A discontinuous Galerkin (DG) finite-element interior calculus is used as a common framework to describe various DG approximation methods for second-order elliptic problems. Using the framework, symmetric interior-penalty methods, local discontinuous Galerkin methods, and dual-wind discontinuous Galerkin methods will be compared by expressing all of the methods in primal form. The penalty-free nature of the dual-wind discontinuous Galerkin method will be both motivated and used to better understand the analytic properties of the various DG methods. Consideration will be given to Neumann boundary conditions with numerical experiments that support the theoretical results. Many norm equivalencies will be derived laying the foundation for applying dual-winding techniques to other problems. Copyright © 2015 John Wiley & Sons, Ltd.

]]>It is well known that a spherically symmetric wave speed problem in a bounded spherical region may be reduced, by means of Liouville transform, to the Sturm–Liouville problem *L*(*q*) in a finite interval. In this work, a uniqueness theorem for the potential *q* of the derived Sturm–Liouville problem *L*(*q*) is proved when the data are partial knowledge of the given spectra and the potential. Copyright © 2015 John Wiley & Sons, Ltd.

W. Sprößig In this paper, we propose a new adaptive method for frequency-domain identification problem of discrete LTI systems. It is based on a dictionary that is consisting of normalized reproducing kernels. We prove that the singular values of the matrix generated by this dictionary converge to zero rapidly; this makes it quite efficient in representing the original systems with only a few elements. For different systems, it results in different selected sequences from the dictionary, that is, its adaptivity. Meanwhile, the stability of results is automatically guaranteed according to the structure of the dictionary. Two examples are presented to illustrate the idea. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, a new numerical method for solving the fractional Bagley-Torvik equation is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is introduced. This operator is then utilized to reduce the solution of the initial and boundary value problems for the fractional Bagley-Torvik differential equation 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.

]]>We consider the Cauchy problem for the Vlasov–Maxwell–Fokker–Planck system in the plane. It is shown that for smooth initial data, as long as the electromagnetic fields remain bounded, then their derivatives do also. Glassey and Strauss have shown this to hold for the relativistic Vlasov–Maxwell system in three dimensions, but the method here is totally different. In the work of Glassey and Strauss, the relativistic nature of the particle transport played an essential role. In this work, the transport is nonrelativistic, and smoothing from the Fokker–Planck operator is exploited. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper is devoted to the attraction–repulsion chemotaxis system with nonlinear diffusion:

where *χ* > 0, *ζ* > 0, *α*_{i}>0, *β*_{i}>0 (*i* = 1,2) and *f*(*s*)≤*κ* − *μ**s*^{τ}. In two-space dimension, we prove the global existence and uniform boundedness of the classical solution to this model for any *μ* > 0. Copyright © 2015 John Wiley & Sons, Ltd.

In a previous study, we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width *d*. We impose the Neumann boundary condition on a disk window of radius *a* and Dirichlet boundary conditions on the remained part of the boundary of the strip. We proved that such system exhibits discrete eigenvalues below the essential spectrum for any *a* > 0. In the present work, we study the effect of the presence of a magnetic field of Aharonov–Bohm type on this system. Precisely, we prove that in the presence of such field, there is some critical values of *a*_{0}>0, for which we have absence of the discrete spectrum for . We give a sufficient condition for the existence of discrete eigenvalues. Copyright © 2015 John Wiley & Sons, Ltd.

We consider the Fisher–KPP equation with advection: *u*_{t}=*u*_{xx}−*β**u*_{x}+*f*(*u*) on the half-line *x*∈(0,*∞*), with no-flux boundary condition *u*_{x}−*β**u* = 0 at *x* = 0. We study the influence of the advection coefficient −*β* on the long time behavior of the solutions. We show that for any compactly supported, nonnegative initial data, (i) when *β*∈(0,*c*_{0}), the solution converges locally uniformly to a strictly increasing positive stationary solution, (ii) when *β*∈[*c*_{0},*∞*), the solution converges locally uniformly to 0, here *c*_{0} is the minimal speed of the traveling waves of the classical Fisher–KPP equation. Moreover, (i) when *β* > 0, the asymptotic positions of the level sets on the right side of the solution are (*β* + *c*_{0})*t* + *o*(*t*), and (ii) when *β*≥*c*_{0}, the asymptotic positions of the level sets on the left side are (*β* − *c*_{0})*t* + *o*(*t*). Copyright © 2015 John Wiley & Sons, Ltd.

In this work, we study the existence of positive solutions for a class of fractional differential equation given by

- (1)

where . Using the mountain pass theorem and comparison argument, we prove that (1) at least has one nontrivial solution. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this work, we develop the negative-order modified Korteweg–de Vries (nMKdV) equation. By means of the recursion operator of the modified KdV equation, we derive negative order forms, one for the focusing branch and the other for the defocusing form. Using the Weiss–Tabor–Carnevale method and Kruskal's simplification, we prove the Painlevé integrability of the nMKdV equations. We derive multiple soliton solutions for the first form and multiple singular soliton solutions for the second form. Copyright © 2015 John Wiley & Sons, Ltd.

