In this paper, three conservative finite volume element schemes are proposed and compared for the modif ied Korteweg–de Vries equation, especially with regard to their accuracy and conservative properties. The schemes are constructed basing on the discrete variational derivative method and the finite volume element method to inherit the properties of the original equation. The theoretical analysis show that three schemes are conservative under suitable boundary conditions as well as unconditionally linear stability. Numerical experiments are given to confirm the theoretical results and the capacity of the proposed methods for capturing the solitary wave phenomena. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study the properties of the Lagrange multiplier for an Allen–Cahn equation with a double obstacle potential. Here, the dynamic boundary condition, including the Laplace–Beltrami operator on the boundary, is investigated. We then establish the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier of our problem. We present remarks on a trace problem as well as on the Neumann boundary condition. Moreover, we describe a numerical experiment for a problem with Neumann boundary condition using the Lagrange multiplier. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We point out some mistakes in a known paper. Some existence results for solutions of two classes of boundary value problems for nonlinear impulsive fractional differential equations are established. Our analysis relies on the well-known Schauder fixed point theorem. Examples are given to illustrate the main results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The aim of this paper is to propose optimal strategies for dengue reduction and prevention in Cali, Colombia. For this purpose, we consider two variants of a simple dengue transmission model, epidemic and endemic, each of which is amended with two control variables. These variables express feasible control actions to be taken by an external decision-maker. First control variable stands for the insecticide spraying and thus targets to suppress the vector population. The second one expresses the protective measures (such as use of repellents, mosquito nets, and insecticide-treated clothes) that are destined to reduce the number of contacts (bites) between female mosquitoes (principal dengue transmitters) and human individuals. We use the Pontryagin's maximum principle in order to derive the optimal strategies for dengue control and then perform the cost-effectiveness analysis of these strategies in order to choose the most sustainable one in terms of cost–benefit relationship. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we establish some new sufficient conditions on the existence of homoclinic solution for a class of second-order impulsive Hamiltonian systems. By using the mountain pass theorem, we demonstrate that the limit of a 2*k**T*-periodic approximation solution is a homoclinic solution of our problem. We also present some examples to illustrate the applications of our main results. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, a new impulsive Lasota–Wazewska model with patch structure and forced perturbed terms is proposed and analyzed on almost periodic time scales. For this, we introduce the concept of matrix measure on time scales and obtain some of its properties. Then, sufficient conditions are established which ensure the existence and exponential stability of positive almost periodic solutions of the proposed biological model. Our results are new even when the time scale is almost periodic, in particular, for periodic time scales on or . An example is given to illustrate the theory. Finally, we present some phenomena which are triggered by almost periodic time scales. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper deals with constructing generalized ‘fractional’ power series representation for solutions of fractional order differential equations. We present a brief review of generalized Taylor's series and generalized differential transform methods. Then, we study the convergence of fractional power series. Our emphasis is to address the sufficient condition for convergence and to estimate the truncated error. Numerical simulations are performed to estimate maximum absolute truncated error when the generalized differential transform method is used to solve non-linear differential equations of fractional order. The study highlights the power of the generalized differential transform method as a tool in obtaining fractional power series solutions for differential equations of fractional order. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The Melan beam equation modeling suspension bridges is considered. A slightly modified equation is derived by applying variational principles and by minimizing the total energy of the bridge. The equation is nonlinear and nonlocal, while the beam is hinged at the endpoints. We show that the problem always admits at least one solution whereas the uniqueness remains open although some numerical results suggest that it should hold. We also emphasize the qualitative difference with some simplified models. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In a recent paper, the notion of *quantum perceptron* has been introduced in connection with projection operators. Here, we extend this idea, using these kind of operators to produce a *clustering machine*, that is, a framework that generates different clusters from a set of input data. Also, we consider what happens when the orthonormal bases first used in the definition of the projectors are replaced by frames and how these can be useful when trying to connect some noised signal to a given cluster. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider the one-dimensional Schrödinger operator on bounded time scales. We construct a space of boundary values of the minimal operator and describe all maximal dissipative, maximal accretive, self-adjoint, and other extensions of the dissipative Schrödinger operators in terms of boundary conditions. In particular, using Lidskii's theorem, we prove a theorem on completeness of the system of root vectors of the dissipative Schrödinger operators on bounded time scales. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The present contribution is concerned with an analytical presentation of the low-frequency electromagnetic fields, which are scattered off a highly conductive ring torus that is embedded within an otherwise lossless ambient and interacting with a time-harmonic magnetic dipole of arbitrary orientation, located nearby in the three-dimensional space. Therein, the particular 3-D scattering boundary value problem is modeled with respect to the solid impenetrable torus-shaped body, where the toroidal geometry fits perfectly. The incident, the scattered, and the total non-axisymmetric magnetic and electric fields are expanded in terms of positive integral powers of the real-valued wave number of the exterior medium at the low-frequency regime, whereas the static Rayleigh approximation and the first three dynamic terms provide the most significant part of the solution, because all the additional terms are small contributors and, hence, they are neglected. Consequently, the typical Maxwell-type physical problem is transformed into intertwined either Laplace's or Poisson's potential-type boundary value problems with the proper conditions, attached to the metallic surface of the torus. The fields of interest assume representations via infinite series expansions in terms of standard toroidal eigenfunctions, obtaining in that way analytical closed-form solutions in a compact fashion. Although this mathematical procedure leads to infinite linear systems for every single case, these can be readily and approximately solved at a certain level of desired accuracy through standard cut-off techniques. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study a multiphasic incompressible fluid model, called the Kazhikhov–Smagulov model, with a particular viscous stress tensor, introduced by Bresch and co-authors, and a specific diffusive interface term introduced for the first time by Korteweg in 1901. We prove that this model is globally well posed in a 3D bounded domain. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The current article devoted on the new method for finding the exact solutions of some time-fractional Korteweg–de Vries (KdV) type equations appearing in shallow water waves. We employ the new method here for time-fractional equations viz. time-fractional KdV-Burgers and KdV-mKdV equations for finding the exact solutions. We use here the fractional complex transform accompanied by properties of local fractional calculus for reduction of fractional partial differential equations to ordinary differential equations. The obtained results are demonstrated by graphs for the new solutions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a class of cellular neural networks with neutral proportional delays and time-varying leakage delays is considered. Some results on the finite-time stability for the equations are obtained by using the differential inequality technique. In addition, an example with numerical simulations is given to illustrate our results, and the generalized exponential synchronization is also established. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we introduced a summation-integral type modification of Szász–Mirakjan operators. Calculation of moments, density in some space, a direct result and a Voronvskaja-type result, are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we establish finite-region stability (FRS) and finite-region boundedness analysis methods to investigate the transient behavior of discrete two-dimensional Roesser models. First, by building special recursive formulas, a sufficient FRS condition is built via solvable linear matrix inequalities constraints. Next, by designing state feedback controllers, the finite-region stabilization issue is analyzed for the corresponding two-dimensional closed-loop system. Similar to FRS analysis, the finite-region boundedness problem is addressed for Roesser models with exogenous disturbances and corresponding criteria, and linear matrix inequalities conditions are reported. To conclude the paper, we provide numerical examples to confirm the validity of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study a cantilevered Euler–Bernoulli beam fixed to a base in a translational motion at one end and to a tip mass at its free end. The beam is subject to undesirable vibrations, and it is made of a viscoelastic material that permits a certain weak damping. By applying a control force at the base, we shall attenuate these vibrations in a fast manner. In fact, we establish the exponential stability of the system. Our method is based on the multiplier technique. Copyright © 2016 John Wiley & Sons, Ltd.

