In this paper, we present a method for solving the Emden–Fowler equation using the Adomian polynomials and the operational calculus introduced by the authors. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We consider a logistic growth model with a predation term and a stochastic perturbation yielding constant elasticity of variance. The resulting stochastic differential equation does not satisfy the standard assumptions for existence and uniqueness of solutions, namely, linear growth and the Lipschitz condition. Nevertheless, for any positive initial condition, we prove that a solution exists and is unique up to the first time it hits zero. Additionally, we provide alternative criteria for population extinction depending on the choice of parameters. More precisely, we provide criteria that guarantee the following: (i) population extinction with positive probability for a set of initial conditions with positive Lebesgue measure; (ii) exponentially fast population extinction with full probability for any positive initial condition; and (iii) population extinction in finite time with full probability for any positive initial condition. 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 following: the predator growth function is of a logistic type, and a weak Allee effect acting in the prey growth function and the functional response is of hyperbolic type.

By making a change of variables and a time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one, in which the nonhyperbolic 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*-limit sets, as an 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.

In this paper, we consider a quite general class of reaction-diffusion equations with cubic nonlinearities and with random Neumann boundary conditions. We derive rigorously amplitude equations, using the natural separation of time-scales near a change of stability and investigate whether additive degenerate noise and random boundary conditions can lead to stabilization of the solution of the stochastic partial differential equation or not. The nonlinear heat equation (Ginzburg–Landau equation) is used to illustrate our result. Copyright © 2015 John Wiley & Sons, Ltd.

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

]]>In some species, the population may decline to zero; that is, the species becomes extinct if the population falls below a given threshold. This phenomenon is well known as an Allee effect. In most Allee models, the model parameters are constants, and the population tends either to a nonzero limiting state (survival) or to zero (extinction). However, when environmental changes occur, these parameters may be slowly varying functions of time. Then, application of multitiming techniques allows us to construct approximations to the evolving population in cases where the population survives to a slowly varying surviving state and those where the population declines to zero. Here, we investigate the solution of a logistic population model exhibiting an Allee effect, when the carrying capacity and the limiting density interchange roles, via a transition point. We combine multiscaling analysis with local asymptotic analysis at the transition point to obtain an overall expression for the evolution of the population. We show that this shows excellent agreement with the results of numerical computations. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we propose a new scheme that combines weighted essentially non-oscillatory (WENO) procedures together with monotone upwind schemes to approximate the viscosity solution of the Hamilton–Jacobi equations. In one-dimensional (1D) case, first, we obtain an optimum polynomial on a four-point stencil. This optimum polynomial is third-order accurate in regions of smoothness. Next, we modify a second-order ENO polynomial by choosing an additional point inside the stencil in order to obtain the highest accuracy when combined with the Harten–Osher reconstruction-evolution method limiter. Finally, the optimum polynomial is considered as a symmetric and convex combination of three polynomials with ideal weights. Following the methodology of the classic WENO procedure, then, we calculate the non-oscillatory weights with the ideal weights. Numerical experiments in 1D and 2D are performed to compare the capability of the hybrid scheme to WENO schemes. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we introduce a class of predator–prey system with general functional response, whose harvesting policy is modeled by a discontinuous function. Based on the differential inclusions theory, topological degree theory in set-valued analysis and generalized Lyapunov approach, we analyze the existence, uniqueness and global asymptotic stability of positive periodic solution. In particular, a series of useful criteria on existence, uniqueness and global asymptotic stability of the positive equilibrium point are established for the autonomous system corresponding to the non-autonomous biological and mathematical model with a discontinuous right-hand side. Moreover, some new sufficient conditions are provided to guarantee the global convergence in measure of harvesting solution and convergence in finite time of any positive solution for the autonomous discontinuous biological system. The obtained results, which improve and generalize previous works on dynamical behavior in the literature, are of interest for understanding and designing biological system with not only continuous or even Lipschitz continuous but also discontinuous harvesting function. Finally, we give three examples with numerical simulations to show the applicability and effectiveness of our main results. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We obtain explicit formulas for the scattering of plane waves with arbitrary profile by a wedge under Dirichlet, Neumann and Dirichlet-Neumann boundary conditions. The diffracted wave is given by a convolution of the profile function with a suitable kernel corresponding to the boundary conditions. We prove the existence and uniqueness of solutions in appropriate classes of distributions and establish the Sommerfeld type representation for the diffracted wave.