]]>A fluid–particles system of the compressible Navier-Stokes equations and Vlasov-Fokker-Planck equation (including the case of Vlasov equation) in three-dimensional space is considered in this paper. The coupling arises from a drag force exerted by the fluid onto the particles. We study a Cauchy problem with large data, and establish the existence of global weak solutions through an approximation scheme, energy estimates, and weak convergence. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we study the time-decay rates of the solution to the Cauchy problem for a nematic liquid crystals system via a refined pure energy method. In particular, the optimal decay rates of the higher-order spatial derivatives of the solution are obtained. The negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We consider the parabolic chemotaxis model

in a smooth, bounded, convex two-dimensional domain and show global existence and boundedness of solutions for *χ*∈(0,*χ*_{0}) for some *χ*_{0}>1, thereby proving that the value *χ* = 1 is not critical in this regard.

Our main tool is consideration of the energy functional

for *a* > 0, *b*≥0, where using nonzero values of *b* appears to be new in this context. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we study a non-local fractional Laplace equation, depending on a parameter, with asymptotically linear right-hand side. Our main result concerns the existence of weak solutions for this equation, and it is obtained using variational and topological methods. We treat both the non-resonant case and the resonant one. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, by using the variational method and the critical point theorem because of Brezis and Nirenberg, we investigate the existence of solutions to a class of second-order impulsive Hamiltonian system. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we introduce some integral transforms that map slice monogenic functions to monogenic functions. We then show that one of these integral transforms, which is based on the Cauchy formula of slice monogenic functions, is useful to define a functional calculus depending on a parameter for *n*-tuples of bounded operators. Copyright © 2015 John Wiley & Sons, Ltd.

By application of Green's function and some fixed-point theorems, that is, Leray–Schauder alternative principle and Schauder's fixed-point theorem, we establish two new existence results of positive periodic solutions for nonlinear fourth-order singular differential equation, which extend and improve significantly existing results in the literature. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we consider the analytical solutions of fractional partial differential equations (PDEs) with Riesz space fractional derivatives on a finite domain. Here we considered two types of fractional PDEs with Riesz space fractional derivatives such as Riesz fractional diffusion equation (RFDE) and Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order *α*∈(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order *β*∈(0,1] and of order *α*∈(1,2] respectively. Here the analytic solutions of both the RFDE and RFADE are derived by using modified homotopy analysis method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. Here the space fractional derivatives are defined as Riesz fractional derivatives. Copyright © 2015 John Wiley & Sons, Ltd.

This paper deals with the model proposed for nonsimple materials with heat conduction of type III. We analyze first the general system of equations, determine the behavior of its solutions with respect to the time, and show that the semigroup associated with the system is not analytic. Two limiting cases of the model are studied later. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The standard obstacles in developing a bifurcation theory for nonautonomous differential equations are the lack of steady-state equilibria and the insignificance of eigenvalues in stability investigations. For this reason, various techniques have been proposed to specify changes in the qualitative behavior of time-dependent dynamical systems. In this paper, we investigate and compare several approaches to nonautonomous bifurcations using SIR-like models from epidemiology as a paradigm. These models are sufficiently simple to allow explicit solutions to a large extent and consequently enable a detailed discussion of the different results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>This paper applies Nevanlinna theory of value distribution to discuss existence of solutions of certain types of non-linear differential-difference equations such as (5) and (8) given in the succeeding paragraphs. Existence of solutions of differential equations and difference equations can be said to have been well studied, that of differential-difference equations, on the other hand, have been paid little attention. Such mixed type equations have great significance in applications. This paper, in particular, generalizes the Rellich–Wittich-type theorem and Malmquist-type theorem about differential equations to the case of differential-difference equations. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We consider a class of compressible fluids with nonlinear constitutive equations that guarantee that the divergence of the velocity field remains bounded. We study mathematical properties of unsteady three-dimensional flows of such fluids in bounded domains. In particular, we show the long-time and large-data existence result of weak solutions with strictly positive density. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we present a new coupled modified (1 + 1)-dimensional Toda equation of BKP type (Kadomtsev-Petviashvilli equation of B-type), which is a reduction of the (2 + 1)-dimensional Toda equation. Two-soliton and three-soliton solutions to the coupled system are derived. Furthermore, the *N*-soliton solution is presented in the form of Pfaffian. The asymptotic analysis of two-soliton solutions is studied to explain their collision properties. It is shown that the coupled system exhibit richer interaction phenomena including soliton fission, fusion, and mixed collision. Copyright © 2015 John Wiley & Sons, Ltd.

