the weighted small initial data. We then obtain the global existence of *H*^{i}(*i* = 2,4) solutions and their asymptotic behavior for the system. Copyright © 2015 John Wiley & Sons, Ltd.

We reprove global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in five dimensions. Inspired by the recent work of Killip and Visan, we adapt the Dodson's strategy ‘long-time Strichartz estimate’ used in the work on mass-critical nonlinear Schrödinger equation sets. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Accuracy in roads geometry is an objective to be achieved by surveyors and cartographers when they obtain their data by GPS, Lidar, or photogrammetry. Nevertheless, those capture methods are expensive. Nowadays, cheap and collaborative methods can produce big datasets, which need to be processed in order to get accuracy axis from not accurate original data. Because a roads network is composed of several points, the resulting dataset could become a large-sized file, difficult to manage, and slow in consultancy for the users. In this paper, we expose our previous solutions for estimating a representative axis and propose a novel B-spline least square method governed by a genetic algorithm. The genetic algorithm minimizes the number of knots necessary to define the B-spline representative axis while keeping the axis' original shape. We know the original shape because we have computed it using a large number of knots by an iterative and convergent method developed in a well-contrasted previous study. This paper shows that our approaches are suitable to be deployed in a web-based application in order to support collaborative digital cartography. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper is concerned with the Cauchy problem of a class of fully nonlinear degenerate parabolic equations with reaction sources. After establishing the necessary local existence theorems of strong solutions, we investigate the blow-up and global existence profile. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this article, we study the geometric relation between two different types of initial conditions (IC) of a class of singular linear systems of fractional nabla difference equations whose coefficients are constant matrices. For these kinds of systems, we analyze how inconsistent and consistent IC are related to the column vector space of the finite and the infinite eigenvalues of the pencil of the system and analyze the geometric connection between these two different types of IC. Numerical examples are given to justify the results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The discrete fractional calculus is used to fractionalize difference equations. Simulations of the fractional logistic map unravel that the chaotic solution is conveniently obtained. Then a Riesz fractional difference is defined for fractional partial difference equations on discrete finite domains. A lattice fractional diffusion equation of random order is proposed to depict the complicated random dynamics and an explicit numerical formulae is derived directly from the Riesz difference. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We present solutions to both trifurcated and pentafurcated spaced waveguides using the mode matching (or eigenfunction expansion) method. While the trifurcated problem with mean fluid flow has been solved previously using the Wiener–Hopf technique, we solve this problem to validate and demonstrate our method. We then show how we can easily generalize the method to the pentafurcated problem that has not been solved previously. We observe that mode matching method is easier to derive and generalize than the Wiener–Hopf technique. We also investigate the numerical solution in detail for various geometries to model practical exhaust systems. Copyright © 2015 John Wiley & Sons, Ltd.

]]>D. Zeidan In this paper, we consider the existence of traveling waves in a generalized nonlinear dispersive–dissipative equation, which is found in many areas of application including waves in a thermoconvective liquid layer and nonlinear electromagnetic waves. By using the theory of dynamical systems, specifically based on geometric singular perturbation theory and invariant manifold theory, Fredholm theory, and the linear chain trick, we construct a locally invariant manifold for the associated traveling wave equation and use this invariant manifold to obtain the traveling waves for the nonlinear dispersive–dissipative equation. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we study *P*-type, *P**I*^{α}-type, and *D*-type iterative learning control for fractional impulsive evolution equations in Banach spaces. We present triple convergence results for open-loop iterative learning schemes in the sense of *λ*-norm through rigorous analysis. The proposed iterative learning control schemes are effective to fractional hybrid infinite-dimensional distributed parameter systems. Finally, an example is given to illustrate our theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we consider a class of parabolic partial differential equations with a time delay. The first model equation is the mixed problems for scalar generalized diffusion equation with a delay, whereas the second model equation is a delayed reaction-diffusion equation. Both of these models have inherent complex nature because of which their analytical solutions are hardly obtainable, and therefore, one has to seek numerical treatments for their approximate solutions. To this end, we develop a fitted Galerkin spectral method for solving this problem. We derive optimal error estimates based on weak formulations for the fully discrete problems. Some numerical experiments are also provided at the end. Copyright © 2015 John Wiley & Sons, Ltd.

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

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

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

]]>In this paper, we consider a nonlinear viscoelastic inverse problem with memory in the boundary. Under some suitable conditions on the coefficients, relaxation function, and initial data, we proved stability of solutions when the integral overdetermination tends to zero as time goes to infinity. Furthermore, we show that there are solutions under some conditions on initial data that blow up in finite time. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We discuss analytically the stationary viscous quantum hydrodynamic model including a barrier potential, which is a nonlinear system of partial differential equations of mixed order in the sense of Douglis–Nirenberg. Combining a reformulation by means of an adjusted Fermi level, a variational functional, and a fixed point problem, we prove the existence of a weak solution. There are no assumptions on the size of the given data or their variation. We also provide various estimates of the solution that are independent of the quantum parameters. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We propose and analyze a mathematical model of the mechanics of gels, consisting of the laws of balance of mass and linear momentum of the polymer and liquid components of the gel. We consider a gel to be an immiscible and incompressible mixture of a nonlinearly elastic polymer and a fluid. The problems that we study are motivated by predictions of the life cycle of body-implantable medical devices. Scaling arguments suggest neglecting inertia terms, and therefore, we consider the quasi-static approximation to the dynamics. We focus on the linearized system about stress-free states, uniform expansions, and compressions and derive sufficient conditions for the solvability of the time-dependent problems. These turn out to be conditions that guarantee local stability of the equilibrium solutions. We also consider non-stress free equilibria and states with residual stress and derive an energy law for the corresponding time-dependent system. The conditions that guarantee stability of solutions provide a selection criteria of the material parameters of devices. The boundary conditions that we consider are of two types, displacement-traction and permeability of the gel surface to the fluid. We address the cases of viscous and inviscid solvent, assume Newtonian dissipation for the polymer component, and establish existence of weak solutions for the different boundary permeability conditions and viscosity assumptions. We present two-dimensional, finite element numerical simulations to study stress concentration on edges, this being the precursor to debonding of the gel from its substrate. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The Bhatnagar–Gross–Krook model of the Boltzmann equation is of great importance in the kinetic theory of rarefied gases. Various existence and uniqueness results have been built under the boundedness of energy. In this paper, we will establish several global existence results to the Bhatnagar–Gross–Krook equation with infinite energy. It heavily relies on a new moments lemma and a new existence and uniqueness theorem of weighted velocity-spatial *L*^{∞} solutions. Copyright © 2015 John Wiley & Sons, Ltd.

