In this paper, we study a system of nonlinear coupled wave equations with damping, source, and nonlinear strain terms. We obtain several results concerning local existence, global existence, and finite time blow-up property with positive initial energy by using Galerkin method and energy method, respectively. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we perform global stability analysis of a multi-group SEIR epidemic model in which we can consider the heterogeneity of host population and the effects of latency and nonlinear incidence rates. For a simpler version that assumes an identical natural death rate for all groups, and with a gamma distribution for the latency, the basic reproduction number is defined by the theory of the next generation operator and proved to be a sharp threshold determining whether or not disease spread. Under certain assumptions, the disease-free equilibrium is globally asymptotically stable if *R*_{0}≤1 and there exists a unique endemic equilibrium which is globally asymptotically stable if *R*_{0}>1. The proofs of global stability of equilibria exploit a matrix-theoretic method using Perron eigenvetor, a graph-theoretic method based on Kirchhoff's matrix tree theorem and Lyapunov functionals. Copyright © 2015 John Wiley & Sons, Ltd.

This work is a natural continuation of an earlier one in which a mathematical model has been studied. This model is based on an age–cycle length structured cell population. The cellular mitosis is mathematically described by a non-compact boundary condition. We investigate the spectral properties of the generated semigroup, and we give an explicit estimation of its type. Copyright © 2015 John Wiley & Sons, Ltd.

]]>H. Ammari In this article, an innovative technique so-called spectral meshless radial point interpolation (SMRPI) method is proposed and, as a test problem, is applied to a classical type of two-dimensional time-fractional telegraph equation defined by Caputo sense for (1 < *α*≤2). This new methods is based on meshless methods and benefits from spectral collocation ideas, but it does not belong to traditional meshless collocation methods. The point interpolation method with the help of radial basis functions is used to construct shape functions, which play as basis functions in the frame of SMRPI method. These basis functions have Kronecker delta function property. Evaluation of high-order derivatives is not difficult by constructing operational matrices. In SMRPI method, it does not require any kind of integration locally or globally over small quadrature domains, which is essential of the finite element method (FEM) and those meshless methods based on Galerkin weak form. Also, it is not needed to determine strict value for the shape parameter, which plays an important role in collocation method based on the radial basis functions (Kansa's method). Therefore, computational costs of SMRPI method are less expensive. Two numerical examples are presented to show that SMRPI method has reliable rates of convergence. Copyright © 2015 John Wiley & Sons, Ltd.

This paper presents a shifted fractional-order Jacobi orthogonal function (SFJF) based on the definition of the classical Jacobi polynomial. A new fractional integral operational matrix of the SFJF is presented and derived. We propose the spectral Tau method, in conjunction with the operational matrices of the Riemann–Liouville fractional integral for SFJF and derivative for Jacobi polynomial, to solve a class of time-fractional partial differential equations with variable coefficients. In this algorithm, the approximate solution is expanded by means of both SFJFs for temporal discretization and Jacobi polynomials for spatial discretization. The proposed tau scheme, both in temporal and spatial discretizations, successfully reduced such problem into a system of algebraic equations, which is far easier to be solved. Numerical results are provided to demonstrate the high accuracy and superiority of the proposed algorithm over existing ones. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In the paper entitled ‘A novel chaotic system and its topological horseshoe’ in [Nonlinear Analysis: Modelling and Control 18 (1) (2013) 66–77], proposed the 3D chaotic system,
, and discussed some of its dynamics according to theoretical and numerical analysis of its parameters
. The present work is devoted to giving some new insights into the system for *b*≥0. Combining theoretical analysis and numerical simulations, some new results are formulated. On the one hand, after some known errors, mainly the distribution of its equilibrium point which is pointed out, correct results are formulated. On the other hand, some of its more rich dynamical properties hiding and not found previously, such as the stability, fold bifurcation, pitchfork bifurcation, degenerated pitchfork bifurcation, and Hopf bifurcation of its isolated equilibria, the dynamics of non-isolated equilibria, the singularly degenerate heteroclinic cycle, the heteroclinic orbit, and the dynamics at infinity are clearly revealed. Using these results, one can easily explain those interesting phenomena for invariant Lyapunov exponent spectrum and amplitude control that are presented in the known literature. What is more important, we probably demonstrate a new route to chaos. Copyright © 2015 John Wiley & Sons, Ltd.

Consider the following two critical nonlinear Schrödinger systems:

- (0.1)

- (0.2)

where
is a smooth bounded domain, *N*≥3,−*λ*(Ω) < *λ*_{1},*λ*_{2}<0,*μ*_{1},*μ*_{2}>0,*α*,*β*≥1 with *α* + *β* = 2^{∗},*γ* ≠ 0,*λ*(Ω) is the first eigenvalue of −Δ with the Dirichlet boundary condition and

For *N* = 3,*λ*_{1}=*λ*_{2},*γ* > 0 small, we obtain the existence of positive least energy solution of (0.1) and (0.2). For *N*≥5,*γ* > 0, the existence of positive least energy solution of (0.2) is established. For *N*≥5,*γ* ≠ 0, we prove that (0.1) possesses a positive least energy solution. The limit behavior of the positive least energy solutions when *γ*−*∞* and phase separation for (0.1) are also considered. Copyright © 2015 John Wiley & Sons, Ltd.