We consider the inverse problem of determining the time-dependent diffusivity in one-dimensional heat equation with periodic boundary conditions and nonlocal over-specified data. The problem is highly nonlinear and it serves as a mathematical model for the technological process of external guttering applied in cleaning admixtures from silicon chips. First, the well-posedness conditions for the existence, uniqueness, and continuous dependence upon the data of the classical solution of the problem are established. Then, the problem is discretized using the finite-difference method and recasts as a nonlinear least-squares minimization problem with a simple positivity lower bound on the unknown diffusivity. Numerically, this is effectively solved using the *l**s**q**n**o**n**l**i**n* routine from the MATLAB toolbox. In order to investigate the accuracy, stability, and robustness of the numerical method, results for a few test examples are presented and discussed. Copyright © 2015 John Wiley & Sons, Ltd.

We prove a logarithmic regularity criterion for the 3D generalized magnetohydrodynamics (MHD) system with diffusion terms −Δ*u* and (−Δ)^{β}*b*, with . Copyright © 2015 John Wiley & Sons, Ltd.

J. Banasiak In this paper, we deal with spectral properties of a weighted Laplacian in the half-space when a Dirichlet or a Neumann boundary condition is imposed. After proving that the spectrum is discrete under suitable assumptions, we give explicit formulae of eigenvalues and eigenfunctions in a specific case. In particular, the obtained eigenfunctions are rational or pseudo-rational and have remarkable orthogonality properties. These results suggest the use of the discovered functions for approximating solutions of elliptic problems in the half-space. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The aim of this paper is to investigate the dynamic of two SEIVS models, which incorporate an imperfect vaccination compartment. In this paper, we focus on the psychological inhibition effect of vaccinated individuals and the efficacy of vaccine on the spread of disease. For the susceptible individuals, we consider the psychological inhibition effect through the nonmonotone incidence rate. We find the disease-free and the disease persistent conditions. We also give some numerical simulations to demonstrate the effect of behavioral change of the vaccinated individuals and the efficiency of vaccine. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we consider the nonlinear oscillation of the following second-order neutral delay dynamic equations with distributed delay

on a time scale , where *Z*(*t*) = *x*(*t*) + *p*(*t*)*x*(*τ*(*t*)),*α*,*β* > 0 are constants. By using some new techniques, we obtain oscillation criteria for the equation when *β* > *α*,*β* = *α*, and *β* < *α*, respectively. Those results established here complete and develop the oscillation criteria in the literature. Also, our main results are illustrated with some examples. Copyright © 2015 John Wiley & Sons, Ltd.

We introduce a nabla, a delta, and a symmetric fractional calculus on arbitrary nonempty closed subsets of the real numbers. These fractional calculi provide a study of differentiation and integration of non-integer order on discrete, continuous, and hybrid settings. Main properties of the new fractional operators are investigated and some fundamental results presented, illustrating the interplay between discrete and continuous behaviors. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we establish sufficient conditions for the global relative controllability of nonlinear neutral fractional Volterra integro-differential systems with distributed delays in control. The results are obtained by using the Mittag–Leffler functions and the Schauder fixed-point theorem. Examples are presented to illustrate the results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>E. Sanchez-Palencia We derive a linearized prestressed elastic shell model from a nonlinear Kirchhoff model of elastic plates. The model is given in terms of displacement and micro-rotation of the cross-sections. In addition to the standard membrane, transverse shear, and flexural terms, the model also contains a nonstandard prestress term. The prestress is of the same order as flexural effects; hence, the model is appropriate when flexural effects dominate over membrane ones. We prove the existence and uniqueness of the solutions by Lax–Milgram theorem and compare solution with the solution of the standard shell model via numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