In this research article, the inverse problem of finding a time-dependent coefficient in a second-order elliptic equation is investigated. The existence and the uniqueness of the classical solution of the problem under consideration are established. Numerical tests using the finite-difference scheme combined with an iteration method are presented, and the sensitivity of this scheme with respect to noisy overdetermination data is illustrated. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we focus on the approximate controllability of control systems described by a large class of fractional evolution hemivariational inequalities. Firstly, we introduce the concept of mild solutions and present the existence of mild solutions for this kind of problems. Next, we show the approximate controllability of the corresponding linear control system. Finally, the approximate controllability of the fractional evolution hemivariational inequalities is formulated and proved under some appropriate conditions. An example demonstrates the applicability of our results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper deals with the solutions, stability character, and asymptotic behavior of the difference equation

where
and the initial values *x*_{−k},*x*_{−k + 1},…,*x*_{0} are nonzero real numbers, such that their solutions are associated to Horadam numbers. Copyright © 2015 John Wiley & Sons, Ltd.

E. Study found that there is a one-to-one correspondence between the oriented lines in Euclidean three space and the dual points of the dual unit sphere in dual three space, and it has wide applications in Engineering. In this paper, we investigate a ruled surface as a curve on the dual unit sphere by using E. Study's theory. Then we define the notion of evolutes of dual spherical curves for ruled surfaces and establish the relationships between singularities of these subjects and geometric invariants of dual spherical curves. Finally, we give an example to illustrate our findings. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Dissipativity theory is a very important concept in the field of control system. In this paper, we pay attention to the problem of dissipativity analysis of memristive neural networks with time-varying delay and randomly occurring uncertainties(ROUs). Under the framework of Filippov solution, differential inclusion theory, by employing a proper Lyapunov functional, and some inequality techniques, the dissipativity criteria are obtained in terms of LMIs. It should be noteworthy that the uncertainty terms as well as the ROUs are separately taken into consideration, in which the uncertainties are norm-bounded and the ROUs obey certain mutually uncorrelated Bernoulli-distributed white noise sequences. Finally, the effectiveness of the proposed method will be verified via numerical example. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we study the longtime dynamics of the non-autonomous Boussinesq-type equation with critical nonlinearity, and time-dependent external forcing, which is translation bounded but not translation compact. We prove the existence of a uniform attractor in . Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, a numerical procedure involving Chebyshev wavelet method has been implemented for computing the approximate solution of Riesz space fractional sine-Gordon equation (SGE). Two-dimensional Chebyshev wavelet method is implemented to calculate the numerical solution of space fractional SGE. The fractional SGE is considered as an interpolation between the classical SGE (corresponding to *α* = 2) and nonlocal SGE (corresponding to *α* = 1). As a consequence, the approximate solutions of fractional SGE obtained by using Chebyshev wavelet approach were compared with those derived by using modified homotopy analysis method with Fourier transform. Copyright © 2015 John Wiley & Sons, Ltd.

The aim of this paper is to investigate the pathwise numerical solution of semilinear parabolic stochastic partial differential equations (SPDEs) with colored noise instead of the usual space–time white noise. We estimate the numerical solution in the *L*^{∞} topology by a method that takes advantages of the smoothing effect of the dominant linear operator. We consider the case the covariance operator of the forcing does not necessarily commute with the linear operator of the SPDE because of the fact that the Brownian motions are not necessarily independent. We show convergence of this method, and numerical examples give insight into the reliability of the theoretical study. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we present a practical exponential stability result for impulsive dynamic systems depending on a parameter. Stability theorem and converse stability theorem are established by employing the second Lyapunov method. These theorems are used to analyze the practical exponential stability of the solution of perturbed impulsive systems and cascaded impulsive systems, depending on a parameter. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we consider a class of impulsive Hamiltonian systems with a *p*-Laplacian operator. Under certain conditions, we establish the existence of homoclinic orbits by means of the mountain pass theorem and an approximation technique. In some special cases, the homoclinic orbits are induced by the impulses in the sense that the associated non-impulsive systems admit no non-trivial homoclinic orbits. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we apply the boundary integral equation technique and the dual reciprocity boundary elements method (DRBEM) for the numerical solution of linear and nonlinear time-fractional partial differential equations (TFPDEs). The main aim of the present paper is to examine the applicability and efficiency of DRBEM for solving TFPDEs. We employ the time-stepping scheme to approximate the time derivative, and the method of linear radial basis functions is also used in the DRBEM technique. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity that appears in the nonlinear problems under consideration. To confirm the accuracy of the new approach, several examples are presented. The convergence of the DRBEM is studied numerically by comparing the exact solutions of the problems under investigation. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we study the qualitative behavior of following two systems of higher-order difference equations:

and

where the parameters *α*,*β*,*γ*,*α*_{1},*β*_{1},*γ*_{1},*a*,*b*,*c*,*a*_{1},*b*_{1},and*c*_{1} and the initial conditions *x*_{0}, *x*_{−1}, ⋯, *x*_{−k}, *y*_{0}, *y*_{−1} ,⋯, *y*_{−k} are positive real numbers. More precisely, we study the equilibrium points, local asymptotic stability, instability, global asymptotic stability of equilibrium points, and rate of convergence of positive solutions that converges to the equilibrium point *P*_{0}=(0,0) of these systems. Some numerical examples are given to verify our theoretical results. These examples are experimental verification of our theoretical discussions. Copyright © 2015 John Wiley & Sons, Ltd.