As an application, we establish (i) stability of long-time asymptotic local perturbations of the profile functions and (ii) the limiting amplitude principle in the case of a harmonic incident wave. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we prove existence of global in time weak solutions for a highly nonlinear PDE system arising in the context of damage phenomena in thermoviscoelastic materials. The main novelty of the present contribution with respect to the ones already present in the literature consists in the possibility of taking into account a damage-dependent thermal expansion coefficient. This term implies the presence of nonlinear coupling terms in the PDE system, which make the analysis more challenging. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Many models of asymmetric distributions proposed in the statistical literature are obtained by transforming an arbitrary symmetric distribution by means of a skewing mechanism. In certain important cases, the resultant skewed distribution shares some properties of its symmetric antecedent. Because of this inheritance, it would be interesting to test if the symmetric generator belongs to a certain family, that is to say, testing goodness-of-fit for the symmetric component. This work proposes a test of such hypothesis. Taking into account that the normal law is perhaps the most studied distribution, as a particular case of unquestionable interest, the generalized skew-normal family is studied in detail, because the symmetric component of the distributions in this family is normal. Copyright © 2015 John Wiley & Sons, Ltd.

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

]]>This article investigates a key information-theoretic performance metric in multiple-antenna wireless communications, the so-called outage probability. The article is partly a review, with the methodology based mainly on [10], while also presenting some new results. The outage probability may be expressed in terms of a moment generating function, which involves a Hankel determinant generated from a perturbed Laguerre weight. For this Hankel determinant, we present two separate integral representations, both involving solutions to certain non-linear differential equations. In the second case, this is identified with a particular *σ*-form of Painlevé V. As an alternative to the Painlevé V, we show that this second integral representation may also be expressed in terms of a non-linear second order difference equation. Copyright © 2014 John Wiley & Sons, Ltd.

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

The dynamics of normal and leukemic stem cell clones competing for resources in the same bone marrow niche can be modeled in a systems biology approach. We derive three related models describing the time evolution of normal and chronic myelogenous leukemia (CML) stem cells. The approach is based on the idea that stem cell proliferation is regulated by feedback mechanisms. We assume here that the growth of stem cell clones can be approximated by Tsallis functions, which depend on population numbers to ensure the communication between different stem cell strains. Of particular interest is the influence of medication, for example, by the CML drug *imatinib*, on the stem cell dynamics, which can be simulated by a variation of the parameterization of the differential equation systems. The basic 2D model represents the contest between cycling normal and wild-type CML stem cells. In addition, extension of the basic model (i) by coupled reservoirs of quiescent stem cells and (ii) by a third stem cell species, which could be interpreted as an *imatinib*-resistant mutant, is investigated. We analyze the global dynamics of the corresponding 2D, 3D, and 4D equation systems by analytic means and provide a complete map of the bifurcation landscape that occurs for changing values of the respective signaling strengths and growth or death parameters. This includes a complete classification of equilibria and their linear and non-linear Lyapunov stability. Copyright © 2014 John Wiley & Sons, Ltd.

A family of conjugated distributions for a given type of copulas is defined in this paper. Those copulas can be written as a mixture of *d*-dimensional parameter exponential functions. The generalized Farlie–Gumbel–Morgenstern copula is an example of this representation. This family is used to illustrate the estimation technique with real data. Also, the applicability of Bayesian predictive approach is shown in an education policy issue by defining goals for the number of students per class that leads to improve their performance at school. Copyright © 2014 John Wiley & Sons, Ltd.