]]>S. Nicaise We consider the symmetric finite element–boundary element coupling that connects two linear elliptic second-order partial differential equations posed in a bounded domain Ω and its complement, where the exterior problem is restated as an integral equation on the coupling boundary *Γ* = *∂*Ω. Under the assumption that the corresponding transmission problem admits a shift theorem for data in *H*^{−1 + s},*s*∈[0,*s*_{0}],*s*_{0}>1/2, we analyze the discretization by piecewise polynomials of degree *k* for the domain variable and piecewise polynomials of degree *k* − 1 for the flux variable on the coupling boundary. Given sufficient regularity, we show that (up to logarithmic factors) the optimal convergence *O*(*h*^{k + 1/2}) in the *H*^{−1/2}(*Γ*)-norm is obtained for the flux variable, whereas classical arguments by Céa-type quasi-optimality and standard approximation results provide only *O*(*h*^{k}) for the overall error in the natural product norm on *H*^{1}(Ω) × *H*^{−1/2}(*Γ*). Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we develop a theoretical framework to investigate the influence of impulsive periodic disturbance on the evolutionary dynamics of a continuous trait, such as body size, in a general Lotka–Volterra-type competition model. The model is formulated as a system of impulsive differential equations. First, we derive analytically the fitness function of a mutant invading the resident populations when rare in both monomorphic and dimorphic populations. Second, we apply the fitness function to a specific system of asymmetric competition under size-selective harvesting and investigate the conditions for evolutionarily stable strategy and evolutionary branching by means of critical function analysis. Finally, we perform long-term simulation of evolutionary dynamics to demonstrate the emergence of high-level polymorphism. Our analytical results show that large harvesting effort or small impulsive harvesting period inhibits branching, while large impulsive harvesting period promotes branching. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The effects of wall impedances on the radiation of the dominant transverse electromagnetic wave by an impedance loaded parallel-plate waveguide radiator immersed in a cold plasma have been analyzed. The solution to the governing mathematical model in cold plasma is determined while using the Wiener–Hopf technique. It is observed that the amplitude of the radiated field increases with increasing permittivity of the plasma. The work presented may be of great interest to quantify the effects of ionosphere plasma on the communicating signals between Earth station and an artificial satellite in the Earth's atmosphere. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In digital signal processing, it is a well know fact that a causal signal of finite energy is front loaded if and only if the corresponding analytic signal, or the physically realizable signal, is a minimum phase signal, or an outer function in the complex analysis terminology. Based on this fact, a series expansion method, called unwinding adaptive Fourier decomposition (AFD), to give rise to positive frequency representations with rapid convergence was proposed several years ago. It appears to be a promising positive frequency representation with great potential of applications. The corresponding algorithm, however, is complicated due to consecutive extractions of outer functions involving computation of Hilbert transforms. This paper is to propose a practical algorithm for unwinding AFD that does not depend on computation of Hilbert transform, but, instead, factorizes out the Blaschke product type of inner functions. The proposed method significantly improves applicability of unwinding AFD. As an application, we give the associated Dirac-type time-frequency distribution of physically realizable signals. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we consider one-dimensional compressible viscous and heat-conducting micropolar fluid, being in a thermodynamical sense perfect and polytropic. The homogenous boundary conditions for velocity, microrotation, and temperature are introduced. This problem has a global solution with a priori estimates independent of time; with the help of this result, we first prove the exponential stability of solution in (*H*^{1}(0,1))^{4}, and then we establish the global existence and exponential stability of solutions in (*H*^{2}(0,1))^{4} under the suitable assumptions for initial data. The results in this paper improve those previously related results. Copyright © 2015 John Wiley & Sons, Ltd.

In the present research article, we introduce the King's type modification of *q*-Bernstein–Kantorovich operators and investigate some approximation properties. We show comparisons and present some illustrative graphics for the convergence of these operators to some function. Copyright © 2015 John Wiley & Sons, Ltd.

The problem of determining the manner in which an incoming acoustic wave is scattered by an elastic body immersed in a fluid is one of the central importance in detecting and identifying submerged objects. The problem is generally referred to as a fluid-structure interaction and is mathematically formulated as a time-dependent transmission problem. In this paper, we consider a typical fluid-structure interaction problem by using a coupling procedure that reduces the problem to a nonlocal initial-boundary problem in the elastic body with a system of integral equations on the interface between the domains occupied by the elastic body and the fluid. We analyze this nonlocal problem by the Lubich approach via the Laplace transform, an essential feature of which is that it works directly on data in the time domain rather than in the transformed domain. Our results may serve as a mathematical foundation for treating time-dependent fluid-structure interaction problems by convolution quadrature coupling of FEM and BEM. Copyright © 2015 John Wiley & Sons, Ltd.

]]>The analytic signal method via the circular Hilbert transform is a critical tool in the time–frequency analysis of signals of finite duration. The circular Bedrosian identity is of major theoretical and practical value in the method. The identity holds whenever the Fourier coefficients of *f*,*g*∈*L*^{2}([−*π*,*π*]) are respectively supported on *A* = [−*n*,*m*] and for some non-negative integers 0≤*n*,*m*≤+*∞*. In this note, we investigate the existence of such an identity for a general-bounded linear translation-invariant operator on *L*^{2}([−*π*,*π*]^{d}) and for general support sets . We give an insightful geometric characterization of the support sets for the existence. In addition, we find all the support sets for the partial Hilbert transforms. Copyright © 2015 John Wiley & Sons, Ltd.

