The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we first utilize the vanishing diffusivity method to prove the existence of global quasi-strong solutions and get some higher order estimates, and then prove the global well-posedness of the two-dimensional Boussinesq system with variable viscosity for *H*^{3} initial data. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we study the two-mode Korteweg–de Vries (TKdV) equation, which describes the propagation of two different waves modes simultaneously. We show that the TKdV equation gives multiple soliton solutions for specific values of the nonlinearity and dispersion parameters involved in the equation. We also derive other distinct exact solutions for general values of these parameters. We apply the simplified Hirota's method to study the specific of the parameters, which gives multiple soliton solutions. We also use the tanh/coth method and the tan/cot method to obtain other set of solutions with distinct physical structures. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is focused on higher order differentiation of Fourier series of functions. By means of Stokes's transformation, the recursion relations between the Fourier coefficients in Fourier series of different order (partial) derivatives of the functions as well as the general formulas for Fourier series of higher order (partial) derivatives of the functions are acquired. And then, the sufficient conditions for term-by-term differentiation of Fourier series of the functions are presented. These findings are subsequently used to reinvestigate the Fourier series methods for linear elasto-dynamical systems. The results given in this paper on the constituent elements, together with their combinatorial modes and numbering, of the sets of coefficients concerning 2*r*th order linear differential equation with constant coefficients are found to be different from the results deduced by Chaudhuri back in 2002. And it is also shown that the displacement solution proposed by Li in 2009 is valid only when the second order mixed partial derivative of the displacement vanishes at all of the four corners of the rectangular plate. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we investigate a Mimura–Tsujikawa model with nonlocal delay effect under the homogeneous Neumann boundary condition. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability, and Hopf bifurcation of nontrivial steady-state solutions bifurcating from the nonzero steady-state solution. Moreover, we illustrate our general results by applications to models with a one-dimensional spatial domain. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with a class of neutral-type BAM neural networks with distributed leakage delays. By applying the exponential dichotomy of linear differential equations, Lyapunov functional method and contraction mapping principle, we establish some sufficient conditions which ensure the existence and exponential stability of almost periodic solutions for such BAM neural networks. An example is given to illustrate the effectiveness of the theoretical findings. The results obtained in this article are completely new and complement the previously known studies. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the attraction–repulsion chemotaxis system with rotational flux terms

where
is a bounded domain with smooth boundary. Here, *S*_{1} and *S*_{2} are given parameter functions on [0,*∞*)^{2}×Ω with values in
. It is shown that for any choice of suitably regular initial data (*u*_{0},*v*_{0},*w*_{0}) fulfilling a smallness condition on the norm of *v*_{0},*w*_{0} in *L*^{∞}(Ω), the corresponding initial-boundary value problem possesses a global bounded classical solution. Copyright © 2016 John Wiley & Sons, Ltd.

Tsunamis are rare events compared with other extreme natural hazards, but the growth of population along coastlines has increased their potential impact. Tsunamis are most often generated by earthquake-induced dislocations of the seafloor, which displace large water masses. They can be simulated effectively as long waves whose propagation is modeled by the nonlinear shallow water equations. In this note, we present a brief assessment of earthquake-generated tsunami hazards for the city of Heraklion, Crete. We employ numerical hydrodynamic simulations, including inundation computations with the model MOST, and use high-resolution bathymetry and topography data for the area of interest. MOST implements a splitting method in space to reduce the system of shallow water equations in two successive systems, one for each spatial variable, and it uses a dispersive, Godunov-type finite difference method to solve the equations in characteristic form. We perform probabilistic analysis to assess the effects of the earthquake epicenter location on the tsunami, for time windows of 100, 500, and 1000years. The tsunami hazard is assessed through computed values of the maximum inundation range and maximum flow depth. Finally, we present a brief vulnerability analysis for the city of Heraklion, Crete. The data needed to identify tsunami-vulnerable areas are obtained by combining remote sensing techniques and geographic information system technology with surveyed observations and estimates of population data. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Completing previous work, a new class of interior solutions for compact static fluid spheres exhibiting pressure anisotropy, admitting conformal motion, and having 7, 8, 9, and 10 spacetime dimensions, respectively, is presented. Einstein's field equations without cosmological constant are solved for a particular energy density distribution function, assuming non-commutative geometry of spacetime. The behavior of the physical quantities obtained does not exclude the possible existence of ultra-compact, though rather exotic, stars in higher spacetime dimensions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a multistage susceptible-infectious-recovered model with distributed delays and nonlinear incidence rate is investigated, which extends the model considered by Guo *et al*. *[H. Guo, M. Y. Li and Z. Shuai, Global dynamics of a general class of multistage models for infectious diseases, SIAM J. Appl. Math., 72 (2012), 261–279]*. Under some appropriate and realistic conditions, the global dynamics is completely determined by the basic reproduction number *R*_{0}. If *R*_{0}≤1, then the infection-free equilibrium is globally asymptotically stable and the disease dies out in all stages. If *R*_{0}>1, then a unique endemic equilibrium exists, and it is globally asymptotically stable, and hence the disease persists in all stages. The results are proved by utilizing the theory of non-negative matrices, Lyapunov functionals, and the graph-theoretical approach. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study the following generalized quasilinear Schrödinger equation