We investigate the properties of the operator , where *σ* is a given parameter whose sign can change on the bounded domain Ω. Here, denotes the subspace of H^{2}(Ω) made of the functions *v* such that *v* = *ν*·∇*v* = 0 on *∂*Ω. The study of this problem arises when one is interested in some configurations of the interior transmission eigenvalue problem. We prove that is a Fredholm operator of index zero as soon as *σ*∈L^{∞}(Ω), with *σ*^{−1}∈L^{∞}(Ω), is such that *σ* remains uniformly positive (or uniformly negative) in a neighbourhood of *∂*Ω. We also study configurations where *σ* changes sign on *∂*Ω, and we prove that Fredholm property can be lost for such situations. In the process, we examine in details the features of a simpler problem where the boundary condition *ν*·∇*v* = 0 is replaced by *σ*Δ*v* = 0 on *∂*Ω. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we establish the existence and non-existence of positive solutions for *p*-Kirchhoff type problems with a parameter on without assuming the usual compactness conditions. We show that the *p*-Kirchhoff type problems have at least one positive solution when the parameter is small, while the *p*-Kirchhoff type problems have no positive solutions when the parameter is large. Our argument is based on variational methods, monotonicity methods, cut-off functional techniques, and a priori estimates techniques. Copyright © 2014 John Wiley & Sons, Ltd.

The close connection between Maxwell's equations and the Dirac equation is under consideration in a rigorous functional analytical framework. The two systems are linked via the so-called extended Maxwell system Picard R. A structural observation for linear material laws in classical mathematical physics. Mathematical Methods in the Applied Sciences 2009;32(14):1768–1803, which is here re-considered in the time-dependent case as an evolutionary space-time operator equation. This structural observation is then applied to recover and generalize the equations of gravito-electromagnetism and to reformulate theMaxwell-Dirac system as a system of three coupled extended Maxwell systems. This reformulation rests on the observation that what in electrodynamics is commonly known as ‘potential’ is actually a solution to another extended Maxwell system. Copyright © 2014 John Wiley & Sons, Ltd.

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

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

]]>In this paper, we study the qualitative behavior of a competitive system of second-order rational difference equations. More precisely, we investigate the boundedness character, existence and uniqueness of positive equilibrium point, local asymptotic stability and global stability of the unique positive equilibrium point and rate of convergence of positive solutions of the system. Some numerical examples are given to verify our theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.

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

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

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

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

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

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

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

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

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

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

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

]]>We present a mathematical model describing an electrochemical system involving electrode–electrolyte interaction. The model is governed by a system of advection–diffusion equations with a nonlinear reaction term at the boundary. Our calculations based on such model demonstrate the dynamics of ionic currents in the electrolyte. The model allows us to predict the effect of varying flow rates, scan rates, and electrolyte concentration of the electrochemical system. Copyright © 2014 John Wiley & Sons, Ltd.

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

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

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

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

The very weak solution for the Stokes, Oseen and Navier–Stokes equations has been studied by several authors in the last decades in domains of . The authors studied the Oseen and Navier–Stokes problems assuming a solenoidal convective velocity in a bounded domain of class for **v**∈**L**^{s}(Ω) for *s*≥3 in some previous papers. The results for the Navier–Stokes equations were obtained by using a fixed-point argument over the Oseen problem. These results improve those of Galdi *et al.* , Farwig *et al.* and Kim for the Navier–Stokes equations, because a less regular domain and more general hypothesis on the data are considered. In particular, the external forces must not be small.

In this work, existence of weak, strong, regularised and very weak solution for the Oseen problem are proved, mainly assuming that **v**∈**L**^{3}(Ω) and its divergence ∇·**v** are sufficiently small in the *W*^{−1,3}(Ω)-norm. In this sense, one extends the analysis made by the authors for a given solenoidal **v** in some previous papers. As a consequence, the existence of very weak solution for the Navier–Stokes problem for a non-zero divergence condition is obtained in the 3D case. Copyright © 2014 John Wiley & Sons, Ltd.

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

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

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

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

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

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

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

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

Dengue is a vector-borne disease transmitted from an infected human to an *Aedes* mosquito, during a blood meal. Dengue is still a major public health problem. A model for the disease transmission is presented, composed by human and mosquitoes compartments. The aim is to simulate the effects of seasonality, on the vectorial capacity and, consequently, on the disease development. Using entomological information about the mosquito behavior under different temperatures and rainfall, simulations are carried out, and the repercussions analyzed. The basic reproduction number of the model is given, as well as a sensitivity analysis of model's parameters. Finally, an optimal control problem is proposed and solved, illustrating the difficulty of making a trade-off between reduction of infected individuals and costs with insecticide. Copyright © 2014 John Wiley & Sons, Ltd.