Interval-valued fuzzy sets are an extension of fuzzy sets and are helpful when there is not enough information to define a membership function. This paper studies the behavior of a construction method for an interval-valued fuzzy relation built from a fuzzy relation. The behavior of this construction method is analyzed depending on the used t-norms and t-conorms, showing that different combinations of them produce a big variation in the results. Furthermore, a hybrid construction method that considers weight functions and a smoothing procedure is also introduced. Among the different applications of this method, the detection of edges in images is one of the most challenging. Thus, the performance of the proposal in detecting image edges is tested, showing that the hybrid approach that combines weights and a smoothing procedure provides better results than the non-weighted methods. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, for 3D modified Swift–Hohenberg equation, the optimal control problem is considered, the existence of optimal solution is proved, and the optimality system is established. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We analyse the dependence with respect to a given domain of the solution of a fluid-structure interaction scattering problem. We establish that the scattered field is continuously Fréchet differentiable with respect to the shape of the elastic scatterer. Our proof assumes that the boundary of the scatterer as well as the admissible perturbations to be only Lipschitzian. Given the applied nature of this problem and its prevalence in engineering and numerical literature, this new result appears to be of practical interest. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we prove the Saint-Venant compatibility conditions in *L*^{p} for *p*∈(1,+*∞*), in a simply connected domain of any space dimension. As a consequence, alternative, simple, and direct proofs of some classical Korn inequalities in *L*^{p} are provided. We also use the Helmholtz decomposition in *L*^{p} to show that every symmetric tensor in a smooth domain can be decomposed in a compatible part, which is the symmetric part of a displacement gradient, and in an incompatible part, which is the incompatibility of a certain divergence-free tensor. Moreover, under a suitable Dirichlet boundary condition, this Beltrami-type decomposition is proved to be unique. This decomposition result has several applications, one of which being in dislocation models, where the incompatibility part is related to the dislocation density and where 1 < *p* < 2. This justifies the need to generalize and prove these rather classical results in the Hilbertian case (*p* = 2), to the full range *p*∈(1,+*∞*). Copyright © 2015 John Wiley & Sons, Ltd.

In this work, a modified Holling–Tanner predator–prey model is analyzed, considering important aspects describing the interaction such as the predator growth function is of a logistic type; a weak Allee effect acting in the prey growth function, and the functional response is of hyperbolic type. Making a change of variables and time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one in which the non-hyperbolic equilibrium point (0,0) is an attractor for all parameter values. An important consequence of this property is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane, and the system exhibits the bistability phenomenon, because the trajectories can have different *ω* − *l**i**m**i**t* sets; as example, the origin (0,0) or a stable limit cycle surrounding an unstable positive equilibrium point. We show that, under certain parameter conditions, a positive equilibrium may undergo saddle-node, Hopf, and Bogdanov–Takens bifurcations; the existence of a homoclinic curve on the phase plane is also proved, which breaks in an unstable limit cycle. Some simulations to reinforce our results are also shown. Copyright © 2015 John Wiley & Sons, Ltd.

Chemostat refers to a laboratory device used for growing microorganisms in a cultured environment and has been regarded as an idealization of nature to study competition modeling in mathematical biology. The simple form of chemostat model assumes that the availability of nutrient and its supply rate are both fixed. In addition, the tendency of microorganisms to adhere to surfaces is neglected by assuming the flow rate is fast enough. However, these assumptions largely limit the applicability of chemostat models to realistic competition systems. In this paper, we relax these assumptions and study chemostat models with random nutrient supplying rate or random input nutrient concentration, with or without wall growth. This leads to random dynamical systems and requires the concept of random attractors developed in the theory of random dynamical systems. Our results include existence of uniformly bounded non-negative solutions, existence of random attractors, and geometric details of random attractors for different value of parameters. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Holomorphic function theory is an effective tool for solving linear elasticity problems in the complex plane. The displacement and stress field are represented in terms of holomorphic functions, well known as Kolosov–Muskhelishvili formulae. In , similar formulae were already developed in recent papers, using quaternionic monogenic functions as a generalization of holomorphic functions. However, the existing representations use functions from to , embedded in . It is not completely appropriate for applications in . In particular, one has to remove many redundancies while constructing basis solutions. To overcome that problem, we propose an alternative Kolosov–Muskhelishvili formula for the displacement field by means of a (paravector-valued) monogenic, an anti-monogenic and a *ψ*-hyperholomorphic function. Copyright © 2015 John Wiley & Sons, Ltd.

In this research article, we investigated the existence of local smooth solutions for relativistic radiation hydrodynamic equations in one spatial variable. The proof is based on a classical iteration method and the Banach contraction mapping principle. However, because of the complexity of relativistic radiation hydrodynamics equations, we first rewrite this system into a semilinear form to construct the iteration scheme and then use left eigenvectors to decouple the system instead of applying standard energy method on symmetric hyperbolic systems. Different from multidimensional case, we just use the characteristic method, which can keep the properties of the initial data. Copyright © 2015 John Wiley & Sons, Ltd.