In this research, the non-relativistic particle scattering has been investigated for an alternative pseudo-Coulomb potential plus ring-shaped and an energy-dependent potentials in *D*-dimensional space. The normalized wave functions of continuous states on the *k*/2*π* scale are expressed in terms of the hyper-geometric series, and formula of phase shifts is presented. Analytical properties of the scattering amplitude and thermodynamics properties are discussed. Some of the numerical results of energy levels have been calculated too. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we study a class of damped vibration problems with nonlinearities being sublinear at both zero and infinity, and we obtain infinitely many nontrivial periodic solutions by using a variant fountain theorem. To the best of our knowledge, there is no published result concerning this case by this method. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Pre-operative planning of percutaneous thermal ablations is a difficult but decisive task for a safe and successful intervention. The purpose of our research is to assist surgeons in preparing cryoablation with an automatic pre-operative path planning algorithm able to propose a placement for multiple needles in three dimensions. The aim is to optimize the tumor coverage problem while taking into account a precise computation of the frozen area. Using an implementation of the precise estimation of the ice balls, this study focuses on the optimization in an acceptable time of multiple probes positions with 5 degrees of freedom, regarding the constraint of optimal volumetric coverage of the tumor by the combined necrosis. Pennes equation was used to solve the propagation of cold within the tissues, and included in an objective function of the optimization process. The propagation computation being time-consuming, seven optimization algorithms from the literature were experimented under different conditions and compared, in order to reduce overall computation time while preserving precision. Moreover, several hybrid algorithms were tested to reduce required time for the computations. Some of these methods were found suitable for the conditions of our cryosurgery planning. We conclude that this combination of bioheat simulation and optimization can be appropriate for a use by practitioners in acceptable conditions of time and precision. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper deals with a fractional two-times evolution equation associated with initial and purely boundary integral conditions. The existence and uniqueness of generalized solution are proved. The classical functional method based on a priori estimates and density used by many authors in the case of nonfractional differential equations is applied for the time fractional case. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we consider a discrete fractional boundary value problem of the form

where 0 < *α*,*β*≤1, 1 < *α* + *β*≤2, 0 < *γ*≤1,
, *ρ* is a constant,
and
denote the Caputo fractional differences of order *α* and *β*, respectively,
is a continuous function, and *ϕ*_{p} is the *p*-Laplacian operator. The existence of at least one solution is proved by using Banach fixed point theorem and Schaefer's fixed point theorem. Some illustrative examples are also presented. Copyright © 2015 John Wiley & Sons, Ltd.

We show well posedness for a model of nonlinear reactive transport of chemical in a deformable channel. The channel walls deform due to fluid–structure interaction between an unsteady flow of an incompressible, viscous fluid inside the channel and elastic channel walls. Chemical solutes, which are dissolved in the viscous, incompressible fluid, satisfy a convection–diffusion equation in the bulk fluid, while on the deforming walls, the solutes undergo nonlinear adsorption–desorption physico-chemical reactions. The problem addresses scenarios that arise, for example, in studies of drug transport in blood vessels. We show the existence of a unique weak solution with solute concentrations that are non-negative for all times. The analysis of the problem is carried out in the context of semi-linear parabolic PDEs on moving domains. The arbitrary Lagrangian–Eulerian approach is used to address the domain movement, and the Galerkin method with the Picard–Lindelöf theorem is used to prove existence and uniqueness of approximate solutions. Energy estimates combined with the compactness arguments based on the Aubin–Lions lemma are used to prove convergence of the approximating sequences to the unique weak solution of the problem. It is shown that the solution satisfies the positivity property, that is, that the density of the solute remains non-negative at all times, as long as the prescribed fluid domain motion is ‘reasonable’. This is the first well-posedness result for reactive transport problems defined on moving domains of this type. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we continue the study of the hyperbolic relaxation of the Cahn–Hilliard–Oono equation with the sub-quintic non-linearity in the whole space started in our previous paper and verify that under the natural assumptions on the non-linearity and the external force, the fractal dimension of the associated global attractor in the natural energy space is finite. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We study the following nonlinear Schrödinger system with magnetic potentials in :

where *μ*_{1}>0, *μ*_{2}>0, and
is a coupling constant. Under some weak symmetry conditions on *A*(*y*), *P*(*y*), and *Q*(*y*), which are given in the introduction, we prove that the nonlinear Schrödinger system has infinitely many non-radial complex-valued segregated and synchronized solutions. Copyright © 2015 John Wiley & Sons, Ltd.