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

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

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

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

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

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

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

]]>In this paper, the Bäcklund transformation of fractional Riccati equation is presented to establish traveling wave solutions for two nonlinear space–time fractional differential equations in the sense of modified Riemann–Liouville derivatives, namely, the space–time fractional generalized reaction duffing equation and the space–time fractional diffusion reaction equation with cubic nonlinearity. The proposed method is effective and convenient for solving nonlinear evolution equations with fractional order. Copyright © 2014 John Wiley & Sons, Ltd.

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

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

The record of the oscillations of the electric potential of the human brain provides useful information about the mind activity at rest and during the achievement of sensory and cognitive processing tasks. The use of appropriate quantitative tools assigning numerical values to the observed variable is necessary to define good descriptors of the electroencephalogram, allowing comparisons between different recordings. In this line, we propose a numerical method for the spectral and temporal reconstruction of a brain signal. The convergence of the procedure is analyzed, providing results of the concerned approximation error. In a second part of the text, we use the methodology described to the quantification of the bioelectric variations produced in the brain waves for the execution of a test of attention related to military simulation. Copyright © 2014 JohnWiley & Sons, Ltd.

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

We investigate a time-domain Galerkin boundary element method for the wave equation outside a Lipschitz obstacle in an absorbing half-space. A priori estimates are presented for both closed surfaces and screens, and we discuss the relevant properties of anisotropic Sobolev spaces and the boundary integral operators between them. Copyright © 2014 John Wiley & Sons, Ltd.

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

]]>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.

]]>The paper deals with the two-dimensional problem of the elastic equilibrium of a body containing a thin rigid inclusion. The rigid inclusion is considered to be soldered into elastic medium. The volume forces act on the body; at the external boundary the body is fixed. The first and second variations of the solution with respect to the shape of the domain are calculated. As an illustrative application of the results, the optimal shape design problem is considered. The necessary and sufficient conditions for local optimality are written. Copyright © 2014 John Wiley & Sons, Ltd.

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

In this per, we consider a special class of initial data for the three-dimensional incompressible Navier–Stokes equations with gravity. We show that, under such conditions, the incompressible Navier-Stokes equations with gravity are globally well posed, and the velocity minus gravity term has finite energy. The important features of the initial data is that the velocity fields minus gravity term are almost parallel to the corresponding vorticity fields in a very large space domain. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we consider the dynamical von Kármán equations of memory type with acoustic boundary conditions. We show an exponential decay result of solutions under weaker assumption than the ones frequently used in the literature. In particular, the kernel we are considering is not necessarily exponentially decaying to zero as was assumed before. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Threshold noise reduction methods of vibration signals have been widely researched and used. However, these methods are less efficient in such situation, including requiring a time-consuming and subjective to manual editing because different degree of noise signal requires selecting different characterization for filtering. In this paper, an efficient denoising method based on PDE for mechanical vibration signals time-frequency distribution is investigated, in which, a one-dimensional vibration signal is transformed into 2D time-frequency domain by using Gabor transform. This enables (i) simultaneously utilize both time and frequency characteristic for effectively multiple dimension signal denosing and (ii) isotropic and anisotropic characteristics to be imposed by employing PDE, which explicitly fit with the local structure of time-frequency signal. This paper analyzes the basic methods of isotropic and anisotropic diffusion filtering, investigates the anisotropic diffusion method based on local feature structure of 2D information, and conducts a set of comparative tests. Experiments show that this proposed method has a better performance of denoising than that of thresholding. At the same time, it is more handy than that of other methods, such as independent component analysis. Finally, problems and ways of improving the PDE-based filter method are analyzed in this paper. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Topological persistence has proven to be a promising framework for dealing with problems concerning the analysis of data. In this context, it was originally introduced by taking into account 1-dimensional properties of data, modeled by real-valued functions. More recently, topological persistence has been generalized to consider multidimensional properties of data, coded by vector-valued functions. This extension enables the study of *multidimensional persistent Betti numbers*, which provide a representation of data based on the properties under examination. In this contribution, we establish a new link between multidimensional topological persistence and Pareto optimality, proving that discontinuities of multidimensional persistent Betti numbers are necessarily pseudocritical or special values of the considered functions. Copyright © 2014 John Wiley & Sons, Ltd.