]]>A novel approach for mechanical vibration signal de-noising filter using PDE and its numerical solution were presented. The proposed method is computationally fast compared with other conventional PDE-based de-noising methods. It enables: (i) by incorporating unconditional stable finite difference backward Euler scheme, the de-noising process has no requirements of grid ratio; (ii) developing variational matrix-based fast filter while the de-noising process can be completed instantly, which will be accomplished by only one iteration; and (iii) effective de-noising method for mechanical vibration signal interfered by Gauss white noise. The method is performed efficiently, and the de-noising tests on different artificial Gauss white noise as well as natural mechanical noise are conducted. Experimental tests have been rigorously compared with different de-noising methods to verify the efficacy of the proposed method. Copyright © 2015 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.

]]>The paper deals with the oscillation analysis of numerical solution in the *θ*-methods for differential equations with piecewise constant arguments of advanced type. The conditions of the oscillation for the *θ*-method are obtained. It is proved that the oscillation of the analytic solution is preserved by the *θ*- method. Some numerical experiments are given. Copyright © 2015 John Wiley & Sons, Ltd.

In this work, the residual-type posteriori error estimates of stabilized finite volume method are studied for the steady Stokes problem based on two local Gauss integrations. By using the residuals between the source term and numerical solutions, the computable global upper and local lower bounds for the errors of velocity in *H*^{1} norm and pressure in *L*^{2} norm are derived. Furthermore, a global upper bound of *u* − *u*_{h} in *L*^{2}-norm is also derived. Finally, some numerical experiments are provided to verify the performances of the established error estimators. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, a class of virus infection model with Beddington–DeAngelis infection function and cytotoxic T-lymphocyte immune response is investigated. Time delay in the immune response term is incorporated into the model. We show that the dynamics of the model are determined by the basic reproduction number and the immune response reproduction number . If , then the infection-free equilibrium is globally asymptotically stable. If , then the immune-free equilibrium is globally asymptotically stable. If , then the stability of the interior equilibrium is investigated. We conclude that Hopf bifurcation occurs as the time delay passes through a critical value. Numerical simulations are carried out to support our theoretical conclusion well. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the dynamical behaviors of three human immunodeficiency virus infection models with two types of cocirculating target cells and distributed intracellular delay. The models take into account both short-lived infected cells and long-lived chronically infected cells. In the two types of target cells, the drug efficacy is assumed to be different. The incidence rate of infection is given by bilinear and saturation functional responses in the first and second models, respectively, while it is given by a general function in the third model. Lyapunov functionals are constructed and LaSalle invariance principle is applied to prove the global asymptotic stability of all equilibria of the models. We have derived the basic reproduction number *R*_{0} for the three models. For the first two models, we have proven that the disease-free equilibrium is globally asymptotically stable (GAS) when *R*_{0}≤1, and the endemic equilibrium is GAS when *R*_{0}>1. For the third model, we have established a set of sufficient conditions for global stability of both equilibria of the model. We have checked our theoretical results with numerical simulations. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we study an a posteriori error indicator introduced in E. Dari, R.G. Durán, and C. Padra, *Appl. Numer. Math*., 2012, for the approximation of the Laplace eigenvalue problem with Crouzeix–Raviart nonconforming finite elements. In particular, we show that the estimator is robust also in presence of eigenvalues of multiplicity greater than one. Some numerical examples confirm the theory and illustrate the convergence of an adaptive algorithm when dealing with multiple eigenvalues. Copyright © 2015 John Wiley & Sons, Ltd.

Both discrete and distributed delays are considered in a two-neuron system. We analyze the influence of interaction coefficient and time delay on the Hopf-pitchfork bifurcation. First, we obtain the codimension-2 unfolding with original parameters for Hopf-pitchfork bifurcation by using the center manifold reduction and the normal form method. Next, through analyzing the unfolding structure, we give complete bifurcation diagrams and phase portraits, in which multistability and other dynamical behaviors of the original system are found, such as a stable periodic orbit, the coexistence of two stable nontrivial equilibria, and the coexistence of a stable periodic orbit and two stable equilibria. In addition, the obtained theoretical results are verified by numerical simulations. Finally, we perform the comparisons of the obtained results of Hopf-pitchfork bifurcation with other Hopf-fold bifurcation results in some biological neural systems and give the obtained mathematical results corresponding to the physical states of neurons. Copyright © 2015 JohnWiley & Sons, Ltd.

]]>Diffusion processes have traditionally been modeled using the classical parabolic advection-diffusion equation. However, as in the case of tracer transport in porous media, significant discrepancies between experimental results and numerical simulations have been reported in the literature. Therefore, in order to describe such anomalous behavior, known as non-Fickian diffusion, some authors have replaced the parabolic model with the continuous time random walk model, which has been very effective. Integro-differential models (IDMs) have been also proposed to describe non-Fickian diffusion in porous media. In this paper, we introduce and test a particular type of IDM by fitting breakthrough curves resulting from laboratory tracer transport. Comparisons with the traditional advection-diffusion equation and the continuous time random walk are also presented. Moreover, we propose and numerically analyze a stable and accurate numerical procedure for the two-dimensional IDM composed by a integro-differential equation for the concentration and Darcy's law for flow. In space, it is based on the combination of mixed finite element and finite volume methods over an unstructured triangular mesh. Copyright © 2015 John Wiley & Sons, Ltd.