A steady-state Poisson–Nernst–Planck system is investigated, which is conformed into a nonlinear Poisson equation by means of the Boltzmann statistics. It describes the electrostatic potential generated by multiple concentrations of ions in a heterogeneous (porous) medium with diluted (solid) particles. The nonlinear elliptic problem is singularly perturbed with the Debye length as a small parameter related to the electric double layer near the solid particle boundary. For star-shaped solid particles, we prove rigorously that the solution of the problem in spatial dimensions 1d, 2d and 3d is uniformly and super-asymptotically approximated by a constant reference state. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper deals with optimal control problems constrained by linear elliptic partial differential equations. The case where the right-hand side of the Neumann boundary is controlled, is studied. The variational discretization concept for these problems is applied, and discretization error estimates are derived. On polyhedral domains, one has to deal with edge and corner singularities, which reduce the convergence rate of the discrete solutions, that is, one cannot expect convergence order two for linear finite elements on quasi-uniform meshes in general. As a remedy, a local mesh refinement strategy is presented, and a priori bounds for the refinement parameters are derived such that convergence with optimal rate is guaranteed. As a by-product, finite element error estimates in the *H*^{1}(Ω)-norm, *L*^{2}(Ω)-norm and *L*^{2}(Γ)-norm for the boundary value problem are obtained, where the latter one turned out to be the main challenge. Copyright © 2015 John Wiley & Sons, Ltd.

In this note, a critical point result for differentiable functionals is exploited in order to prove that a suitable class of one-dimensional fractional problems admits at least one non-trivial solution under an asymptotical behaviour of the nonlinear datum at zero. A concrete example of an application is then presented. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The band preserving and phase retrieval problems have long been interested and studied. In this paper, we, for the first time, give solutions to these problems in terms of backward shift invariant subspaces. The backward shift method among other methods seems to be direct and natural. We show that a function
, with
, that makes the band of *fg* to be within that of *f* if and only if *g* divided by an inner function related to *f*, belongs to some backward shift invariant subspace in relation to *f*. By the construction of backward shift invariant space, the solution *g* is further explicitly represented through the span of the rational function system whose zeros are those of the Laplace transform of *f*. As an application, we also use the backward shift method to give a characterization for the solutions of the phase retrieval problem. Copyright © 2015 John Wiley & Sons, Ltd.

We first reformulate and expand with several novel findings some of the basic results in the integrability of Abel equations. Next, these results are applied to Vein's Abel equation whose solutions are expressed in terms of the third-order hyperbolic functions, and a phase space analysis of the corresponding nonlinear oscillator is also provided. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In 2002–2003, Paumier studied the Signorini problem with friction in the linear Kirchhoff–Love theory of plates using the convergence method. In 2008, Léger and Miara generalized this study to the case of linearized shallow shell but without friction. The purpose of this paper is to extend those results to the case of linearized shallow shell with a Coulomb friction law. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper investigates the existence of solutions for multi-point boundary value problems of higher-order nonlinear Caputo fractional differential equations with p-Laplacian. Using the five functionals fixed-point theorem, the existence of multiple positive solutions is proved. An example is also given to illustrate the effectiveness of ourmain result. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, the general exact implicit solution of the second-order nonlinear ordinary differential equation governing heat transfer in rectangular fin is obtained using Lie point symmetry method. General relationship among the fin efficiency, the rate of heat transfer from the entire fin, the fin effectiveness, and the thermo-geometric fin parameter is obtained for any value of the mode of heat transfer *n* and the constant *β*. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we study the existence of infinitely many solutions to *p*-Kirchhoff-type equation

- (0.1)

where *f*(*x*,*u*) = *λ**h*_{1}(*x*)|*u*|^{m − 2}*u* + *h*_{2}(*x*)|*u*|^{q − 2}*u*,*a*≥0,*μ* > 0,*τ* > 0,*λ*≥0 and
. The potential function
verifies
, and *h*_{1}(*x*),*h*_{2}(*x*) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists *λ*_{0}>0 such that problem (0.1) admits infinitely many nonnegative high-energy solutions provided that *λ*∈[0,*λ*_{0}) and
. Also, we prove that problem (0.1) has at least a nontrivial solution under the assumption *f*(*x*,*u*) = *h*_{2}|*u*|^{q − 2}*u*,*p* < *q*< min{*p**,*p*(*τ* + 1)} and has infinitely many nonnegative solutions for *f*(*x*,*u*) = *h*_{1}|*u*|^{m − 2}*u*,1 < *m* < *p*. Copyright © 2015 John Wiley & Sons, Ltd.