In the present article, the authors have studied the dynamical behavior of delay-varying computer virus propagation (CVP) model with fractional order derivative, and it is found that the chaotic attractor exists in the considered fractional order system. In order to eliminate the chaotic behavior of fractional order delay-varying CVP model, feedback controlmethod is used. This article also dealswith the synchronization between controlled and chaotic delay-varying CVPmodel via active controlmethod. The fractional derivative is described in the Caputo sense. Numerical simulation results are carried out by means of Adams-Boshforth-Moultonmethod with the help ofMATLAB, and the results are successfully depicted through graphs .Copyright © 2015 John Wiley & Sons, Ltd.

]]>We prove in this article that there is in the set of all problems we consider a subset, which is residual. Every problem in this subset is shown to be structurally stable and defines a dynamical system, which looks like the graph of the figure given in Section 1, contrary to what happens for ordinary dynamical systems, that is, the ones associated with ODEs. There, the initial value problem (in the smooth case) is uniquely solvable; the structurally stable systems look like the figure given in Section 1, but sources are equilibria. Copyright © 2015 John Wiley & Sons, Ltd.

]]>T. Wanner We investigate global dynamics of the equation

where the parameters *b*,*c*, and *f* are nonnegative numbers with condition *b* + *c* > 0,*f* ≠ 0 and the initial conditions *x*_{−1},*x*_{0} are arbitrary nonnegative numbers such that *x*_{−1}+*x*_{0}>0. We obtain precise characterization of basins of attraction of all attractors of this equation and describe the dynamics in terms of bifurcations of period-two solutions. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we study a new class of 3-point boundary value problems of nonlinear fractional difference equations. Our problems contain difference and fractional sum boundary conditions. Existence and uniqueness of solutions are proved by using the Banach fixed-point theorem, and existence of the positive solutions is proved by using the Krasnoselskii's fixed-point theorem. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We use a modification of Krasnoselskii's fixed point theorem introduced by Burton to show the periodicity and non-negativity of solutions for the nonlinear neutral differential equation with variable delay

We invert this equation to construct the sum of a compact map and a large contraction, which is suitable for applying the modification of Krasnoselskii's theorem. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We consider the short time strong solutions to the compressible magnetohydrodynamic equations with initial vacuum, in which the velocity field satisfies the Navier-slip condition. The Navier-slip condition differs in many aspects from no-slip conditions, and it has attracted considerable attention in nanoscale and microscale flows research. Inspired by Kato and Lax's idea, we use the Lax–Milgram theorem and contraction mapping argument to prove local existence. Moreover, under the Navier-slip condition, we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of *L*^{∞} norm of the deformation tensor *D*(*u*). Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we study the following anisotropic problem with singularity:

where is a bounded domain with smooth boundary. Using the critical point theory, we obtain the existence of weak solutions for the problem under suitable conditions. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Numerical solution and chaotic behaviors of the fractional-order simplified Lorenz hyperchaotic system are investigated in this paper. The solution of the fractional-order hyperchaotic system is obtained by employing Adomian decomposition method. Lyapunov characteristic exponents algorithm for the fractional-order chaotic system is designed. Dynamics of the fractional-order hyperchaotic system are analyzed by means of bifurcation diagrams, Lyapunov characteristic exponents, C_{0} complexity, and chaos diagram. It shows that this system has rich dynamical behaviors, and it is more complex when the fractional order *q* is small. It lays a foundation for the practical application of the fractional-order hyperchaotic systems. Copyright © 2015 John Wiley & Sons, Ltd.

This paper mainly considers the problem of reconstructing a reproducing kernel stochastic signal from its average samples. First, a uniform convergence result for reconstructing the deterministic reproducing kernel signals by an iterative algorithm is established. Then, we prove that the quadratic sum of the corresponding reconstructed functions is uniformly bounded. Moreover, the reconstructed functions provide a frame expansion in the special case *p* = 2. Finally, the mean square convergence for recovering a weighted reproducing kernel stochastic signal from its average samples is given under some decay condition for the autocorrelation function, which can be removed for the case *p* = 2. Copyright © 2015 John Wiley & Sons, Ltd.

This paper deals with a parabolic–parabolic Keller–Segel-type system in a bounded domain of
, {*N* = 2;3}, under different boundary conditions, with time-dependent coefficients and a positive source term. The solutions may blow up in finite time *t*^{∗}; and under appropriate assumptions on data, explicit lower bounds for blow-up time are obtained when blow up occurs. Copyright © 2015 John Wiley & Sons, Ltd.

An Human Immunodeficiency Virus/Acquired Immuno-Deficiency Syndrome (HIV/AIDS) epidemic model for sexual transmission with asymptomatic and symptomatic phase is proposed as a system of differential equations. The threshold and steady state for the model are determined and stabilities of disease free steady state is investigated. We use the model and study the effect of public health education on the spread of HIV/AIDS as a single-strategy in HIV prevention. The education, including basic reproduction number for the model with public health education, is compared with the basic reproduction number for the HIV/AIDS in the absence of public health education. By comparing these two values, influence of public health education appears. According to property of , threshold proportion of educated adolescents, education rate for susceptible individuals and education efficacy is obtained. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The purpose of this paper is to show existence of a solution of the Dirichlet problem for a nonlinear generalized Darcy–Forchheimer–Brinkman system in a bounded Lipschitz domain in
, with small boundary datum in *L*^{2}-based Sobolev spaces. A useful intermediary result is the well-posedness of the Poisson problem for a generalized Brinkman system in a bounded Lipschitz domain in
, with Dirichlet boundary condition and data in *L*^{2}-based Sobolev spaces. We obtain this well-posedness result by showing that the matrix type operator associated with the Poisson problem is an isomorphism. Then, we combine the well-posedness result from the linear case with a fixed point theorem in order to show the existence of a solution of the Dirichlet problem for the nonlinear generalized Darcy–Forchheimer–Brinkman system. Some applications are also included. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we consider a three dimensional quantum Navier-Stokes-Poisson equations. Existence of global weak solutions is obtained, and convergence toward the classical solution of the incompressible Navier-Stokes equation is rigorously proven for well prepared initial data. Furthermore, the associated convergence rates are also obtained. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we consider the Bresse system coupled with the Fourier law of heat conduction. We prove that the decay rate of the solution is very slow. In fact, we show that the *L*^{2}-norm of the solution decays with the rate of (1 + *t*)^{−1/12} similar to the one obtained for the Timoshenko system. In addition, we found that the wave speed of the first two equations still control the decay rate of the solution with respect to the regularity of the initial data. This seems to be the first result dealing with the behavior of the Cauchy problem in the Bresse–Fourier model. Copyright © 2014 John Wiley & Sons, Ltd.

