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.

]]>By using the Bell polynomials method and symbolic computation, we study the integrability of the KdV6 equation. We develop, in this work, new results regarding the integrability concept. We show that the newly developed bilinear representation and bilinear Bäcklund transformation are different from those reported in the literature. Moreover, we firstly present the infinite conservation laws, and the conserved densities and fluxes are given in explicit recursion formulas. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this work, we study the well-posedness and the asymptotic stability of a one-dimensional linear thermoelastic Timoshenko system, where the heat conduction is given by Cattaneo's law and the coupling is via the displacement equation. We prove that the system is exponentially stable provided that the stability number *χ*_{τ}=0. Otherwise, we show that the system lacks exponential stability. Furthermore, in the latter case, we show that the solution decays polynomially. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we study the well-posedness and exact controllability of a physical model for an extrusion process in the isothermal case. The model expresses the mass balance in the extruder chamber and consists of a hyperbolic partial differential equation (PDE) and a nonlinear ordinary differential equation (ODE) whose dynamics describes the evolution of a moving interface. By suitable change of coordinates and fixed point arguments, we prove the existence, uniqueness, and regularity of the solution and finally, the exact controllability of the coupled system. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper presents an MLP-type neural network with some fixed connections and a backpropagation-type training algorithm that identifies the full set of solutions of a complete system of nonlinear algebraic equations with *n* equations and *n* unknowns. The proposed structure is based on a backpropagation-type algorithm with bias units in output neurons layer. Its novelty and innovation with respect to similar structures is the use of the hyperbolic tangent output function associated with an interesting feature, the use of adaptive learning rate for the neurons of the second hidden layer, a feature that adds a high degree of flexibility and parameter tuning during the network training stage. The paper presents the theoretical aspects for this approach as well as a set of experimental results that justify the necessity of such an architecture and evaluate its performance. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, a class of fourth-order nonlinear difference equation is considered. By making use of the critical point theory, we establish various sets of sufficient conditions for the existence of homoclinic solutions and give some new results. One of our results generalizes and improves the results in the literature. Copyright © 2015 John Wiley & Sons, Ltd.

]]>A theoretical eco-epidemiological model of a prey–predator interaction system with disease in prey species is studied. Predator consumes both susceptible and infected prey population, but predator also feeds preferentially on many numerous species, which are over represented in the predator's diet. Equilibrium points of the system are determined, and the dynamic behaviour of the system is investigated around equilibrium points. Death rate of predator species is considered as a bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighbourhood of the coexisting equilibria. Numerical simulations are carried out to support the analytical results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we study the asymptotic profile of the solution for a *σ*-evolution equation with a time-dependent structural damping. We introduce a classification of the damping term, which clarifies whether the solution behaves like the solution to an anomalous diffusion problem. We call this damping effective, whereas we say that the damping is noneffective when the solution shows oscillations in its asymptotic profile that cannot be neglected. Our classification shows a completely new interplay between the strength of the damping and the long time behavior of its coefficient. Copyright © 2015 John Wiley & Sons, Ltd.

The purpose of the paper is to introduce Stancu-type linear positive operators generated by Dunkl generalization of exponential function. We present approximation properties with the help of well-known Korovkin-type theorem and weighted Korovkin-type theorem and also acquire the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's K-functional, and second-order modulus of continuity by Dunkl analogue of Szász operators. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The process of flow of a thin plastic layer enclosed between two approaching surfaces of the tool bodies is considered. The new results of theoretical and experimental researches are presented. 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.

Our aim in this paper is to study, in term of finite dimensional exponential attractors, the Willmore regularization, (depending on a small regularization parameter *β* > 0), of two phase-field equations, namely, the Allen–Cahn and the Cahn–Hilliard equations. In both cases, we construct robust families of exponential attractors, that is, attractors that are continuous with respect to the perturbation parameter. Copyright © 2015 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.

]]>We propose a new numerical method for the solution of the Bernoulli free boundary value problem for harmonic functions in a doubly connected domain *D* in where an unknown free boundary Γ_{0} is determined by prescribed Cauchy data on Γ_{0} in addition to a Dirichlet condition on the known boundary Γ_{1}. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar, and Kress for the solution of a related inverse boundary value problem. For this, we interpret the free boundary Γ_{0} as the unknown boundary in the inverse problem to construct Γ_{0} from the Dirichlet condition on Γ_{0} and Cauchy data on the known boundary Γ_{1}. Our method for the Bernoulli problem iterates on the missing normal derivative on Γ_{1} by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet–Neumann boundary value problem in *D*. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach. Copyright © 2015 John Wiley & Sons, Ltd.