]]>It is well known that the damping term will give more smooth effect to obtain global solutions. In this paper, we consider the effect of damping term on the solutions to system of inhomogeneous wave equation with damping term. We can obtain the singularity that will be formed in finite time for some large initial data. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider a version of the stationary phase method in one dimension of A. Erdélyi, allowing the phase to have stationary points of non-integer order and the amplitude to have integrable singularities. After having completed the original proof and improved the error estimate in the case of regular amplitude, we consider a modification of the method by replacing the smooth cut-off function employed in the source by a characteristic function, leading to more precise remainder estimates. We exploit this refinement to study the time-asymptotic behaviour of the solution of the free Schrödinger equation on the line, where the Fourier transform of the initial data is compactly supported and has a singularity. We obtain asymptotic expansions with respect to time in certain space-time cones as well as uniform and optimal estimates in curved regions, which are asymptotically larger than any space-time cone. These results show the influence of the frequency band and of the singularity on the propagation and on the decay of the wave packets. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with the mathematical analysis of the electromagnetic wave scattering by an unbounded dielectric medium, which is mounted on a perfectly conducting infinite plane. By introducing a transparent boundary condition on a plane surface confining the medium, the scattering problem is modeled as a boundary value problem of Maxwell's equations. Based on a variational formulation, the problem is shown to have a unique weak solution for a wide class of dielectric permittivity and magnetic permeability by using the generalized Lax–Milgram theorem. Copyright © 2016 John Wiley & Sons, Ltd.