This manuscript presents the HIV-1 infection model along with cause of differentiation of cytotoxic T lymphocyte response, the total carrying capacity of CD4C^{+} T-cells, logistic growth term, effect of combination of antiretroviral therapy and discrete type immune response delay. The possibility of existence of multiple equilibriums for the proposed model is analyzed. Asymptotic stability of the non-delayed infection model is proved from the roots of characteristic equation which are obtained by employing the Jacobian matrix method. The existence of Hopf bifurcation due to immune activation delay is proved. The stability switching is studied by choosing immune activation delay as a bifurcation parameter. Utilizing normal formtheory and centermanifold , we derive the explicit formulae for determining the stability and direction of the periodic solutions bifurcating from Hopf bifurcations. Numerical simulations are executed to verify the derived analytical results. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper,we applied the Painlevé property test on Krook-Wu model of the nonlinear Boltzmann equation (*p* = 1). As a result, by using Bäcklund transformation, we obtained three solutions two of them were known earlier, while the third one is new and more general, we have also two reductions one of them is Abel's equation. Also, Lie-group method is applied to the (p + 1)th Boltzmann equation. The complete Lie algebra of infinitesimal symmetries is established. Three nonequivalent sub-algebraic of the complete Lie algebra are used to investigate similarity solutions and similarity reductions in the form of nonlinear ordinary equations for (p + 1)th Boltzmann equation; we obtained two general solutions for (p + 1)th Boltzmann equation and new solutions for Krook-Wu model of Boltzmann equation (p = 1). Copyright © 2014 John Wiley & Sons, Ltd.

In this article, we employ the complex method to obtain all meromorphic exact solutions of complex Klein–Gordon (KG) equation, modified Korteweg-de Vries (mKdV) equation, and the generalized Boussinesq (gB) equation at first, then find all exact solutions of the Equations KG, mKdV, and gB. The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic solutions are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions *w*_{2r,2}(*z*) and simply periodic solutions *w*_{1s,2}(*z*),*w*_{2s,1}(*z*) in these equations such that they are not only new but also not degenerated successively by the elliptic function solutions. We have also given some computer simulations to illustrate our main results. Copyright © 2014 John Wiley & Sons, Ltd.

We study the semilinear equation

where 0 < *s* < 1,
, *V*(*x*) is a sufficiently smooth non-symmetric potential with
, and *ϵ* > 0 is a small number. Letting *U* be the radial ground state of (−Δ)^{s}*U* + *U* − *U*^{p}=0 in
, we build solutions of the form

for points *ϑ*_{j},*j* = 1,⋯,*m*, using a Lyapunov–Schmidt variational reduction. Copyright © 2014 John Wiley & Sons, Ltd.

The present note presents some errors in the aforementioned paper published in Mathematical Methods in the Applied Sciences. Two errors are found in the definition of the non-dimensional parameters and correct results are presented for temperature profiles included in figure 10 of the previous paper. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In the present paper, an iteration regularization method for solving the Cauchy problem of the modified Helmholtz equation is proposed. The *a priori* and *a posteriori* rule for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method. Copyright © 2014 John Wiley & Sons, Ltd.

The aim of this article is to derive an asymptotic two-scale model for the propagation of a fungal disease over a large vineyard. The original model is based on a singularly perturbed system of two linear reaction-diffusion equations coupled with a set of nonlinear ordinary differential equations in a highly heterogeneous medium. We prove the well-posedness of the asymptotic model and obtain a convergence result confirmed by numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we study the global existence of solution for the quasilinear chemotaxis system with Dirichlet boundary conditions, and further we show that the blow up properties of the solution depend only on the first eigenvalue. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, the authors give sufficient conditions for the boundedness and global asymptotic stability of solutions to certain nonlinear multi-delay functional differential equations of the third order. The technique of proof involves defining an appropriate Lyapunov-Krasovskii functional and applying LaSalle's invariance principle. An example is included to illustrate the results. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Bacterial biofilms are microbial depositions on immersed surfaces. Their mathematical description leads to degenerate diffusion-reaction equations with two non-Fickian effects: (i) a porous medium equation like degeneracy where the biomass density vanishes and (ii) a super-diffusion singularity if the biomass density reaches its threshold density. In the case of multispecies interactions, several such equations are coupled, both in the reaction terms and in the nonlinear diffusion operator. In this paper, we generalize previous work on existence and uniqueness of solutions of this type of models and give a general, relatively easy to apply criterion for well-posedness. The use of the criterion is illustrated in several examples from the biofilm modeling literature. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We study the dynamics of three-dimensional Vlasov-Poisson system in the presence of a point charge with attractive interaction. Compared to the repulsive interaction,we cannot get a priori conversation law. Nevertheless,we are able to obtain bound of kinetic energy by introducing a Lyapunov functional. Combining this result with the concept of Diperna-Lions flow, we establish global existence of weak solutions for this system. Copyright © 2014 John Wiley & Sons, Ltd.