This work is concerned with the periodic problem for compressible non-isentropic Euler–Maxwell systems with a temperature damping term arising in plasmas. For this problem, we prove the global in time existence of a smooth solution around a given non-constant steady state with the help of an induction argument on the order of the mixed time-space derivatives of solutions in energy estimates. Moreover, we also show the convergence of the solution to this steady state as the time goes to the infinity. This phenomenon on the charge transport shows the essential relation of the systems with the non-isentropic Euler–Maxwell and the isentropic Euler–Maxwell systems. Copyright © 2015 John Wiley & Sons, Ltd.

]]>A nonlinear evolution equation of second order with damping is studied. The quasilinear damping term is monotone and coercive but exhibits anisotropic and nonpolynomial growth. The appropriate setting for such equations is that of monotone operators in Orlicz spaces. Global existence of solutions in the sense of distributions is shown via convergence of the backward Euler scheme combined with an internal approximation. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper investigates the steady subsonic inviscid flows with large vorticities through two-dimensional infinitely long nozzles. We establish the existence and uniqueness of the smooth subsonic ideal flows, which are governed by a two-dimensional complete Euler system. More precisely, given the horizontal velocity with possible large oscillation and the entropy of the incoming flows at the entrance of the nozzles, it was shown that there exists a critical value; if the mass flux of the incoming flows is larger than the critical one, then there exists a unique smooth subsonic polytropic gas through the given smooth infinitely long nozzles. Furthermore, the maximal speed of the flows approaches to the sonic speed, as the mass flux goes to the critical value. The results improve the previous work for steady subsonic flows with small vorticities and for subsonic irrotational flows and indicate that the large vorticity is admissible for the smooth subsonic ideal flows in nozzles. This paper gives a rigorous proof to the well posedness of the smooth subsonic problem first posed back in the basic survey of Lipman Bers for inviscid flows with large vorticities. John Wiley & Sons, Ltd.

]]>In this paper, a high-order accurate numerical method for two-dimensional semilinear parabolic equations is presented. We apply a Galerkin–Legendre spectral method for discretizing spatial derivatives and a spectral collocation method for the time integration of the resulting nonlinear system of ordinary differential equations. Our formulation can be made arbitrarily high-order accurate in both space and time. Optimal a priori error bound is derived in the *L*^{2}-norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence property of the method, show our formulation have spectrally accurate in both space and time. 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.

]]>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.

]]>Milling chatter leads to a poor surface finish, premature tool wear, and potential damage to the machine or tool. Thus, it is desirable to predict and avoid the onset of this instability. Considering that the stability of milling with variable pitch cutter or tool runout case is characterized by multiple delays, in this paper, an improved semi-discretization method is proposed to predict the stability lobes for milling processes with multiple delays. Taking the variable pitch milling, for example, a comparisonwith prior methods is conducted to verify the accuracy and efficiency of the proposed approach for the stability prediction both in low and high radial immersion ratios. In addition, the rate of convergence of the proposed method is also evaluated. The results show that the proposed method has high computational efficiency. Copyright©2015 JohnWiley & Sons, Ltd.

This paper deals with the blow-up phenomena for a class of fourth-order nonlinear wave equations with a viscous damping term

with Ω = (0,1) and *α* > 0. Here, *f*(s) is a given nonlinear smooth function.

For 0 < *α* < *p* – 1, we prove that the blow-up occurs in finite time for arbitrary positive initial energy and suitable initial data. This result extends the recent results obtained by Xu *et al*. (Applicable Analysis)(2013) and Chen and Lu (J. Math. Anal. Appl.)(2009). Copyright © 2015 John Wiley & Sons, Ltd.

K. Guerlebeck In this paper, we consider the following nonlinear Dirac equation

By applying the variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou, we prove the existence of nontrivial and ground state solutions for the aforementioned system under conditions weaker than those in Zhang *et al.* (Journal of Mathematical Physics, 2013). John Wiley & Sons, Ltd.