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

]]>In this study, by using the concepts and results on spherical curves in dual Lorentzian space, we give the criterions for ruled surfaces with non-lightlike ruling to be closed (periodic). Moreover, we obtain the necessary and sufficient conditions to guarantee that a timelike or a spacelike ruled surface is closed (periodic). Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the stability problems and *L*_{2}-gain analysis for switched singular linear systems with jumps. Based on the concept of average impulsive interval, some novel sufficient conditions on the stability and *L*_{2}-gain for switched singular linear systems with jumps are developed. Two examples are given to illustrate the effectiveness of the results. Copyright © 2016 John Wiley & Sons, Ltd.

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

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

]]>This paper deals with the existence of positive solutions of the discrete counterpart of nonlinear elliptic problems. We apply our methods to the study of positive solutions under different hypotheses about the nonlinearities. For example, we consider the cases that are superlinear and sublinear at infinity. Copyright © 2016 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.

]]>An important question of electroencephalography and magnetoencephalography is associated with the possibility to identify the number of simultaneously activated areas in the brain. In the present paper, employing a homogeneous spherical conductor, serving as a geometrical model of the brain, we provide a criterion that determines whether the measured surface potential is caused by a single or a double localized neuronal excitations. We present the necessary and sufficient conditions, which decided whether the collected data originates from a single or from a set of two or more dipoles. Furthermore, we investigate the impossibility of deciding when we cannot be sure for the existence of one or two excitation centres. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we establish new Hartman–Wintner-type inequalities for a class of nonlocal fractional boundary value problems. As an application, we obtain a lower bound for the eigenvalues of corresponding equations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We provide sufficient conditions for the nonexistence of nontrivial nonnegative solutions for some nonlinear elliptic inequalities involving the fractional Laplace operator and variable exponents. The used techniques are based on the test function method. 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 this paper, we study the global existence and asymptotic behavior of the Boussinesq-Burgers system subject to the Dirichlet boundary conditions. Based on the *L*^{p}(*p* > 2) estimates of the solution, which are different from the standard *L*^{2}-based energy methods, we show that the classical solutions exist globally and converge to their boundary data at an exponential decay rate as time goes to infinity for large initial data. Copyright © 2016 John Wiley & Sons, Ltd.

In a singular limit, the Klein–Gordon (KG) equation can be derived from the Klein–Gordon–Zakharov (KGZ) system. We point out that for the original system posed on a *d*-dimensional torus, the solutions of the KG equation do not approximate the solutions of the KGZ system. The KG system has to be modified to make correct predictions about the dynamics of the KGZ system. We explain that this modification is not necessary for the approximation result for the whole space
with *d*≥3. Copyright © 2016 John Wiley & Sons, Ltd.

The present paper first introduces the notion of quaternion infinite series of positive term and establishes its several tests. Next, we give the definitions of the positive-definite quaternion sequence and the positive semi-definite quaternion function, and we extend the classical Herglotz's theorem to the quaternion linear canonical transform setting. Then we investigate the properties of the two-sided quaternion linear canonical transform, such as time shift characteristics and differential characteristics. Finally, we derive its several basic properties of the quaternion linear canonical transform of a probability measure, in particular, and establish the Bochner–Minlos theorem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, the ultimate bound for a new chaotic system is derived based on stability theory of dynamical systems. The meaningful contribution of this article is that the results presented in this paper contain the existing results as special cases. Finally, numerical simulations are given to verify the effectiveness and correctness of the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a nonperiodic discrete nonlinear equation with Jacobi operators is considered. By using the critical point theory, we establish some new sufficient conditions on the existence and multiplicity of homoclinic solutions. Recent results are generalized and significantly improved. Furthermore, our results greatly improve some existing ones even for some special cases. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We obtain in this paper the expression of the solutions of the following recursive sequences:

where the initial conditions are arbitrary real numbers. Also, we study the behavior of the solution of these equations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we will propose a Durrmeyer variant of *q*-Bernstein–Schurer operators. A Bohman–Korovkin-type approximation theorem of these operators is considered. The rate of convergence by using the first modulus of smoothness is computed. The statistical approximation of these operators is also studied. Copyright © 2016 John Wiley & Sons, Ltd.