The purpose of the present paper is to propose an efficient numerical method for solving the differential equations of Bratu-type with fractional order in reproducing kernel Hilbert space. The exact solution is calculated in the form of a convergent series with easily computable components. Finally, some examples are given to illustrate the efficiency and applicability of the method. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The aim of this note is to investigate the existence of signed and sign-changing solutions to the Kirchhoff type problem

- (0.1)

where Ω is a bounded smooth domain in
(*N* = 1,2,3), *a*,*b* > 0 and 2 < *p* < 2^{⋆}, with 2^{⋆}=+*∞* if *N* = 1,2 and 2^{⋆}=6 if *N* = 3. Using variational methods, we show that (0.1) possesses three solutions of mountain pass type (one positive, one negative and one sign-changing) and infinitely many high-energy sign-changing solutions. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, a robust visual tracking method is proposed based on local spatial sparse representation. In the proposed approach, the learned target template is sparsely and compactly expressed by forming local spatial and trivial samples dynamically. An adaptive multiple subspaces appearance model is developed to describe the target appearance and construct the candidate target templates during the tracking process. An effective selection strategy is then employed to select the optimal sparse solution and locate the target accurately in the next frame. The experimental results have demonstrated that our method can perform well in the complex and noisy visual environment, such as heavy occlusions, dramatic illumination changes, and large pose variations in the video. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, the influence of magnetic field on the dispersion of a solute in peristaltic flow of an incompressible micropolar fluid is studied as a model of fluid transport in the human intestinal system with wall properties. Long wavelength approximation, Taylor's limiting condition, and dynamic boundary conditions at the flexible walls are used to obtain the average effective dispersion coefficient in the presence of combined homogeneous and heterogeneous chemical reactions. The effects of various pertinent parameters on the effective dispersion coefficient are discussed. Average effective dispersion coefficient increases with amplitude ratio, which implies that dispersion is more in the presence of peristalsis. It also increases with the cross-viscosity coefficient, heterogeneous chemical reaction rate, and wall parameters. Further, dispersion decreases with micropolar parameter, magnetic parameter, and homogeneous chemical reaction rates. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper is concerned with neutral bidirectional associative memory neural networks with time-varying delays in leakage terms on time scales. Some sufficient conditions on the existence, uniqueness, and global exponential stability of almost-periodic solutions are established. An example is presented to illustrate the feasibility and effectiveness of the obtained results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Quartic C-Bézier curves possess similar properties with the traditional Bézier curves including terminal property, convex hull property, affine invariance, and approaching the shape of their control polygons as the shape parameter *α* decreases. In this paper, by adjusting the shape parameter *α* on the basis of the utilization of the least square approximation and nonlinear functional minimization together with fairing of a quartic C-Bézier curve with *G*^{1} continuity of quartic C-Bézier curve segments, we develop a fairing and *G*^{1} continuity algorithm for any given stitching coefficients *λ*_{k}(*k* = 1,2,*…*,*n* − 1). The shape parameters *α*_{i}(*i*=1, 2, …, *n*) can be adjusted by the value of control points. The curvature of the resulting quartic C-Bézier curve segments after fairing is more uniform than before. Moreover, six examples are provided in the paper to demonstrate the efficacy of the algorithm and illustrate how to apply this algorithm to the computer-aided design/computer-aided manufacturing modeling systems. Copyright © 2015 John Wiley & Sons, Ltd.

Y. Wang In this paper, the finite time extinction of solutions to the fast diffusion system *u*_{t}=div(|∇*u*|^{p − 2}∇*u*) + *v*^{m}, *v*_{t}=div(|∇*v*|^{q − 2}∇*v*) + *u*^{n} is investigated, where 1 < *p*,*q* < 2, *m*,*n* > 0 and
is a bounded smooth domain. After establishing the local existence of weak solutions, the authors show that if *m**n* > (*p* − 1)(*q* − 1), then any solution vanishes in finite time provided that the initial data are ‘comparable’; if *m**n* = (*p* − 1)(*q* − 1) and Ω is suitably small, then the existence of extinction solutions for small initial data is proved by using the De Giorgi iteration process and comparison method. On the other hand, for 1 < *p* = *q* < 2 and *m**n* < (*p* − 1)^{2}, the existence of at least one non-extinction solution for any positive smooth initial data is proved. Copyright © 2015 John Wiley & Sons, Ltd.

S. G. Georgiev, Complete orthogonal systems of monogenic polynomials over 3D prolate spheroids have recently experienced an upsurge of interest because of their many remarkable properties. These generalized polynomials and their applications to the theory of quasi-conformal mappings and approximation theory have played a major role in this development. In particular, the underlying functions of three real variables take on values in the reduced quaternions (identified with ) and are generally assumed to be null-solutions of the well-known Riesz system in . The present paper introduces and explores a new complete orthogonal system of monogenic functions as solutions to this system for the space exterior of a 3D prolate spheroid. This will be made in the linear spaces of square integrable functions over . The representations of these functions are explicitly given. Some important properties of the system are briefly discussed, from which several recurrence formulae for fast computer implementations can be derived. Copyright © 2015 John Wiley & Sons, Ltd.