]]>By means of nonsmooth critical point theory, we obtain existence of infinitely many weak solutions of the fractional Schrödinger equation with logarithmic nonlinearity. We also investigate the Hölder regularity of the weak solutions. Copyright © 2015 JohnWiley & Sons, Ltd

]]>In this paper, the photothermal effects of plasmon resonance are investigated. Metal nanoparticles efficiently generate heat in the presence of electromagnetic radiation. The process is strongly enhanced when a fixed frequency of the incident wave illuminates on nanoparticles such that plasmon resonance happens. We introduce the electromagnetic radiation model and show exactly how and when the plasmon resonance happens. We then construct the heat generation and transfer theory and derive the heat effect induced by plasmon resonance. Finally, the heat generation under plasmon resonance in a concentric nanoshell structure is considered specially, and excited result is obtained. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, the global asympotic behavior of solutions of a class of continuous-time dynamical system is studied. Not only do we obtain the ultimate boundedness of solutions of the system but we also obtain the rate of the trajectories of the system going from the exterior of the trapping set to the interior of the trapping set, which can be applied to study chaotic control and chaotic synchronization of the system. Copyright © 2015 John Wiley & Sons, Ltd.

]]>A simple proof is given of a new summation formula recently added in the literature for a terminating _{r + 3}*F*_{r + 2}(1) hypergeometric series for the case when *r* pairs of numeratorial and denominatorial parameters differ by positive integers. This formula represents an extension of the well-known Saalschütz summation formula for a _{3}*F*_{2}(1) series. Two applications of this extended summation formula are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde–Chu summation theorem for the _{2}*F*_{1} series, extends certain reduction formulas for the Kampé de Fériet function of two variables given by Exton and Cvijović & Miller. Copyright © 2015 John Wiley & Sons, Ltd.

This paper deals with the Cauchy problem for a doubly degenerate parabolic equation with variable coefficient

For the case *λ* + 1 ≥ *N*, one proves that depending on the behavior of the variable coefficient at infinity, the Cauchy problem either possesses the property of finite speed of propagation of perturbation or the support blows up in finite time. This completes a result by Tedeev (A.F.Tedeev, The interface blow-up phenomenon and local estimates for doubly degenerate parabolic equations, Appl. Anal. 86 (2007) 755–782.), which asserts the same result under the condition *λ* + 1 < *N*. Copyright © 2014 John Wiley & Sons, Ltd.

Well-posed boundary-value problems in multiply-connected regions are targeted for some sets of two-dimensional Laplace equations written in geographical coordinates on joint surfaces of revolution. Those are problems that simulate potential fields induced by point sources in joint perforated thin shell structures consist of fragments of different geometry. A semi-analytical approach is proposed to accurately compute solutions of such problems. The approach is based on the matrix of Green's type formalism. The elements of required matrices of Green's type are obtained analytically and expressed in closed computer-friendly form. This makes it possible to efficiently deal with the targeted class of problems. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with the nonlinear Dirac equations

Under suitable assumptions on the nonlinearity, we establish the existence of infinitely many large energy solutions by the generalized variant fountain theorem developed recently by Batkam and Colin. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with the simple waves for the general quasilinear strictly hyperbolic systems in two independent variables. By using the method of characteristic decomposition, we first establish a more general sufficient condition for the existence of characteristic decompositions. These decompositions allow us to extend a well-known result on reducible equations by Courant and Friedrichs to the non-reducible equations. Then we construct a simple wave solution, in which the characteristics may be a set of non-straight curves, around a given curve under the obtained sufficient condition. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we will obtain that there exists a maximizer for the non-endpoint Strichartz inequalities for the fourth-order Schrödinger equation with initial data in the *L*^{2}(**R**^{d}) space in all dimensions, and then we obtain a maximizer also for the non-endpoint Sobolev–Strichartz inequality for the fourth-order Schrödinger equation with initial data in the homogeneous Sobolev space. Our analysis derived from the linear profile decomposition. Copyright © 2014 John Wiley & Sons, Ltd.

We study the initial boundary value problem of a class of fourth order semilinear parabolic equations. Global existence and nonexistence of solutions with initial data in the potential well are derived. Moreover, by using the iteration technique for regularity estimates, we obtain that for any *k* ≥ 0, the semilinear parabolic possesses a global attractor in *H*^{k}(Ω), which attracts any bounded subsets of *H*^{k}(Ω) in the *H*^{k}-norm. Copyright © 2014 John Wiley & Sons, Ltd.