Analytical study to an alpha-regularization of the Boussinesq system is performed using Fourier theory. Existence and uniqueness of strong solution are proved. Convergence results of the unique strong solution for the regularized Boussinesq system to the unique strong solution for the Boussinesq system are established as the regularizing parameter vanishes. The proofs are performed in the frequency space. We use energy methods, Friedrichs's approximation scheme, and Arselà–Ascoli compactness theorem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Dynamical behavior of many nonlinear systems can be described by fractional-order equations. This study is devoted to fractional differential–difference equations of rational type. Our focus is on the construction of exact solutions by means of the (G'/G)-expansion method coupled with the so-called fractional complex transform. The solution procedure is elucidated through two generalized time-fractional differential–difference equations of rational type. As a result, three types of discrete solutions emerged: hyperbolic, trigonometric, and rational. Copyright © 2016 John Wiley & Sons, Ltd.

]]>A rate of *rational decay* is obtained for smooth solutions of a PDE model, which has been used in the literature to describe structural acoustic flows. This structural acoustics model is composed of two distinct PDE systems: (i) a wave equation, to model the interior acoustic flow within the given cavity Ω and (ii) a structurally damped elastic equation, to describe time-evolving displacements along the flexible portion Γ_{0} of the cavity walls. Moreover, the extent of damping in this elastic component is quantified by parameter *η*∈[0,1]. The coupling between these two distinct dynamics occurs across the boundary interface Γ_{0}. Our main result is the derivation of uniform decay rates for classical solutions of this particular structural acoustic PDE, decay rates that are obtained without incorporating any additional boundary dissipative feedback mechanisms. In particular, in the case that full Kelvin–Voight damping is present in fourth-order elastic dynamics, that is, the structural acoustics system as it appears in the literature, solutions that correspond to smooth initial data decay at a rate of
. By way of deriving these stability results, necessary *a priori* inequalities for a certain static structural acoustics PDE model are generated here; these inequalities ultimately allow for an application of a recently derived resolvent criterion for rational decay. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we consider a model with one basal resource and two species of predators feeding by the same resource. There are three non-trivial boundary equilibria. One is the saturated state *E*_{K} of the prey without any predator. Other two equilibria, *E*_{1} and *E*_{2}, are the coexistence states of the prey with only one species of predators. Using a high-dimensional shooting method, the Wazewski' principle, we establish the conditions for the existence of traveling wave solutions from *E*_{K} to *E*_{2} and from *E*_{1} to *E*_{2}. These results show that the advantageous species *v*_{2} always win in the competition and exclude species *v*_{1} eventually. Finally, some numerical simulations are presented, and biological interpretations are given. 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, we deal with two-point boundary value problems for nonlinear impulsive Hamiltonian systems with sub-linear or linear growth. A theorem based on the Schauder fixed point theorem is established, which gives a result that yields existence of solutions without implications that solutions must be unique. An upper bound for the solution is also established. Examples are given to illustrate the main result. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Spaces of spinor-valued homogeneous polynomials, and in particular spaces of spinor-valued spherical harmonics, are decomposed in terms of irreducible representations of the symplectic group Sp(*p*). These Fischer decompositions involve spaces of homogeneous, so-called
-monogenic polynomials, the Lie super algebra
being the Howe dual partner to the symplectic group Sp(*p*). In order to obtain Sp(*p*)-irreducibility, this new concept of
-monogenicity has to be introduced as a refinement of quaternionic monogenicity; it is defined by means of the four quaternionic Dirac operators, a scalar Euler operator
underlying the notion of symplectic harmonicity and a multiplicative Clifford algebra operator *P* underlying the decomposition of spinor space into symplectic cells. These operators
and *P*, and their Hermitian conjugates, arise naturally when constructing the Howe dual pair
, the action of which will make the Fischer decomposition multiplicity free. Copyright © 2016 John Wiley & Sons, Ltd.

A virus infection model with time delays and humoral immunity has been investigated. Mathematical analysis shows that the global dynamics of the model is fully determined by the basic reproduction numbers of the virus and the immune response, *R*_{0} and *R*_{1}. The infection-free equilibrium *P*_{0} is globally asymptotically stable when *R*_{0}≤1. The infection equilibrium without immunity *P*_{1} is globally asymptotically stable when *R*_{1}≤1 < *R*_{0}. The infection equilibrium with immunity *P*_{2} is globally asymptotically stable when *R*_{1}>1. The expression of the basic reproduction number of the immune response *R*_{1} implies that the immune response reduces the concentration of free virus as *R*_{1}>1. Copyright © 2016 John Wiley & Sons, Ltd.