]]>A new systematic method to find the relative invariant differentiation operators is developed. We incorporate this new approach with Lie's infinitesimal method to study the general class *y*′′′ = *f*(*x*, *y*, *y*′, y′′) under general point equivalence transformations in the generic case.As a result, all third-order differential invariants, relative and absolute invariant differentiation operators are determined for third-order ODEs *y*′′′ = *f*(*x*, *y*, *y*′, *y*′′), which are not quadratic in the second-order derivative. These relative invariant differentiation operators are used to determine the fourth-order differential invariants and absolute invariant differentiation operators in a degenerate case of interest. As an application, invariant descriptions of all the canonical forms in the complex planewith four infinitesimal symmetries for third-order ODEs *y*′′′ = *f*(*x*, *y*, *y*′, y′′), which are not quadratic in the second-order derivative, are provided. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we analyze the energy-conserved splitting finite-difference time-domain (FDTD) scheme for variable coefficient Maxwell's equations in two-dimensional disk domains. The approach is energy-conserved, unconditionally stable, and effective. We strictly prove that the EC-S-FDTD scheme for the variable coefficient Maxwell's equations in disk domains is of second order accuracy both in time and space. It is also strictly proved that the scheme is energy-conserved, and the discrete divergence-free is of second order convergence. Numerical experiments confirm the theoretical results, and practical test is simulated as well to demonstrate the efficiency of the proposed EC-S-FDTD scheme. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The topological derivative concept has been successfully applied in many relevant physics and engineering problems. In particular, the topological asymptotic analysis has been fully developed for a wide range of problems modeled by partial differential equations. In this paper, the topological asymptotic analysis of the energy shape functional associated with a diffusive/convective steady-state heat equation is developed. The topological derivative with respect to the nucleation of a circular inclusion is derived in its closed form with help of a non-standard adjoint state. Finally, we provide the estimates for the remainders of the topological asymptotic expansion and perform a complete mathematical justification for the derived formulas. The obtained result is new and can be applied in the context of topology design of heat sinks, for instance. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we consider a non-local stochastic parabolic equation that actually serves as a mathematical model describing the adiabatic shear banding formation phenomena in strained metals. We first present the derivation of the mathematical model. Then we investigate under which circumstances a finite-time explosion for this non-local stochastic partial differential equation, corresponding to shear banding formation, occurs. For that purpose, some results related to the maximum principle for this non-local stochastic partial differential equation are derived, and afterwards the Kaplan eigenfunction method is employed. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this article, the existence of positive solutions of a boundary value problem for nonlinear singular fractional-order elastic beam equation is established. Here, *f* depends on *t*,*x*, and *x*′; *f* may be singular at *t* = 0 and *t* = 1; and *f* is a non-Carathéodory function. The results obtained are based upon fixed-point theorems in a cone in Banach space. An example is included to illustrate the main results. Copyright © 2015 John Wiley & Sons, Ltd.

The singular solution of the Laplace equation with a straight crack is represented by a series of eigenpairs, shadows, and their associated edge flux intensity functions (EFIFs). We address the computation of the EFIFs associated with the integer eigenvalues by the quasi-dual function method (QDFM). The QDFM is based on the dual eigenpairs and shadows, and we exhibit the presence of logarithmic terms in the dual singularities associated with the integer eigenvalues. These are then used with the QDFM to extract EFIFs from *p*-version finite element solutions. Numerical examples are provided. Copyright © 2015 John Wiley & Sons, Ltd.

T. Monovasilis Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, a numerical method for solving Lane-Emden type equations, which are nonlinear ordinary differential equations on the semi-infinite domain, is presented. The method is based upon the modified rational Bernoulli functions; these functions are first introduced. Operational matrices of derivative and product of modified rational Bernoulli functions are then given and are utilized to reduce the solution of the Lane-Emden type equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we consider the solution to Wente's problem with the fractional Laplace operator (−Δ)^{α/2}, where 0 < *α* < 2. We derive a Wente-type inequality for this problem. Next, we compute the optimal constant in such inequality. Copyright © 2015 John Wiley & Sons, Ltd.

T. Qian In this paper, we prove two theorems on normal families of meromorphic functions, which improve a few of results from several authors. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper deals with the quasilinear Keller–Segel system with rotation

where
is a bounded domain with smooth boundary, *D*(*u*) is supposed to be sufficiently smooth and satisfies *D*(*u*)≥*D*_{0}*u*^{m − 1}(*m*≥1) and *D*(*u*)≤*D*_{1}(*u* + 1)^{K − m}*u*^{m − 1}(*K*≥1) for all *u*≥0 with some positive constants *D*_{0} and *D*_{1}, and *f*(*u*) is assumed to be smooth enough and non-negative for all *u*≥0 and *f*(0) = 0, while *S*(*u*,*v*,*x*) = (*s*_{ij})_{n × n} is a matrix with
and
with *l*≥2, where
is nondecreasing on [0,*∞*). It is proved that when
, the system possesses at least one global and bounded weak solution for any sufficiently smooth non-negative initial data. Copyright © 2015 John Wiley & Sons, Ltd.