We study Smoluchowski–Poisson equation in two space dimensions provided with Dirichlet boundary condition for the Poisson part. For this equation, several profiles of blowup solution have been noticed: blowup threshold on *L*^{1} norm of the initial value, finiteness of blowup points, formation of delta singularities called collapses, occurrence of type II blowup, exclusion of the boundary blowup point, and so forth. Here, we show the collapse mass quantization with possible residual terms. Copyright © 2015 John Wiley & Sons, Ltd.

In the present paper, we study the vector potential problem in exterior domains of . Our approach is based on the use of weighted spaces in order to describe the behavior of functions at infinity. As a first step of the investigation, we prove important results on the Laplace equation in exterior domains with Dirichlet or Neumann boundary conditions. As a consequence of the obtained results on the vector potential problem, we establish useful results on weighted Sobolev inequalities and Helmholtz decompositions of weighted spaces. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we discretize techniques for the construction of axially monogenic functions to the setting of discrete Clifford analysis. Wherefore, we work in the discrete Hermitian Clifford setting, where each basis vector *e*_{j} is split into a forward and backward basis vector:
. We prove a discrete version of Fueter's theorem in odd dimension by showing that for a discrete monogenic function *f*(*ξ*_{0},*ξ*_{1}) left-monogenic in two variables *ξ*_{0} and *ξ*_{1} and for a left-monogenic *P*_{k}(*ξ*), the *m*-dimensional function
is in itself left monogenic, that is, a discrete function in the kernel of the discrete Dirac operator. Closely related, we consider a Vekua-type system for the construction of axially monogenic functions. We consider some explicit examples: the discrete axial-exponential functions and the discrete Clifford–Hermite polynomials. Copyright © 2015 John Wiley & Sons, Ltd.

The treatment of human immunodeficiency virus (HIV) remains a major challenge, even if significant progress has been made in infection treatment by ‘drug cocktails’. Nowadays, research trend is to minimize the number of pills taken when treating infection. In this paper, an HIV-1 within host model where healthy cells follow a simple logistic growth is considered. Basic reproduction number of the model is calculated using next generation matrix method, steady states are derived; their local, as well as global stability, is discussed using the Routh–Hurwitz criteria, Lyapunov functions and the Lozinskii measure approach. The optimal control policy is formulated and solved as an optimal control problem. Numerical simulations are performed to compare several cases, representing a treatment by Interleukin2 alone, classical treatment by multitherapy drugs alone, then both treatments at the same time. Objective functionals aim to (i) minimize infected cells quantity; (ii) minimize free virus particles number; and (iii) maximize healthy cells density in blood. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, a new rational approximation based on a rational interpolation and collocation method is proposed for the solutions of generalized pantograph equations. A comprehensive error analysis is provided. The first part of the error analysis gives an upper bound for the absolute error. The second part is based on residual error procedure that estimates the absolute error. Some numerical examples are given to illustrate the method. The theoretical results support the numerical results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The purpose of this paper is to study the global stability properties of equilibria for age-dependent epidemiological models in presence of recurrence phenomenon. In these systems, the recurrence rate depends on asymptomatic–infection–age. The models are appropriate for human herpes virus (HSV-1 and HSV-2) and varicella-zoster virus. We derived explicit formulas for the basic reproductive number, which completely characterizes the global behaviour of solutions to the models: if the basic reproductive number is less than or equal to unity, the disease will die out; if the basic reproductive number is greater than unity, the disease will be persistent. Volterra-type Lyapunov functions are constructed to establish the global asymptotic stability of the infection-free and endemic steady states. Copyright © 2015 John Wiley & Sons, Ltd.