This paper deals with the Cauchy problem for the degenerate parabolic equation with a strongly nonlinear source

where *N* ≥ 1,* p* > 2,* q* ≥ *p* − 1, and the blow-up time *T* < ∞ . It has been shown that the solution *u*(*x*,*t*) is strictly localized for *q* ≥ *p* − 1, provided that the initial function *u*_{0}(*x*) has a compact support by Liang and Zhao. In addition, if *q* > 2*p* − 1, an upper estimate on the localization in terms of the initial support and the blow-up time *T* is partially derived by Liang. In this work, by using the De Giorgi-type iteration technique, we give a complete estimate on the localization for all *q* ≥ *p* − 1. Copyright © 2014 John Wiley & Sons, Ltd.

In the present paper, we construct exact solutions to a system of partial differential equations *iu*_{x} + *v* + *u* | *v* | ^{2} = 0, *iv*_{t} + *u* + *v* | *u* | ^{2} = 0 related to the Thirring model. First, we introduce a transform of variables, which puts the governing equations into a more useful form. Because of symmetries inherent in the governing equations, we are able to successively obtain solutions for the phase of each nonlinear wave in terms of the amplitudes of both waves. The exact solutions can be described as belonging to two classes, namely, those that are essentially linear waves and those which are nonlinear waves. The linear wave solutions correspond to waves propagating with constant amplitude, whereas the nonlinear waves evolve in space and time with variable amplitudes. In the traveling wave case, these nonlinear waves can take the form of solitons, or solitary waves, given appropriate initial conditions. Once the general solution method is outlined, we focus on a number of more specific examples in order to show the variety of physical solutions possible. We find that radiation naturally emerges in the solution method: if we assume one of *u* or *v* with zero background, the second wave will naturally include both a solitary wave and radiation terms. The solution method is rather elegant and can be applied to related partial differential systems. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we study a virus dynamics model with logistic mitosis, cure rate, and intracellular delay. By means of construction of a suitable Lyapunov functionals, obtained by linear combinations of Volterra—type functions, composite quadratic functions and Volterra—type functionals, we provide the global stability for this model. If *R*_{0}, the basic reproductive number, satisfies *R*_{0} ≤ 1, then the infection-free equilibrium state is globally asymptotically stable. Our system is persistent if *R*_{0} > 1. On the other hand, if *R*_{0} > 1, then infection-free equilibrium becomes unstable and a unique infected equilibrium exists. The local stability analysis is carried out for the infected equilibrium, and it is shown that, if the parameters satisfy a condition, the infected equilibrium can be unstable and a Hopf bifurcation can occur. We also have that if *R*_{0} > 1, then the infected equilibrium state is globally asymptotically stable if a sufficient condition is satisfied. We illustrate our findings with some numerical simulations. Copyright © 2014 John Wiley & Sons, Ltd.

In this article, we show that a technique for showing well-posedness results for evolutionary equations in the sense of Picard and McGhee [Picard, McGhee, Partial Differential Equations: A unified Hilbert Space Approach, DeGruyter, Berlin, 2011] established in [Picard, Trostorff, Wehowski, Waurick, On non-autonomous evolutionary problems. J. Evol. Equ. 13:751-776, 2013] applies to a broader class of non-autonomous integro-differential-algebraic equations. Using the concept of evolutionary mappings, we prove that the respective solution operators do not depend on certain parameters describing the underlying spaces in which the well-posedness results are established. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, impulsive Lasota-Wazewska model with infinite delay is studied. By using fixed point theorem of decreasing operator, we obtain sufficient conditions for the existence of unique almost periodic positive solution. Particularly, we give iterative sequence, which converges to the almost periodic positive solution. Moreover, we investigate exponential stability of the almost periodic positive solution by Liapunov functional. Copyright © 2014 John Wiley & Sons, Ltd.