]]>We study the Rayleigh–Bénard convection in a 2D rectangular domain with no-slip boundary conditions for the velocity. The main mathematical challenge is due to the no-slip boundary conditions, because the separation of variables for the linear eigenvalue problem, which works in the free-slip case, is no longer possible. It is well known that as the Rayleigh number crosses a critical threshold *R*_{c}, the system bifurcates to an attractor, which is an (*m* − 1)-dimensional sphere, where *m* is the number of eigenvalues, which cross zero as *R* crosses *R*_{c}. The main objective of this article is to derive a full classification of the structure of this bifurcated attractor when *m* = 2. More precisely, we rigorously prove that when *m* = 2, the bifurcated attractor is homeomorphic to a one-dimensional circle consisting of exactly four or eight steady states and their connecting heteroclinic orbits. In addition, we show that the mixed modes can be stable steady states for small Prandtl numbers. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, the -expansion method is proposed to establish hyperbolic and trigonometric function solutions for fractional differential-difference equations with the modified Riemann–Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential-difference equation into its differential-difference equation of integer order. We obtain the hyperbolic and periodic function solutions of the nonlinear time-fractional Toda lattice equations and relativistic Toda lattice system. The proposed method is more effective and powerful for obtaining exact solutions for nonlinear fractional differential–difference equations and systems. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this study, a mathematical model previously proposed for the transmission dynamics of brucellosis among bison was mathematically analyzed. Our qualitative and quantitative findings support the general hypothesis that the infection will vanish from the herd when the basic reproduction number *R*_{0}<1 and will persist otherwise. A global sensitivity analysis was conducted, and the results of the Sobol indices indicated that the rate of loss of resistance (*δ*) and the recovery rate (*v*) were responsible for most of the variability in the expected number of infectious bison. On the other hand, according to the partial ranked correlation coefficients, the density-dependent reduction in birth (*φ*), the mortality rate (*m*), the transmission coefficient (*β*), and the recovery rate (*v*) exerted very high (and negative) correlations with the number of infectious bison, whereas the rate of loss of resistance (*δ*) and the calving rate (*a*) exerted very high (and positive) correlations with the number of infectious bison. Control measures should therefore aim at increasing the magnitude of *φ*, *m*, and *v* and reducing those of *δ* and *a*. In addition, experimental studies are needed to precisely estimate the rate of loss of resistance and the recovery rate in order to increase the accuracy of the expected number of infectious bison. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we study a one-dimensional morphogenesis model considered by C. Stinner *et al*. (Math. Meth. Appl. Sci.2012;35: 445–465). Under homogeneous boundary conditions, we prove the existence of nonconstant positive steady states through local bifurcation theories. Then we rigorously study the stability of these nonconstant solutions when the sensitivity functions are chosen to be linear and logarithmic, respectively. Finally, we present numerical solutions to illustrate the formation of stable inhomogeneous spatial patterns. Our numerical simulations show that this model can develop very complicated and interesting structures even over one-dimensional finite domains. Copyright © 2014 John Wiley & Sons, Ltd.

We introduce and study a degenerate reaction-diffusion system that can serve as a model prototype for the pattern formation of a bacterial multicellular community where the bacteria produce biofilm, grow and spread in the presence of a nutrient. Under proper conditions on the reaction terms, we prove the global existence and the uniqueness of solutions and illustrate the possible model behaviour in numerical simulations for a two-dimensional setting. Copyright © 2014 John Wiley & Sons, Ltd.

]]>We consider a class of inverse source problems for the parabolic approximation to the Maxwell equations.We relate this to an exact controllability problem; the regularisation of the considered source problems is studied with an optimal control method. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, it is the first time that some important inequalities are obtained to estimate the exponential type of upper and lower bounds of solutions for the three representative classes of homogeneous impulsive dynamic systems on time scales. Based on these, some new criteria are established for admitting an exponential dichotomy of the impulsive dynamic systems. The obtained results are essentially new, even the time scale or . In addition, in applications, we apply the obtained results to discuss the almost periodic problems of a class of integro-differential systems, and the numerical simulations are given to illustrate that our timescale methods are feasible and effective. Finally, we present the conclusion and further discussion related to this topic. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one-dimensional nonlinear sine-Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth-order for discretizing the spatial derivative and the standard second-order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V-cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one-dimensional sine-Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we study the initial boundary value problem of nonlinear pseudo-parabolic equation with a memory term

with initial conditions and Dirichlet boundary conditions. By the combination of the Galerkin method and Potential well theory, the existence of global solutions is derived. Moreover, not only the finite time blow up of solutions with the negative initial energy (*E*(0) < 0) but also the finite time blow up results with the nonnegative initial energy (0≤*E*(0) < *d*_{k}) are obtained by using Concavity method and Potential well theory. Copyright © 2014 John Wiley & Sons, Ltd.

A delayed predator–prey system with Holling type II functional response and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each feasible equilibrium of the system is discussed, and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of the persistence theory for infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using suitable Lyapunov functions and the LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when both the predator–extinction equilibrium and the coexistence equilibrium do not exist, and that the predator–extinction equilibrium is globally stable when the coexistence equilibrium does not exist. Further, sufficient conditions are obtained for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The Allen–Cahn equation, coupled with dynamic boundary conditions, has recently received a good deal of attention. The new issue of this paper is the setting of a rather general mass constraint, which may involve either the solution inside the domain or its trace on the boundary. The system of nonlinear partial differential equations can be formulated as a variational inequality. The presence of the constraint in the evolution process leads to additional terms in the equation and the boundary condition containing a suitable Lagrange multiplier. A well-posedness result is proved for the related initial value problem. Copyright © 2014 John Wiley & Sons, Ltd.

]]>With methods of potential theory, we develop a representation of a solution of the coupled Stokes–Darcy model in a Lipschitz domain for boundary data in *H*^{−1/2}. Copyright © 2014 John Wiley & Sons, Ltd.

We propose and analyze the finite volume method for solving the variational inequalities of first and second kinds. The stability and convergence analysis are given for this method. For the elliptic obstacle problem, we derive the optimal error estimate in the *H*^{1}-norm. For the simplified friction problem, we establish an abstract *H*^{1}-error estimate, which implies the convergence if the exact solution *u*∈*H*^{1}(Ω) and the optimal error estimate if *u*∈*H*^{1 + α}(Ω),0 < *α*≤2. Copyright © 2015 John Wiley & Sons, Ltd.