]]>When steady supersonic flow hits a slim wedge, there may appear an oblique transonic shock attached to the vertex of the wedge, if the downstream pressure is rather large. This paper studies stability in certain weighted partial Hölder spaces of the oblique transonic shock attached to the vertex of a wedge, which is against steady supersonic flows, under perturbations of the upstream flow and the profile of the wedge. We show that under reasonable conditions on the upcoming supersonic flow and the slope of the wedge, such transonic shocks are structural stable. Mathematically, we solve an elliptic–hyperbolic mixed type in an unbounded domain, and the flow field is proved to be *C*^{1}. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we present five classes/categories of time scales. Then, on each class, we introduce and analyze delays that not only lead to new types of delay systems on time scales but also reveal the limitations of the known results in the literature. To show the importance and significance of our analysis, several examples are illustrated. We conclude our paper with some interesting open problems. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we prove the existence of mild solutions for a first-order impulsive semilinear stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay. We consider the cases in which the right hand side is convex or nonconvex valued. The results are obtained by using two different fixed point theorems for multivalued mappings. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The task of this paper is to study and analyse *transformed localization* and *generalized localization* for ensemble methods in *data assimilation*. Localization is an important part of *ensemble methods* such as the *ensemble Kalman filter* or *square root filter*. It guarantees a sufficient number of degrees of freedom when a small number of ensembles or particles, respectively, are used. However, when the observation operators under consideration are *non-local*, the localization that is applicable to the problem can be severly limited, with strong effects on the quality of the assimilation step. Here, we study a transformation approach to change non-local operators to local operators in transformed space, such that localization becomes applicable. We interpret this approach as a generalized localization and study its general algebraic formulation. Examples are provided for a compact integral operator and a non-local Matrix observation operator to demonstrate the feasibility of the approach and study the quality of the assimilation by transformation. In particular, we apply the approach to temperature profile reconstruction from infrared measurements given by the infrared atmospheric sounding interferometer (IASI) infrared sounder and show that the approach is feasible for this important data type in atmospheric analysis and forecasting. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, a bifurcation solution's analysis is proposed for an HIV-1 within the host model around its chronic equilibrium point, this is carried out based on Lyapunov–Schmidt approach. It is shown that the coefficient *b*, which represents the healthy CD4^{+} T-cells growth rate, is a bifurcation parameter; this means that the rate of multiplication of healthy cells can have serious effects on the qualitative dynamical properties and structural stability of the infection evolution dynamics. Copyright © 2015 John Wiley & Sons, Ltd.

The (*G*′/*G*,1/*G*)-expansion method and (1/*G*′)-expansion method are interesting approaches to find new and more general exact solutions to the nonlinear evolution equations. In this paper, these methods are applied to construct new exact travelling wave solutions of nonlinear Schrödinger equation. The travelling wave solutions are expressed by hyperbolic functions, trigonometric functions and rational functions. It is shown that the proposed methods provide a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we apply the Jacobi collocation method for solving nonlinear fractional differential equations with integral boundary conditions. Due to existence of integral boundary conditions, after reformulation of this equation in the integral form, the method is proposed for solving the obtained integral equation. Also, the convergence and stability analysis of the proposed method are studied in two main theorems. Furthermore, the optimum degree of convergence in the *L*_{2} norm is obtained for this method. Furthermore, some numerical examples are presented in order to illustrate the performance of the presented method. Finally, an application of the model in control theory is introduced. Copyright © 2015 John Wiley & Sons, Ltd.

In this study, a parameterized split Newton method is derived by using the accelerating technique. Convergence and error estimates of the method are obtained. In practical application, the proposed method can give a better result in view of computational CPU time. Numerical examples on several partial differential equations are shown to illustrate our findings. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this work, we give sufficient conditions for the existence and uniqueness of *μ*−pseudo almost periodic integral solutions for some neutral partial functional differential equations with Stepanov *μ*−pseudo almost periodic forcing functions. Our working tools are based on the variation of constant formula and the spectral decomposition of the phase space. To illustrate our main results, we give applications to a neutral model arising in physical systems, as well as an application to heat equations with discrete and continuous delay. Copyright © 2015 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.

I. Stratis In this work, we investigate the analyticity properties of solutions of Kuramoto–Sivashinsky-type equations in two spatial dimensions, with periodic initial data. In order to do this, we explore the applicability in three-dimensional models of a spectral method, which was developed by the authors for the one-dimensional Kuramoto–Sivashinsky equation. We introduce a criterion, which provides a sufficient condition for analyticity of a periodic function *u*∈*C*^{∞}, involving the rate of growth of ∇^{n}*u*, in suitable norms, as *n* tends to infinity. This criterion allows us to establish spatial analyticity for the solutions of a variety of systems, including Topper–Kawahara, Frenkel–Indireshkumar, and Coward–Hall equations and their dispersively modified versions, once we assume that these systems possess global attractors. Copyright © 2015 John Wiley & Sons, Ltd.

T. Qian As it will turn out in this paper, the recent hype about most of the Clifford–Fourier transforms is not thoroughly worth the pain. Almost everyone that has a real application is separable, and these transforms can be decomposed into a sum of real valued transforms with constant multivecor factors. This fact makes their interpretation, their analysis, and their implementation almost trivial. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The paper deals with a system of parabolic–hyperbolic partial differential equations, which models the diffusion of *N* species of isotopes of the same element, possibly radioactive, in a multidimensional medium. Some qualitative properties of the solutions, such as localization property, are studied together with the asymptotic behavior for large times. Copyright © 2015 John Wiley & Sons, Ltd.