We obtain the existence and the uniqueness at the same time in the reconstruction of orthotropic conductivity in two-space dimensions by using two sets of internal current densities and boundary conductivity. The curl-free equation of Faraday's law is taken instead of the elliptic equation in a divergence form that is typically used in electrical impedance tomography. A reconstruction method based on layered bricks-type virtual-resistive network is developed to reconstruct orthotropic conductivity with up to 40% multiplicative noise. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this article, we establish some new criteria for the oscillation of *n*th-order nonlinear delay differential equations of the form

provided that the second-order equation

is either nonoscillatory or oscillatory. Examples are given to illustrate the results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper is concerned with the initial-boundary value problem for a variable coefficient beam equation with nonlinear damping. Such a model arises from the vertical deflections of a damped extensible elastic inhomogeneous beam whose density depends on time and position. By using the Faedo–Galerkin method and energy method, we obtain the existence and uniqueness of global strong solution. Furthermore, the exponential decay estimate for the total energy is also derived. Copyright © 2015 John Wiley & Sons, Ltd.

]]>J. Banasiak We study Berg's effect on special domains. This effect is understood as a monotonicity of a harmonic function (with respect to the distance from the center of a flat part of the boundary) restricted to the boundary. The harmonic function must satisfy piecewise constant Neumann boundary conditions. We show that Berg's effect is a rare and fragile phenomenon. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we deal with the system that has solutions and the periodicity character of the following systems of rational difference equations with order three

with initial conditions *x*_{−2},*x*_{−1},*x*_{0},*y*_{−2},*y*_{−1}, and *y*_{0} that are arbitrary nonzero real numbers. Some numerical examples will be given to illustrate our results. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we investigate exact traveling wave solutions of the fourth-order nonlinear Schrödinger equation with dual-power law nonlinearity through Kudryashov method and (G'/G)-expansion method. We obtain miscellaneous traveling waves including kink, antikink, and breather solutions. These solutions may be useful in the explanation and understanding of physical behavior of the wave propagation in a highly dispersive optical medium. Copyright © 2015 John Wiley & Sons, Ltd.

]]>By using the method of dynamical systems, for the nonlinear surface wind waves equation, which is given by Manna, we study its dynamical behavior to determine all exact explicit traveling wave solutions. To guarantee the existence of the aforementioned solutions, all parameter conditions are determined. Our procedure shows that the nonlinear surface wind waves equation has no peakon solution. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper is concerned with the stability of positive periodic solutions for the Mackey–Glass model of respiratory dynamics with a control term. We prove the existence, positivity, and permanence of solutions, which help to deduce the global exponential stability of positive periodic solutions for this model. Our method relies upon a differential inequality technique and a Lyapunov functional. At the end, we give an example with numerical simulations to demonstrate the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>J. Banasiak We discuss a mixed-suspension, mixed-product removal crystallizer operated at thermodynamic equilibrium. We derive and discuss the mathematical model based on population and mass balance equations and prove local existence and uniqueness of solutions using the method of characteristics. We also discuss the global existence of solutions for continuous and batch mode. Finally, a numerical simulation of a continuous crystallizer in steady state is presented. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We study the existence and multiplicity of solutions for a nonlinear boundary value problem subject to perturbations of impulsive terms. Under suitable assumptions on the potential of the nonlinearity, the existence of one or two solutions is established. Our approach is based on a local minimum theorem and a two non-zero critical points for differentiable functionals. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, the spectral analysis of a singular dissipative fourth order differential operator in lim-4 case with finite transmission conditions is investigated. For this purpose, the inverse operator with explicit form is used. Finally, with the help of Livšic's theorem, it is proved that all root vectors of the fourth order dissipative operator in lim-4 case with finite transmission conditions are complete in the Hilbert space. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We investigate in this paper the solutions and the periodicity of the following rational systems of difference equations of three-dimensional

with initial conditions *x*_{−2},*x*_{−1},*x*_{0},*y*_{−2},*y*_{−1},*y*_{0},*z*_{−2},*z*_{−1}and*z*_{0} are nonzero real numbers. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, for a coupled system of one-dimensional wave equations with Dirichlet boundary controls, we show that the controllability of classical solutions implies the controllability of weak solutions. This conclusion can be applied in proving some results that are hardly obtained by a direct way in the framework of classical solutions. For instance, we strictly derive the necessary conditions for the exact boundary synchronization by two groups in the framework of classical solutions for the coupled system of wave equations. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we are concerned with the existence of infinitely many solutions for the following fractional Hamiltonian systems

where , , and . The novelty of this paper is that, relaxing the conditions on the potential function *W*(*t*,*x*), we obtain infinitely many solutions via critical point theory. Our results generalize and improve some existing results in the literature. Copyright © 2015 John Wiley & Sons, Ltd.

We study the exact solutions of the Emden–Fowler equations and generalize the *n* = 1 and *n* = 5 Lane–Emden equations. We analyze the series approximations and show that explicit formulas can be written for new classes of equations. Copyright © 2015 John Wiley & Sons, Ltd.