Invoking some estimates obtained in [F.T. Akyildiz *et al*., Mathematical Methods in the Applied Sciences 33 (2010) 601–606] (which presented an alternate method of proof for the present problem), we correct the parameter regime considered in [R.A. Van Gorder, K. Vajravelu, and F. T. Akyildiz, Existence and uniqueness results for a nonlinear differential equation arising in viscous flow over a nonlinearly stretching sheet, Applied Mathematics Letters 24 (2011) 238–242] and add some details, which were omitted in the original proof. After this is done, we formulate a more elegant method of proof, converting the nonlinear ODE into a first nonlinear order system. This gives us a more natural way to view the problem and lends insight into the behavior of the solutions. Finally, we give a new way to approximate the shooting parameter *α* = *f* ′ ′ (0) analytically, through minimization of the *L*^{2}([0, ∞ )) norm of residual errors. This approximation demonstrates the behavior of the parameter *α* we expect from the proved theorems, as well as from numerical simulations. In this way, we obtain a concise analytical approximation to the similarity solution. In summary, from this analysis, we find that monotonicity of solutions and their derivatives is essential in determining uniqueness, and these monotone solutions can be approximated analytically in a fairly simple way. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with a class of fourth-order nonlinear difference equations. By using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of solutions for Dirichlet boundary value problems and give some new results. Our results successfully complement the existing ones. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the Cauchy problem for the 3D viscous incompressible magnetohydrodynamic equations and establish a Beale–Kato–Majda regularity criterion of smooth solutions in terms of the velocity vector in the homogeneous bounded mean oscillations space. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we study the following modified Kirchhoff-type equations of the form:

where *a* > 0, *b* ≥ 0, and . Under appropriate assumptions on *V* (*x*) and *h*(*x*,*u*), some existence results for positive solutions, negative solutions, and sequence of high energy solutions are obtained via a perturbation method. Copyright © 2014 John Wiley & Sons, Ltd.

The present article aims to study the projective synchronization between two identical and non‒identical time‒delayed chaotic systems with fully unknown parameters. Here the asymptotical and global synchronization are achieved by means of adaptive control approach based on Lyapunov–Krasovskii functional theory. The proposed technique is successfully applied to investigate the projective synchronization for the pairs of time‒delayed chaotic systems amongst advanced Lorenz system as drive system with multiple delay Rössler system and time‒delayed Chua's oscillator as response system. An adaptive controller and parameter update laws for unknown parameters are designed so that the drive system is controlled to be the response system. Numerical simulation results, depicted graphically, are carried out using Runge–Kutta Method for delay‒differential equations, showing that the design of controller and the adaptive parameter laws are very effective and reliable and can be applied for synchronization of time‒delayed chaotic systems. Copyright © 2014 John Wiley & Sons, Ltd.

]]>This paper is devoted to the derivation of a Helmholtz decomposition of vector fields in the case ofmixed boundary conditions imposed on the boundary of the domain. This particular decomposition allows to obtain a residual a posteriori error estimator for the approximation ofmagnetostatic problems given in the so-called *A*-formulation, for which the reliability can be established. Numerical tests confirm the obtained theoretical predictions. Copyright © 2014 John Wiley & Sons, Ltd.

Dimension functions play a significant role in the study of wavelets and have been attracting many waveletters’ interest. In recent years, the study of wavelet dimension functions has seen great achievements, but the study of Parseval frame wavelet (PFW) dimension functions has not. Bownik, Rzeszotnik and Speegle in 2001 and Arambašić, Bakić and Rajić in 2007 characterized -periodic functions that are wavelet dimension functions. But it is open what a -periodic function is qualified to be a dimension function of some semi-orthogonal PFW. This paper addresses semi-orthogonal PFW dimension functions associated with expansive matrices of determinant ± 2. We obtain a description of the ranges of semi-orthogonal PFW dimension functions and establish a necessary and sufficient condition for an integer-valued function to be a dimension function of some semi-orthogonal PFW. Copyright © 2014 John Wiley & Sons, Ltd.

]]>We consider a simplified model arising in radiation hydrodynamics based on the incompressible Navier–Stokes–Fourier system describing a macroscopic fluid motion coupled to a transport equation modeling the propagation of radiative intensity. We establish global-in-time existence for the associated initial-boundary value problem in the framework of weak solutions. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with the existence of a global attractor for the nonlinear viscoelastic beam equation with past history memory

where *g*(*u*_{t}) is a damping like | *u*_{t} | ^{r}*u*_{t} and *f*(*u*) is a source term like | *u* | ^{α}*u* − | *u* | ^{β}*u*, by considering 0 ≤ *β* < *α* and *r* > 0. Copyright © 2014 John Wiley & Sons, Ltd.