We study the small-data Cauchy problem for *n*-dimensional Stokes damped Rosenau equation. Under some assumptions, we prove the global existence and uniqueness of the small-amplitude solution by utilizing the contraction mapping principle and study the asymptotic behavior of the solution. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, applying both Lyapunov function techniques and monotone iterative techniques, we establish new sufficient conditions under which the infected equilibrium of an HIV pathogenesis model with cure rate is globally asymptotically stable. By giving an explicit expression for eventual lower bound of the concentration of susceptible CD4^{+} T cells, we establish an affirmative partial answer to the numerical simulations investigated in the recent paper [Liu, Wang, Hu and Ma, Global stability of an HIV pathogenesis model with cure rate, *Nonlinear Analysis RWA* (2011) **12**: 2947–2961]. Our monotone iterative techniques are applicable for the small and large growth rate in logistic functions for the proliferation rate of healthy and infected CD4^{+} T cells. Copyright © 2014 John Wiley & Sons, Ltd.

This paper concerns the 3D Navier-Stokes equations and prove an almost Serrin-type regularity criterion in terms of one directional derivative of the pressure. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Numerical method for a coupled continuum pipe-flow/Darcy model describing flow in porous media with an embedded conduit pipe is considered. Wilson element on anisotropic mesh is used to solve the Darcy equation on porous matrix. The existence and uniqueness of the approximation solution are obtained. Optimal error estimates in *L*^{2} and *H*^{1} norms are established independent of the regularity condition on the mesh. Numerical examples show the efficiency of the presented scheme. With the same number of nodal points, the results using Wilson element on anisotropic mesh are much better than those of the same element and *Q*_{1} element on regular mesh. Copyright © 2014 John Wiley & Sons, Ltd.

We consider the coupled problem describing the motion of a linear array of three-dimensional obstacles floating freely in a homogeneous fluid layer of finite depth. The interaction of time-harmonic waves with the floating objects is analyzed under the usual assumptions of linear water-wave theory. Quasi-periodic boundary conditions and a simplified reduction scheme turn the system into a linear spectral problem for a self-adjoint operator in Hilbert space. Based upon the operator formulation, we derive a sufficient condition for the nonemptiness of its discrete spectrum. Various examples of obstacles that generate trapped modes are given. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The paper deals with the existence and uniqueness of classical solutions of the homogeneous Neumann problem for a class of parabolic–hyperbolic system of partial differential equations in *n* dimensions. The problem arises from a model of the diffusion of *N* species of radioactive isotopes of the same element. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with the Cauchy problem of the modified Hunter-Saxton equation. The local well-posedness of the model equation is obtained in Besov spaces
(which generalize the Sobolev spaces *H*^{s}) by using Littlewood-Paley decomposition and transport equation theory. Moreover, the local well-posedness in critical case (with
) is considered.

We present a nonlinear model for Johnson–Segalman type polymeric fluids in porous media, accounting for thermal effects of Oldroyd-B type. We provide a thermodynamic development of the Darcy's theory, which is consistent with the interlacement between thermal and viscoelastic relaxation effects and diffusion phenomena. The appropriate invariant convected time derivative for the flux vector and the stress tensor is discussed. This is performed by investigating the local balance laws and entropy inequality in the spatial configuration, within the single-fluid approach. For constant parameters, our thermomechanical setting is of Jeffreys type with two delay time parameters, and hence, in the linear/linearized version, it is strictly related to phase-lag theories within first-order Taylor approximations.

A detailed spectral analysis is carried out for the linearized version of the model, with a scrutiny to some significant limit situations, enhancing the stabilizing effects of the dissipative and elastic mechanisms, also for retardation responses.

For polymeric liquids, rheological aspects, wave propagation properties and analogies with other theories with lagging are pointed out. Copyright © 2014 John Wiley & Sons, Ltd.

Integral equations of the first kind for exterior problems arising in the study of the three-dimensional Helmholtz equation are considered. These equations are derived by seeking solutions in the form of layer potentials with modified fundamental solutions. For each first kind equation, existence and uniqueness of solution are proved with the aid of composition relations involving associated modified boundary integral operators. For the Dirichlet problem, an optimal choice of the modification coefficients is considered in order to minimize the condition number of the resulting integral operator. Copyright © 2014 John Wiley & Sons, Ltd.

]]>This paper is concerned with the asymptotic behavior of the decreasing energy solution *u*_{ε} to a p-Ginzburg–Landau system with the initial-boundary data for *p* > 4/3. It is proved that the zeros of *u*_{ε} in the parabolic domain *G* × (0,*T*] are located near finite lines {*a*_{i}}×(0,*T*]. In particular, all the zeros converge to these lines when the parameter *ε* goes to zero. In addition, the author also considers the uniform energy estimation on a domain far away from the zeros. At last, the Hölder convergence of *u*_{ε} to a heat flow of p-harmonic map on this domain is proved when *p* > 2. Copyright © 2014 John Wiley & Sons, Ltd.

We consider initial boundary value problems, including boundary damping, for planar magnetohydrodynamics. We show that global strong solutions exist with large data and no shock wave, mass concentration, or vacuum appear for general equations of state. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we prove two blow-up criteria of smooth solution: one for the generalized incompressible Oldroyd model with fractional Laplacian velocity dissipation (−Δ)^{α}*u* in the space
and one for the inviscid Oldroyd model. Assume that (*u*(*t*,*x*),*F*(*t*,*x*)) is a smooth solution to the generalized Oldroyd model in [0,*T*); then, the solution (*u*(*t*,*x*),*F*(*t*,*x*)) does not develop singularity until *t* = *T* provided
. For the ideal impressible viscoelastic flow, it is shown that the smooth solution (*u*,*F*) can be extended beyond *T* if
, which is an improvement of the result given by Hu and Hynd (A blowup criterion for ideal viscoelastic flow, J. Math. Fluid Mech., 15(2013), 431–437). Copyright © 2015 John Wiley & Sons, Ltd.