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.

]]>A well-posedness result for a time-shift invariant class of evolutionary operator equations involving material laws with fractional time-integrals of order α ϵ ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time-)derivative established as a normal operator in a suitable *L ^{2}* type space. Employing causality, we show that the fractional derivatives thus obtained coincide with the Riemann-Liouville fractional derivative. We exemplify our results by applications to a fractional Fokker-Planck equation, equations describing super-diffusion and sub-diffusion processes, and a Kelvin-Voigt type model in fractional visco-elasticity. Moreover, we elaborate a suitable perspective to deal with initial boundary value problems. Copyright © 2014 John Wiley & Sons, Ltd.

Currently, chaotic systems and chaos-based applications are commonly used in the engineering fields. One of the main structures used in these applications is chaotic control and synchronization. In this paper, the dynamical behaviors of a new hyperchaotic system are considered. Based on Lyapunov Theorem with differential and integral inequalities, the global exponential attractive sets and positively invariant sets are obtained. Furthermore, the rate of the trajectories is also obtained. The global exponential attractive sets of the system obtained in this paper also offer theoretical support to study chaotic control, chaotic synchronization for this system. Computer simulation results show that the proposed method is effective. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we construct some compactly supported orthogonal regular wavelet basis on Heisenberg group . Because of the regularity of wavelets, we could use such wavelets to characterize function spaces on , such as bounded mean oscillation space (BMO) space, Hardy space, Besov spaces and Besov–Morrey spaces. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we study the numerical solution to time-fractional partial differential equations with variable coefficients that involve temporal Caputo derivative. A spectral method based on Gegenbauer polynomials is taken for approximating the solution of the given time-fractional partial differential equation in time and a collocation method in space. The suggested method reduces this type of equation to the solution of a linear algebraic system. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we investigate Poincaré type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. We find exact and easily computable constants in these inequalities for some basic domains (rectangles, cubes, and right triangles) and discuss applications of the inequalities to quantitative analysis of partial differential equations. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The use of asymptotic limits to model heterogeneous plates can be troublesome, because it requires *a priori* knowledge on the ratio between characteristic lengths of heterogeneities and thickness. Moreover, it also relies on some assumption on the inclusions, like periodicity. We propose and analyze here *hierarchical modeling* techniques and show that such approach not only avoids such pitfalls, but it is actually simpler to obtain, and it provably converges to the correct asymptotic limits. Its derivation does not requires any restrictive assumptions on the heterogeneities. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we present a novel numerical scheme to solve a 1D, one-phase extended Stefan problem with fractional Caputo derivative with respect to time. The proposed method is based on a suitable choice of the new space coordinate for the subdiffusion equation and extends the front-fixing method to the subdiffusion case. In the final part, examples of numerical results are discussed. Copyright © 2014 John Wiley & Sons, Ltd.

]]>This paper deals with numerical methods for reconstruction of inhomogeneous conductivities. We use the concept of Generalized Polarization Tensors to do reconstruction. Basic resolution and stability analysis are presented. Least-square norm methods with respect to Generalized Polarization Tensors are used for reconstruction of conductivities. Finally, reconstruction of three different types of conductivities in the plane is demonstrated. Copyright © 2014 John Wiley & Sons, Ltd.