When one characteristic of the system is linearly degenerate, under suitable boundary conditions, we get the existence of traveling wave solutions located on the corresponding characteristic trajectory to the one-sided mixed initial-boundary value problem. When the system is linearly degenerate, by introducing the semi-global normalized coordinates, we derive the related formulas of wave decomposition to prove the stability of traveling wave solutions corresponding to all leftward and the rightmost characteristic trajectories. Finally, for the traveling wave solutions corresponding to other rightward characteristic trajectories, some examples show their possible instability. Copyright © 2014 John Wiley & Sons, Ltd.

This work is focused on the long-time behavior of solutions to the singularly perturbed damped Boussinesq equation in a 3D case

where *ε* > 0 is small enough. Without any growth restrictions on the nonlinearity *f*(*u*), we establish in an appropriate bounded phase space a finite dimensional global attractor as well as an exponential attractor of optimal regularity. The key step is the estimate of the difference between the solutions of the damped Boussinesq equation and the corresponding pseudo-parabolic equation.Copyright © 2014 John Wiley & Sons, Ltd.

We study the inverse conductivity problem with partial data in dimension *n* ≥ 3. We derive stability estimates for this inverse problem if the conductivity has regularity for 0 < *σ* < 1. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we consider a class of stochastic delay fractional evolution equations driven by fractional Brownian motion in a Hilbert space. Sufficient conditions for the existence and uniqueness of mild solutions are obtained. An application to the stochastic fractional heat equation is presented to illustrate the theory. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Picone type formula for half-linear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions is derived. Employing the formula, Leighton and Sturm–Picone type comparison theorems as well as several oscillation criteria for impulsive differential equations are obtained. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Our aim in this work is to find decay integral solutions for a class of neutral fractional differential equations in Banach spaces involving unbounded delays. By constructing a suitable measure of noncompactness on the space of solutions and establishing new estimates for fractional resolvent operators, we prove the existence of a compact set of decay integral solutions to the mentioned problem. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we study the global structure of the positive solutions to a logistic equation with constant yield harvesting under Neumann boundary value conditions. Moreover, we show that the logistic equation with the variable coefficients has exactly either zero, or one, or two solutions depending on the harvesting parameter. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, a novel 3D nonlinear system with chaotic behavior is considered. Our proposed system is a generalized Lorenz-like system, with no reflections about coordinates axes. Although the considered system depends on seven real parameters, its stability is completely analyzed. Also, the presence of Hopf bifurcation is pointed out. Copyright © 2014 John Wiley & Sons, Ltd.

]]>This paper is concerned with a class of dynamic boundary systems with boundary feedback. The well-posedness of the considered systems is proved under some regularity conditions. Moreover, some spectral properties are derived. As an application, the well-posedness and the asymptotic behavior of population dynamical systems with unbounded birth process ‘ ’ are solved. Such population dynamical systems were pointed out in [S. Piazzera, Math. Methods Appl. Sci., 27 (2004), 427-439] to be a current research topic in semigroup theory and still an open problem. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The stochastic stability problem of networked control systems (NCSs) with random time delays and packet dropouts is investigated in this paper. The mathematical NCS model is developed as a stochastic discrete-time jump system with combined integrated stochastic parameters characterized by two identically independently distributed processes, which accommodate the abrupt variations of network uncertainties within an integrated frame. The effective instant is introduced to establish the relationship between the destabilizing transmission factors and stability of NCSs. The stabilizing state feedback controller gain that depends not only on the delay modes but also on the dropouts modes is obtained in terms of the linear matrix inequalities formulation via the Schur complement theory. A numerical example is given to demonstrate the effectiveness of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Using the Krasnoselskii–Zabreiko fixed point theorem, we establish two existence theorems for positive solutions of a coupled system of nonlinear fractional differential equations. Power functions and nonnegative matrices are used to characterize coupling behavior of our nonlinearities, so nonlinearities may grow differently; in fact, one may grow superlinearly, and the other may grow sublinearly. Copyright © 2014 John Wiley & Sons, Ltd.

We prove the global-in-time existence and uniqueness of weak solutions in critical spaces for a mathematical model in superfluidity, with initial data *ψ*_{0},*A*_{0} ∈ *L*^{3},*u*_{0} ∈ *L*^{3 ∕ 2},*u*_{0} ≥ 0 in three dimension and in two dimension. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, by using the Nehari manifold approach in combination with periodic approximations, we obtain the sufficient conditions on the existence of the nontrivial ground state solutions of the periodic discrete coupled nonlinear Schrödinger equations. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The bound of a chaotic system is important for chaos control, chaos synchronization, and other applications. In the present paper, the bounds of the generalized Lorenz system are studied, based on the Lyapunov function theory and the Lagrange multiplier method. We obtain a precise bound for the generalized Lorenz system. The rate of the trajectories is also obtained. Furthermore, we perform the numerical simulations. Numerical simulations are presented to show the effectiveness of the proposed scheme. Copyright © 2014 John Wiley & Sons, Ltd.