M. Otani Dedicated to Messoud Efendiev on the occasion of his 60th birthday

For the deceptively innocent case of monomolecular reactions, only, we embark on a systematic mathematical analysis of the steady-state response to perturbations of reaction rates. Our *structural sensitivity analysis* is based on the directed graph structure of the monomolecular reaction network, only. In fact our function-free approach does not require numerical input. We work with general, not necessarily monotone reaction rate functions. Based on the graph structure alone, we derive which steady-state concentrations and reaction fluxes are sensitive to, and thus affected by, a rate change—and which are not. Moreover we establish a transitivity property for the influence of a rate perturbation, at any reaction, on all reaction fluxes. The results and concepts developed here, from a mathematical view point, are of applied relevance including metabolic networks in biology; see our companion paper quoted below. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we will study the lower bounds of the life span (the maximal existence time) of solutions to the initial-boundary value problems with small initial data and zero Neumann boundary data on exterior domain for one-dimensional general quasilinear wave equations *u*_{tt}−*u*_{xx}=*b*(*u*,*D**u*)*u*_{xx}+*F*(*u*,*D**u*). Our lower bounds of the life span of solutions in the general case and special case are shorter than that of the initial-Dirichlet boundary value problem for one-dimensional general quasilinear wave equations. We clarify that although the lower bounds in this paper are same as that in the case of Robin boundary conditions obtained in the earlier paper, however, the results in this paper are not the trivial generalization of that in the case of Robin boundary conditions because the fundamental Lemmas 2.4, 2.5, 2.6, and 2.7, that is, the priori estimates of solutions to initial-boundary value problems with Neumann boundary conditions, are established differently, and then the specific estimates in this paper are different from that in the case of Robin boundary conditions. Another motivation for the author to write this paper is to show that the well-posedness of problem is the essential precondition of studying the lower bounds of life span of classical solutions to initial-boundary value problems for general quasilinear wave equations. The lower bound estimates of life span of classical solutions to initial-boundary value problems is consistent with the actual physical meaning. Finally, we obtain the sharpness on the lower bound of the life span in the general case and in the special case. Copyright © 2015 John Wiley & Sons, Ltd.

Previous analysis and research on the power option – one of the exotic options – have focused on the interest rate of the stock and its volatility as constant parameters throughout the run of execution. In this paper, we attempt to extend these results to the more practical and realistic case of when these parameters are time dependent. By making no *ansatz* or relying on *ad hoc* methods, we are able to achieve this via an algorithmic method – the Lie group approach – leading to exact solutions for the power option problem. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we study a class of nonlinear obstacle problems with nonstandard growth. We obtain the L∞ estimate on the solutions and prove the existence and uniqueness of solutions to such problems. Our results are generalizations of the corresponding results in the constant exponent case. Copyright © 2014 John Wiley & Sons, Ltd.

]]>This paper focuses on the stability in terms of two measures for functional differential equation with variable-time impulses. Being different from most of existing literatures, the impulses of functional differential equation are assumed to be closely associated to the current state. We propose a new comparison principle for the considered systems and establish several stability criteria in terms of two measures. Also, the theoretical results are applied in a class of delayed neural network systems with variable-tine impulses, and numerical simulations are introduced to illustrate the effectiveness of our results. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we propose a mathematical model to study the dynamics of anorexic and bulimic populations. The model proposed takes into account, among other things, the effects of peers' influence, media influence, and education.

We prove the existence of three possible equilibria that without media influences are disease-free, bulimic-endemic, and endemic. Neglecting media and education effects, we investigate the stability of such equilibria, and we prove that under the influence of media, only one of such equilibria persists and becomes a global attractor. Which of the three equilibria becomes global attractor depends on the other parameters. Copyright © 2014 John Wiley & Sons, Ltd.

In this article, we prove the existence of solutions to singular coagulation equations with multifragmentation. We use weighted *L*^{1} spaces to deal with the singularities and to obtain regular solutions. The Smoluchowski kernel is covered by our proof. The weak *L*^{1} compactness methods are applied to suitably chosen approximating equations as a base of our proof. A more restrictive uniqueness result is also given. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we consider Hamilton–Jacobi equations with homogeneous Neumann boundary condition. We establish some results on noncompact manifold with homogeneous Neumann boundary conditions in view of weak Kolmogorov-Arnold-Moser (KAM) theory, which is a generalization of the results obtained by Fathi under the non-bounded condition. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, a humoral and cellular immunity virus dynamics model with the Beddington-DeAngelis incidence rate is set up. We derive the basic reproductive number R_{0}, the cytotoxic T lymphocytes immune response reproductive number R_{1}, the humoral immune response reproductive number R_{2}, humoral immune response competitive reproductive number R_{3}, and cytotoxic T lymphocytes immune response competitive reproductive number R_{4}, and a full description of the relation between the existence of the equilibria and reproductive numbers is given. The global properties of the five equilibria are obtained by constructing Lyapunov functions. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, a diffusive predator–prey system, in which the prey species exhibits herd behavior and the predator species with quadratic mortality, has been studied. The stability of positive constant equilibrium, Hopf bifurcations, and diffusion-driven Turing instability are investigated under the Neumann boundary condition. The explicit condition for the occurrence of the diffusion-driven Turing instability is derived, which is determined by the relationship of the diffusion rates of two species. The formulas determining the direction and the stability of Hopf bifurcations depending on the parameters of the system are derived. Finally, numerical simulations are carried out to verify and extend the theoretical results and show the existence of spatially homogeneous periodic solutions and nonconstant steady states. Copyright © 2014 John Wiley & Sons, Ltd.