Local phase is now known to carry information about image features or object motions. But it is harder to use directly compared with amplitude, so far. In this paper, we propose that the *relative local phase*, which is a function of scale, position and time, really matters in representing the information of image structures or movements. A unified description of relative phase is given in this paper based on a bilinear representation of natural image series via multi-scale orientated dual tree complex wavelets. Then, the behaviors of *nontrivial* relative phase, especially for their distribution on multi-scale and multi-subband, are investigated. We propose a new generalized model, which is derived from Möbius transform, to describe various relative phases. Numerical experiments for a large amount of test images show that the new model performs best compared with the von Mises or wrapped Cauchy distribution. Especially for those with asymmetric pdf, our function fits with the histogram quite well while the other two may fail. We thus lay a groundwork for relative phase-based image processing methods, such as classification, deblurring and motion perception. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we investigate the vacuum free boundary problem of a compressible Navier–Stokes–Poisson system with density-dependent viscosity. By introducing Eulerian and Lagrange energy, we obtain a local in time well-posedness of the strong solution to the Navier–Stokes–Poisson system in a spherically symmetric case. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper offers conditions ensuring the existence of solutions of linear boundary value problems for systems of dynamic equations on time scales. Utilizing a method of Moore–Penrose pseudo-inverse matrices leads to an analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a system of dynamic equations. As an example of an application of the presented results, the problem of bifurcation of solutions of boundary value problems for systems of dynamic equations on time scales with a small parameter is considered.

]]>In this paper, a time-delayed free boundary problem for tumor growth under the action of external inhibitors is studied. It is assumed that the process of proliferation is delayed compared with apoptosis. By L^{p} theory of parabolic equations, the Banach fixed point theorem and the continuation theorem, the existence and uniqueness of a global solution is proved. The asymptotic behavior of the solution is also studied. The proof uses the comparison principle and the iteration method. Copyright © 2014 John Wiley & Sons, Ltd.

This paper presents new analytical results and the first numerical results for a recently proposed multiscale deconvolution model (MDM) recently proposed. The model involves a large-eddy simulation closure that uses a novel deconvolution approach based on the introduction of two distinct filtering length scales. We establish connections between the MDM and two other models, and, on the basis of one of these connections, we establish an improved regularity estimate for MDM solutions. We also prove that the MDM preserves Taylor-eddy solutions of the Navier–Stokes equations and therefore does not distort this particular vortex structure. Simulations of the MDM are performed to examine the accuracy of the MDM and the effect of the filtering length scales on energy spectra for three-dimensional homogeneous and isotropic flows. Numerical evidence for all tests clearly indicates that the MDM gives very accurate coarse-mesh solutions and that this multiscale approach to deconvolution is effective. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Inductive electromagnetic means, currently employed in real physical applications and dealing with voluminous bodies embedded in lossless media, often call for analytically demanding tools of field calculation at modeling stage and later on at numerical stage. Here, one is considering two closely adjacent perfect conductors, possibly almost touching one another, for which the 3D bispherical geometry provides a good approximation. The particular scattering problem is modeled with respect to the two solid impenetrable metallic spheres, which are excited by a time-harmonic magnetic dipole, arbitrarily orientated in the 3D space. The incident, the scattered, and the total non-axisymmetric electromagnetic fields yield rigorous low-frequency expansions in terms of positive integral powers of the real-valued wave number in the exterior medium. We keep the most significant terms of the low-frequency regime, that is, the static Rayleigh approximation and the first three dynamic terms, while the additional terms are small contributors and they are neglected. The typical Maxwell-type problem is transformed into intertwined either Laplace's or Poisson's potential-type boundary value problem with impenetrable boundary conditions. In particular, the fields are represented via 3D infinite series expansions in terms of bispherical eigenfunctions, obtaining analytical closed-form solutions in a compact fashion. This procedure leads to infinite linear systems, which can be solved approximately within any order of accuracy through a cutoff technique.

]]>Herein, the generalized diffusion equation that encompasses the nonlinear diffusion equation with a source term and the Boussinesq equation in hydrology as its particular form and appears in a wide variety of physical and engineering applications has been analyzed via symmetry method that was developed by Steinberg.

According to physical situations, in each case, the similarity variables obtained have led us to an ordinary differential equation, and we acquire some new solutions by solving the ODEs. Copyright © 2015 John Wiley & Sons, Ltd

This paper is concerned with a periodic two-component *μ*-Hunter–Saxton system. We prove that the solution map of the Cauchy problem of the *μ*-Hunter–Saxton system is not uniformly continuous in
, *s* > 5/2. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we devise a simple way to explicitly construct the Riemann theta function periodic wave solution of the nonlinear partial differential equation. The resulting theory is applied to the Hirota–Satsuma shallow water wave equation. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function. We obtain the one-periodic and two-periodic wave solutions of the equation. The relations between the periodic wave solutions and soliton solutions are rigorously established. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Previously, existence and uniqueness of a class of monotone similarity solutions for a nonlinear differential equation arising in magnetohydrodynamic Falkner–Skan flow were considered in the case of accelerating flows. It was shown that a solution satisfying certain monotonicity properties exists and is unique for the case of accelerated flows and some decelerated flows. In this paper, we show that solutions to the problem can exist for decelerated flows even when the monotonicity conditions do not hold. In particular, these types of solutions have nonmonotone second derivatives and are, hence, a distinct type of solution from those studied previously. By virtue of this result, the present paper demonstrates the existence of an important type of solution for decelerated flows. Importantly, we show that multiple solutions can exist for the case of strongly decelerated flows, and this occurs because of the fact that the solutions do not satisfy the aforementioned monotonicity requirements. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper discusses some basic dynamical properties of the chaotic finance system with parameter switching perturbation, and investigates chaos projective synchronization of the chaotic finance system with the time-varying delayed feedback controller, which are not fully considered in the existing research. Different from the previous methods, in this paper, the delayed feedback controller is not only time-varying, but also the time-varying delay is adaptive. Finally, an illustrate example is provided to show the effectiveness of this method. Copyright © 2014 John Wiley & Sons, Ltd.

]]>We characterize the values of the parameters for which a zero-Hopf equilibrium point takes place at the singular points, namely, *O* (the origin), *P*_{+}, and *P*_{−} in the FitzHugh–Nagumo system.

We find two two-parameter families of the FitzHugh–Nagumo system for which the equilibrium point at the origin is a zero-Hopf equilibrium. For these two families, we prove the existence of a periodic orbit bifurcating from the zero-Hopf equilibrium point *O*.