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

]]>A dimension splitting method (DSM) with Crank–Nicolson time discrete strategy for a three-dimensional heat equation is proposed. The basic idea is to simulate the three-Dimensional problem by numerically solving a series of two-dimensional problems in parallel fashion. Convergence and error estimation for the DSM scheme are derived in the paper. Numerical experiments demonstrate the feasibility and efficiency of the DSM scheme. Copyright © 2016 John Wiley & Sons, Ltd.

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

]]>Macro-hybrid mixed variational models of two-phase flow, through fractured porous media, are analyzed at the mesoscopic and macroscopic levels. The mesoscopic models are treated in terms of nonoverlapping domain decompositions, in such a manner that the porous rock matrix system and the fracture network interact across rock–rock, rock–fracture, and fracture–fracture interfaces, with flux transmission conditions dualized. Alternatively, the models are scaled to a macroscopic level via an asymptotic process, where the width of the fractures tends to zero, and the fracture network turns out to be an interface system of one less spatial dimension, with variable high permeability. The two-phase flow is characterized by a fractional flow dual mixed variational model. Augmented two-field and three-field variational reformulations are presented for regularization, internal approximations, and macro-hybrid mixed finite element implementation. Also abstract proximal-point penalty-duality algorithms are derived and analyzed for parallel computing. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with the control systems of semilinear fractional evolution hemivariational inequalities and their optimal controls in Banach space. Firstly, the existence of mild solutions is obtained and proved mainly by using a well-known fixed point theorem of multivalued maps and the properties of generalized Clarke subdifferential. Then, by applying generally mild conditions of cost functionals, we investigate the existence results of the optimal controls for fractional differential evolution hemivariational inequalities. Finally, an example is given to demonstrate the applicability of the main results. Copyright © 2016 John Wiley & Sons, Ltd.

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

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

]]>The persistence of subsonic phase boundaries in a multidimensional Van der Waals fluid is analyzed. The phase boundary is considered as a sharp free boundary that connects liquid and vapor bulk phase dynamics given by the isothermal Euler equations. The evolution of the boundary is driven by effects of configurational forces as well as surface tension.

To analyze this problem, the equations and trace conditions are linearized such that one obtains a general hyperbolic initial boundary value problem with higher-order boundary conditions. A global existence theorem for the linearized system with constant coefficients is shown. The proof relies on the normal mode analysis and a linear form in suitable spaces that is defined using an associated adjoint problem. Especially, the associated adjoint problem satisfies the uniform backward in time Kreiss–Lopatinskiĭ condition. A new energy-like estimate that also includes surface energy terms leads finally to the uniqueness and regularity for the found solutions of the problem in weighted spaces. Copyright © 2016 John Wiley & Sons, Ltd.

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

In this work, we consider the interior transmission eigenvalue problem for a spherically stratified medium, which can be formulated as *y*^{′′}(*r*) + *k*^{2}*η*(*r*)*y*(*r*) = 0 endowed with boundary conditions
, where the refractive index *η*(*r*) is positive and real. We obtain the distribution of transmission eigenvalues under assumptions that
and one of these conditions that (i) *η*(1) ≠ 1, (ii) *η*(1) = 1,*η*′(1) ≠ 0, and (iii) *η*(1) = 1,*η*′(1) = 0,*η*^{″}(1) ≠ 0, respectively. Moreover, in the case *a* = 1, we prove that if partial information on *η*(*r*) is known on subdomain, then only a part of eigenvalues can uniquely determine *η*(*r*) on the whole interval, and the relationship between the proportion of the missing eigenvalues and the subinterval of the known information on *η*(*r*) is revealed. Copyright © 2016 John Wiley & Sons, Ltd.

We analyze a bounded confidence model, introduced by Krause, on isolated time scales. In this model, each agent takes into account only the assessments of the agents whose opinions are not too far away from its own opinion. We show that the behavior of the model depends strongly on the graininess function *μ*: If *μ* takes values in the interval ]0,1], then our discrete time scale model behaves similarly to the classical one, but if *μ* takes values in ]1,+*∞*[, then the model has different properties. Simulations are performed to validate the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