]]>We give some alternative forms of the generating functions for the Bernstein basis functions. Using these forms,we derive a collection of functional equations for the generating functions. By applying these equations, we prove some identities for the Bernstein basis functions. Integrating these identities, we derive a variety of identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients, Pascal's rule, Vandermonde's type of convolution, the Bernoulli polynomials, and the Catalan numbers. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, a shifted Jacobi–Gauss collocation spectral algorithm is developed for solving numerically systems of high-order linear retarded and advanced differential–difference equations with variable coefficients subject to mixed initial conditions. The spatial collocation approximation is based upon the use of shifted Jacobi–Gauss interpolation nodes as collocation nodes. The system of differential–difference equations is reduced to a system of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The convergence is discussed graphically. The proposed method has an exponential convergence rate. The validity and effectiveness of the method are demonstrated by solving several numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, a collocation method is given to solve singularly perturbated two-point boundary value problems. By using the collocation points, matrix operations and the matrix relations of the Bessel functions of the first kind and their derivatives, the boundary value problem is converted to a system of the matrix equations. By solving this system, the approximate solution is obtained. Also, an error problem is constructed by the residual function, and it is solved by the presented method. Thus, the error function is estimated, and the approximate solutions are improved. Finally, numerical examples are given to show the applicability of the method, and also, our results are compared by existing results. Copyright © 2014 JohnWiley & Sons, Ltd.

]]>This paper is devoted to the study of a nonlinear parabolic p(*x*)-Laplace equation with gradient term and L^{1} data. The authors obtain the existence of renormalized solutions via strong convergence of truncation. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we obtain several extensions to the quaternionic setting of some results concerning the approximation by polynomials of functions continuous on a compact set and holomorphic in its interior. To this end, we prove an analog of the Riemann mapping theorem for a subclass of open sets, whose validity involves precisely the slice regular functions for which the composition remains slice regular. The results include approximation on compact starlike sets and compact axially symmetric sets. The cases of some concrete particular sets are described in details, including also quantitative estimates. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The aim of this article is to propose an optimization strategy for traffic flowon roundabouts using amacroscopic approach. The roundabout is modeled as a sequence of 2 × 2 junctions with one main lane and secondary incoming and outgoing roads. We consider two cost functionals: the total travel time and the total waiting time, which give an estimate of the time spent by drivers on the network section. These cost functionals areminimized with respect to the right ofway parameter of the incoming roads. For each cost functional, the analytical expression is given for each junction. We then solve numerically the optimization problem and show some numerical results. Copyright © 2014 John Wiley & Sons, Ltd.

]]>We consider the nonlinear Schrödinger equation (NLSH) with the convolution combined term

- (CNLSH)

(CNLSH) in the energy space
. We firstly use a variational approach to give a dichotomy of scattering and blow up for the radial solution with the energy below the threshold, which is given by the ground state *W* for the energy-critical NLS: *i**u*_{t}+Δ*u*=−|*u*|^{4}*u*. The basic strategy is the concentration-compactness arguments from Kenig and Merle. We overcome the main difficulties coming from the lack of scaling invariance and the non-local property of the convolution term. Our result shows that the focusing,
-critical term −|*u*|^{4}*u* plays the decisive role of the threshold of the scattering solution of (CNLSH) in the energy space. Copyright © 2015 John Wiley & Sons, Ltd.

The classical orthogonal polynomials (COPs) satisfy a second-order differential equation of the form *σ*(*x*)*y*^{′′}+*τ*(*x*)*y*^{′}+*λ**y* = 0, which is called the equation of hypergeometric type (EHT). It is shown that two numerical methods provide equivalent schemes for the discrete representation of the EHT. Thus, they lead to the same matrix eigenvalue problem. In both cases, explicit closed-form expressions for the matrix elements have been derived in terms only of the zeros of the COPs. On using the equality of the entries of the resulting matrices in the two discretizations, unified identities related to the zeros of the COPs are then introduced. Hence, most of the formulas in the literature known for the roots of Hermite, Laguerre and Jacobi polynomials are recovered as the particular cases of our more general and unified relationships. Furthermore, we present some novel results that were not reported previously. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, the stability problem of impulsive functional differential equations with infinite delays is considered. By using Lyapunov functions and the Razumikhin technique, some new theorems on the uniform stability and uniform asymptotic stability are obtained. The obtained results are milder and more general than several recent works. Two examples are given to demonstrate the advantages of the results. Copyright © 2014 John Wiley & Sons, Ltd.

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