We prove that there exist three two-parameter families of the FitzHugh–Nagumo system for which the equilibrium point at *P*_{+} and at *P*_{−} is a zero-Hopf equilibrium point. For one of these families, we prove the existence of one, two, or three periodic orbits starting at *P*_{+} and *P*_{−}. Copyright © 2014 John Wiley & Sons, Ltd.

People having extreme idealogies affect the process in a region using fear of terror acts, money power, and the word of mouth communication network to change individuals to their way of thinking. This forces government to divert its limited financial resources for controlling extremism and thus affecting development. In this paper, therefore, a nonlinear mathematical model is proposed to study the dynamics of extremism governed by four dependent variables, namely, number of people in the general population having no extreme ideology, number of extreme ideologists, number of isolated ideologists (prisoners), and the cumulative density of government efforts and their interactions. The model is analyzed using the stability theory of differential equations and computer simulation. The analysis shows that if appropriate level of government efforts is applied on extremists, the spread of their ideology can be controlled in the general population. A numerical study of the model is also carried out to investigate the effects of certain parameters on the spread of extremism confirming the analytical results.Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we consider the nonlocal non-autonomous evolution problems where Ω is a bounded smooth domain in
, *N*≥1, *β* is a positive constant, the coefficient *a* is a continuous bounded function on
, and *K* is an integral operator with symmetric kernel
, being *J* a non-negative function continuously differentiable on
and
. We prove the existence of global pullback attractor, and we exhibit a functional to evolution process generated by this problem that decreases along of solutions. Assuming the parameter *β* is small enough, we show that the origin is locally pullback asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.

Immunotherapies are important methods for controlling and curing malignant tumors. Based on recent observations that many tumors have been immuno-selected to evade recognition by the traditional cytotoxic T lymphocytes, we propose mathematical models of tumor–CD4^{+}–cytokine interactions to investigate the role of CD4^{+} on tumor regression. Treatments of either CD4^{+} or cytokine are applied to study their effectiveness. It is found that doses of treatments are critical in determining the fate of the tumor, and tumor cells can be eliminated completely if doses of cytokine are large. Bistability is observed in models with either of the treatment strategies, which signifies that a careful planning of the treatment strategy is necessary for achieving a satisfactory outcome. Copyright © 2014 John Wiley & Sons, Ltd.

A kind of *N* × *N* non-semisimple Lie algebra consisting of triangular block matrices is used to generate multi-component integrable couplings of soliton hierarchies from zero curvature equations. Two illustrative examples are made for the continuous Ablowitz–Kaup–Newell–Segur hierarchy and the semi-discrete Volterra hierarchy, together with recursion operators. Copyright © 2014 John Wiley & Sons, Ltd.

We study in this paper the Q-symmetry and conditional Q-symmetries of Drinfel'd–Sokolov–Wilson equations. The solutions which we obtain in this paper take the form of convergent power series with easily computable components. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Over the past few years, considerable attention has been given to the role played by the Lamé's Wave Functions (LWFs) in various problems of mathematical physics and mechanics. The LWFs arise via the method of separation of variables for the wave equation in ellipsoidal coordinates. The present paper introduces the Lamé's Quaternionic Wave Functions (LQWFs), which extend the LWFs to a non-commutative framework. We show that the theory of the LQWFs is determined by the Moisil-Theodorescu type operator with quaternionic variable coefficients. As a result, we explain the connections between the solutions of the Lamé's wave equation, on one hand, and the quaternionic hyperholomorphic and anti-hyperholomorphic functions on the other. We establish analogues of the basic integral formulas of complex analysis such as Borel-Pompeiu's, Cauchy's, and so on, for this version of quaternionic function theory. We further obtain analogues of the boundary value properties of the LQWFs such as Sokhotski-Plemelj formulae, the -hyperholomorphic extension of a given Hölder function and on the square of the singular integral operator. We address all the text mentioned earlier and explore some basic facts of the arising quaternionic function theory. We conclude the paper showing that the spherical, prolate, and oblate spheroidal quaternionic wave functions can be generated as particular cases of the LQWFs. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we study behavior of the solution of the following max-type difference equation system:

where
, the parameter *A* is positive real number, and the initial values *x*_{0},*y*_{0} are positive real numbers. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we are concerned with the inverse Sturm–Liouville problem with polynomially dependent eigenparameter in discontinuity and boundary conditions. By using a self-adjoint operator-theoretic interpretation for this sort of problem, Ambarzumyan theorem is provided for the mentioned Sturm–Liouville operator. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, a delayed eco-epidemiological model with Holling type II functional response is investigated. By analyzing corresponding characteristic equations, the local stability of each of the feasible equilibria and the existence of Hopf bifurcations at the disease-free equilibrium, the susceptible predator-free equilibrium and the endemic-coexistence equilibrium are established, respectively. By means of Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are derived for the global stability of the endemic-coexistence equilibrium, the disease-free equilibrium, the susceptible predator-free equilibrium and the predator-extinction equilibrium of the system, respectively. Numerical simulations are carried out to illustrate the theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.

]]>First-order systems in
on
with absolutely continuous real symmetric *π*-periodic matrix potentials are considered. A thorough analysis of the discriminant is given. Interlacing of the eigenvalues of the periodic, antiperiodic and Dirichlet-type boundary value problems on [0,*π*] is shown for a suitable indexing of the eigenvalues. The periodic and antiperiodic eigenvalues are characterized in terms of Dirichlet-type eigenvalues. It is shown that all instability intervals vanish if and only if the potential is the product of an absolutely continuous real scalar valued function with the identity matrix. Copyright © 2014 John Wiley & Sons, Ltd.

In a fairly recent paper (2008 American Control Conference, June 11-13, 1035-1039), the problem of dealing with trading in optimal pairs was treated from the viewpoint of stochastic control. The analysis of the subsequent nonlinear evolution partial differential equation was based upon a succession of Ansätze, which can lead to a solution of the terminal-value problem. Through an application of the Lie Theory of Continuous Groups to this equation, we show that the Ansätze are based upon the underlying symmetries of the equation (their (14)). We solve the problem in a more general context by allowing the parameters to be explicitly time dependent. The extension means thatmore realistic problems are amenable to the samemode of solution. Copyright © 2014 JohnWiley & Sons, Ltd.

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