Optical coherence tomography (OCT) and photoacoustic tomography are emerging non-invasive biological and medical imaging techniques. It is a recent trend in experimental science to design experiments that perform photoacoustic tomography and OCT imaging at once. In this paper, we present a mathematical model describing the dual experiment. Because OCT is mathematically modelled by Maxwell's equations or some simplifications of it, whereas the light propagation in quantitative photoacoustics is modelled by (simplifications of) the radiative transfer equation, the first step in the derivation of a mathematical model of the dual experiment is to obtain a unified mathematical description, which in our case are Maxwell's equations. As a by-product, we therefore derive a new mathematical model of photoacoustic tomography based on Maxwell's equations. It is well known by now that without additional assumptions on the medium, it is not possible to uniquely reconstruct all optical parameters from either one of these modalities alone. We show that in the combined approach, one has additional information, compared with a single modality, and the inverse problem of reconstruction of the optical parameters becomes feasible. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study a nonlinear PDE problem motivated by the peculiar patterns arising in myxobacteria, namely, counter-migrating cell density waves. We rigorously prove the existence of Hopf bifurcations for some specific values of the parameters of the system. This shows the existence of periodic solutions for the systems under consideration. Copyright © 2016 John Wiley & Sons, Ltd.

]]>No abstract is available for this article.

]]>The construction of Brownian motion paths is the most important part of simulation methods for option pricing. Particularly, there are several commonly used path generation methods in the context of quasi-Monte Carlo, including the standard method and the Brownian bridge method. To apply each method, an inevitable step is to decide how many points are used to discretize the time interval. This paper implements an iterative algorithm to select a suitable number of time steps by successively adding discretization nodes until a specific convergence criterion is met. Numerical results with this algorithm are presented in the valuation of Asian options. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the coupled system of Kirchhoff-type equations:

where 4 < *τ* < 6, *a*,*c* > 0, *b*,*d*≥0 are constants and *λ* is a positive parameter. The main purpose of this paper is to study the existence of ground state solutions for the aforementioned system with a nonlinearity in the critical growth under some suitable assumptions on *V* and *F*. Recent results from the literature are improved and extended. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we prove the global well-posedness of non-Newtonian viscous fluid flow of the Oldroyd-B model with free surface in a bounded domain of *N*-dimensional Euclidean space
. The assumption of the problem is that the initial data are small enough and orthogonal to rigid motions. Copyright © 2015 John Wiley & Sons, Ltd.

We propose a model for unsaturated poro-plastic flow derived from the thermodynamic principles. For the isothermal case, the problem consists of a degenerate coupled system of two PDEs with two independent hysteresis operators describing hysteresis phenomena in both the solid and the pore fluids. Under natural hypotheses, we prove the existence of a global strong solution for this system. 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.

By application of Green's function and some fixed-point theorems, that is, Leray–Schauder alternative principle and Schauder's fixed point theorem, we establish two new existence results of positive periodic solutions for nonlinear fourth-order singular differential equation with variable-coefficient, which extend and improve significantly existing results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a new finite volume scheme for the numerical solution of the pure aggregation population balance equation, or Smoluchowski equation, on non-uniform meshes is derived. The main feature of the new method is its simple mathematical structure and high accuracy with respect to the number density distribution as well as its moments. The new method is compared with the existing schemes given by Filbet and Laurençot (SIAM J. Sci. Comput., 25 (2004), pp. 2004–2028) and Forestier and Mancini (SIAM J. Sci. Comput., 34 (2012), pp. B840–B860) for selected benchmark problems. It is shown that the new scheme preserves all the advantages of a conventional finite volume scheme and predicts higher-order moments as well as number density distribution with high accuracy. 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.

]]>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.

]]>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.

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.

The paper is devoted to the study of a system of semilinear wave equations associated with the helical flows of Maxwell fluid. First, based on Faedo–Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we establish two local existence theorems of weak solutions. Next, we prove that any weak solutions with negative initial energy will blow up in finite time. Finally, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions via the construction of a suitable Lyapunov functional. 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.

]]>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.

]]>In this paper, we first propose the so-called improved split-step theta methods for non-autonomous stochastic differential equations driven by non-commutative noise. Then, we prove that the improved split-step theta method is convergent with strong order of one for stochastic differential equations with the drift coefficient satisfying a superlinearly growing condition and a one-sided Lipschitz continuous condition. Finally, the obtained results are verified by numerical experiments. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this study, we introduce the Durrmeyer type Jakimoski–Leviatan operators and examine their approximation properties. We study the local approximation properties of these operators. Further, we investigate the convergence of these operators in a weighted space of functions and obtain the approximation properties. Furthermore, we give a Voronovskaja type theorem for the our new operators. Copyright © 2015 John Wiley & Sons, Ltd.

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