]]>A multiple monopole method based on the generalized multipole technique is presented for the calculation of band structures of two-dimensional mixed solid/fluid phononic crystals. In this method, the fields are expanded by using the fundamental solutions with multiple origins. Besides the sources used to expand the wave fields, an extra monopole source is introduced as the external excitation. By varying the frequency of the excitation, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and sweeping the boundary of the irreducible first Brillouin zone, the band structure of the phononic crystals can be obtained. The method can consider the fluid–solid interface conditions and the transverse wave mode in the solid component strictly. Some typical examples are illustrated to discuss the accuracy of the present method. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Direct scattering problems for partially coated piecewise homogenous and inhomogeneous layered obstacles in linear elasticity lead to mixed impedance transmission problems for the steady-state elastic oscillation equations. For a piecewise homogenous isotropic composite body, we employ the potential method and reduce the mixed impedance transmission problem to an equivalent system of boundary pseudodifferential equations. We give a detailed analysis of the corresponding pseudodifferential operators, which live on the interface between the layers and on a proper submanifold of the boundary of the composite elastic body, and establish uniqueness and existence results for the original mixed impedance transmission problem for arbitrary values of the oscillation frequency parameter; this is crucial in the study of inverse elastic scattering problems for partially coated layered obstacles. We also investigate regularity properties of solutions near the collision curves, where the different boundary conditions collide, and establish almost best Hölder smoothness results. Further, we analyze the asymptotic behavior of the stress vector near the collision curve and derive explicit formulas for the stress singularity exponents. The case of Lipschitz surfaces is briefly treated separately. In the case of a composite body containing homogeneous or inhomogeneous finite anisotropic inclusions, we develop an alternative hybrid method based on the so-called nonlocal approach and reduce the mixed transmission problem to an equivalent functional-variational equation with a sesquilinear form that ‘lives’ on a bounded part of the layered composite body and its boundary. We show that this sesquilinear form is coercive and that the corresponding variational equation is uniquely solvable. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper is devoted to study the periodic nature of the solution of the following max-type difference equation:

where the initial conditions *x*_{−2},*x*_{−1},*x*_{0} are arbitrary positive real numbers and
is a periodic sequence of period two. Copyright © 2014 John Wiley & Sons, Ltd.

In this note, a non-standard finite difference (NSFD) scheme is proposed for an advection-diffusion-reaction equation with nonlinear reaction term. We first study the diffusion-free case of this equation, that is, an advection-reaction equation. Two exact finite difference schemes are constructed for the advection-reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection-reaction equation is combined with a finite difference space-approximation of the diffusion term to provide a NSFD scheme for the advection-diffusion-reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.

]]>In this paper, we investigate the existence of solutions to nonlinear fractional order differential coupled systemswith the classical nonlocal initial conditions.We introduce a useful vector norm, named β·B-vector norm,which is not only a novelty but also provides another way to deal with a large number of problems not limit to integer and noninteger differential systems and singular integral systems. Copyright © 2014 John Wiley & Sons, Ltd.

]]>An inverse spectral problem is considered for Dirac operators with parameter-dependent transfer conditions inside the interval, and parameter appears also in one boundary condition. The approach that was used in the investigation of uniqueness theorems of inverse problems for Weyl function or two eigenvalue sets is employed. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Understanding animal movements and modeling the routes they travel can be essential in studies of pathogen transmission dynamics. Pathogen biology is also of crucial importance, defining the manner in which infectious agents are transmitted. In this article, we investigate animal movement with relevance to pathogen transmission by physical rather than airborne contact, using the domestic chicken and its protozoan parasite *Eimeria* as an example. We have obtained a configuration for the maximum possible distance that a chicken can walk through straight and nonoverlapping paths (defined in this paper) on square grid graphs. We have obtained preliminary results for such walks which can be practically adopted and tested as a foundation to improve understanding of nonairborne pathogen transmission. Linking individual nonoverlapping walks within a grid-delineated area can be used to support modeling of the frequently repetitive, overlapping walks characteristic of the domestic chicken, providing a framework to model fecal deposition and subsequent parasite dissemination by fecal/host contact. We also pose an open problem on multiple walks on finite grid graphs. These results grew from biological insights and have potential applications. © 2014 The Authors. *Mathematical Methods in the Applied Sciences* published by John Wiley & Sons Ltd.

The concept of uncertain fractional differential equation is introduced, and solutions of several uncertain fractional differential equations are presented. This kind of equation is a counterpart of stochastic fractional differential equation. By the proposed concept, an interest rate model is considered, and the price of a zero-coupon bond is obtained. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, a parameter identification method to determine surface shortwave fluxes using temperature and thickness measurements of sea ice in CHINARE 2006 is presented. Adopting a new standard for the calculation of the thermodynamic properties of seawater named TEOS-10, the surface shortwave fluxes are calculated by the temperature and thickness observations that were measured at Nella Fjord around Zhongshan Station, Antarctica. New simulations for temperature profiles in a different measurement period are performed by three parameterization schemes including the present method, Zillman and Shine. All numerical results are compared with *in situ* measurements. Results show that better simulations of the surface shortwave radiations and temperature distributions are possible with the identification method than Zillman and Shine. Therefore this method is valid, and the obtained shortwave radiation function can be applied in sea ice modeling. Copyright © 2015 John Wiley & Sons, Ltd.