where *N*≥3,
is a *C*^{1} even function, *g*(0) = 1 and *g*^{′}(*s*) > 0 for all *s* > 0. Under some suitable conditions, we prove that the equation has a ground state solution and infinitely many pairs ±*u* of geometrically distinct solutions. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, the global solvability to the mixed problem involving the wave equation with memory term and acoustic boundary conditions for non-locally reacting boundary is considered. Moreover, the general decay of the energy functionality is established by the techniques of Messaoudi. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The main purpose of this article is to investigate the optimal wholesaler's replenishment decisions for deterioration items under two levels of the trade credit policy and two storage facilities in order to reflect the supply chain management situation within the economic order quantity framework. In this study, each of the following assumptions have been made: (1) The own warehouse with limited capacity always is not sufficient to store the order quantity, so that a rented warehouse is needed to store the excess units over the capacity of the own warehouse; (2) The wholesaler always obtains the partial trade credit, which is independent of the order quantity offered by the supplier, but the wholesaler offers the full trade credit to the retailer; (3) The wholesaler must take a loan to pay his or her supplier the partial payment immediately when the order is received and then pay off the loan with the entire revenue. Under these three conditions, the wholesaler can obtain the least costs. Furthermore, this study models the wholesaler's optimal replenishment decisions under the aforementioned conditions in the supply chain management. Two theorems are developed to efficiently determine the optimal replenishment decisions for the wholesaler. Finally, numerical examples are given to illustrate the theorems that are proven in this study, and the sensitivity analysis with respect to the major parameters in this study is performed. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the model equations for the Timoshenko beam as a first-order system in the framework of evolutionary equations. The focus is on boundary damping, which is implemented as a dynamic boundary condition. A change of material laws allows the inclusion of a large class of cases of boundary damping. By choosing a particular material law, it is shown that the first-order approach to Sturm–Liouville problems with boundary damping is also covered. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We will prove that for piecewise C^{2}-concave domains in
Korn's first inequality holds for vector fields satisfying homogeneous normal or tangential boundary conditions with explicit Korn constant
. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study the following quasilinear chemotaxis–haptotaxis system

- (⋆)

in a bounded smooth domain
under zero-flux boundary conditions, where the nonlinearities *D*,*S*_{1}, and *S*_{2} are supposed to generalize the prototypes

with
, and *f*∈*C*^{1}([0,+*∞*) × [0,+*∞*)) satisfies

with *r* > 0 and *b* > 0. If the nonnegative initial data *u*_{0}(*x*)∈*W*^{1,∞}(Ω),*v*_{0}(*x*)∈*W*^{1,∞}(Ω), and
for some *α*∈(0,1), it is proved that

- For
*n*= 1, if and then (⋆) has a unique nonnegative classical solution, which is globally bounded. - For
*n*= 2, if and then (⋆) has a unique nonnegative classical solution, which is globally bounded. - For
*n*≥3, if and then (⋆) has a unique nonnegative classical solution, which is globally bounded.

Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this work, we consider the question of controllability of a class of integrodifferential equations on Hilbert space with measures as controls. We assume that the linear part has a resolvent operator in the sense given by R. Grimmer. We generalize the original work of N. Ahmed on vector measures, and we use it to develop necessary and sufficient conditions for weak and the exact controllability of the integrodifferential equation. Using the latter, we prove that exact controllability of the integrodifferential equation implies exact controllability of a perturbed integrodifferential equation. Controllability problem for the perturbed system is formulated fixed point problem in the space of vector measures. Our results cover impulsive controls as well as regular controls. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we investigate a class of generalized Burgers–Huxley equation by employing the bifurcation method of planar dynamical systems. Firstly, we reduce the equation to a planar system via the traveling wave solution ansatz; then by computing the singular point quantities, we obtain the conditions of integrability and determine the existence of one stable limit cycle from Hopf bifurcation in the corresponding planar system. From this, some new exact solutions and a special periodic traveling wave solution, which is isolated as a limit, are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We investigate sufficient conditions for existence of multiple solutions to a coupled system of fractional-order differential equations with three-point boundary conditions. By coupling the method of upper and lower solutions together with the method of monotone iterative technique, we develop conditions for iterative solutions. Based on these conditions, we study maximal and minimal solutions to the problem under consideration. We also study error estimates and provide an example. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This article contains an existence result for a class of quasiconvex stored energy functions satisfying the material non-interpenetrability condition, which primarily obstructs applying classical techniques from the vectorial calculus of variations to nonlinear elasticity. The fundamental concept of reversibility serves as the starting point for a theory of nonlinear elasticity featuring the basic duality inherent to the Eulerian and Lagrangian points of view. Motivated by this concept, split-quasiconvex stored energy functions are shown to exhibit properties, which are very alluding from a mathematical point of view. For instance, any function with finite energy is automatically a Sobolev homeomorphism; existence of minimizers can be readily established and first variation formulae hold. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, along the idea of Souplet and Zhang, we deduce a local elliptic-type gradient estimates for positive solutions of the nonlinear parabolic equation:

on
for *α* ≥ 1 and *α* ≤ 0. As applications, related Liouville-type theorem is exported. Our results are complement of known results. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we study two operators that arise in electromagnetic scattering in chiral media. We first consider electromagnetic scattering by a chiral dielectric with a perfectly conducting core. We define a chiral Calderon-type surface operator in order to solve the direct electromagnetic scattering problem. For this operator, we state coercivity and prove compactness properties. In order to prove existence and uniqueness of the problem, we define some other operators that are also related to the chiral Calderon-type operator, and we state some of their properties that they and their linear combinations satisfy. Then we sketch how to use these operators in order to prove the existence of the solution of the direct scattering problem. Furthermore, we focus on the electromagnetic scattering problem by a perfect conductor in a chiral environment. For this problem, we study the chiral far-field operator that is defined on a unit sphere and contains the far-field data, and we state and prove some of its properties that are preliminaries properties for solving the inverse scattering problem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the elliptic boundary blow-up problem

where Ω is a bounded smooth domain of
are positive continuous functions supported in disjoint subdomains Ω_{+},Ω_{−} of Ω, respectively, *a*_{+} vanishes on the boundary of
satisfies *p*(*x*)≥1 in Ω,*p*(*x*) > 1 on *∂*Ω and
, and *ε* is a parameter. We show that there exists *ε*^{∗}>0 such that no positive solutions exist when *ε* > *ε*^{∗}, while a minimal positive solution *u*_{ε} exists for every *ε*∈(0,*ε*^{∗}). Under the additional hypotheses that
is a smooth *N* − 1-dimensional manifold and that *a*_{+},*a*_{−} have a convenient decay near Γ, we show that a second positive solution *v*_{ε} exists for every *ε*∈(0,*ε*^{∗}) if
, where *N*^{∗}=(*N* + 2)/(*N* − 2) if *N* > 2 and
if *N* = 2. Our results extend that of Jorge Garcá-Melián in 2011, where the case that *p* > 1 is a constant and *a*_{+}>0 on *∂*Ω is considered. Copyright © 2016 John Wiley & Sons, Ltd.

Consider the following fractional Kirchhoff equations involving critical exponent:

where (−Δ)^{α} is the fractional Laplacian operator with *α*∈(0,1),
,
, *λ*_{2}>0 and
is the critical Sobolev exponent, *V*(*x*) and *k*(*x*) are functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space, the minimax arguments, Pohozaev identity, and suitable truncation techniques, we obtain the existence of a nontrivial weak solution for the previously mentioned equations without assuming the Ambrosetti–Rabinowitz condition on the subcritical nonlinearity *f*. Copyright © 2016 John Wiley & Sons, Ltd.

The aim of this paper is to propose mixed two-grid finite difference methods to obtain the numerical solution of the one-dimensional and two-dimensional Fitzhugh–Nagumo equations. 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 family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two-grid method, where the two-grid method is used for solving the large-sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of *h* and Δ*t*. The numerical examples show the efficiency of this algorithm for solving the one-dimensional and two-dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a class of nonlinear fractional differential equations on the infinite interval

with the integral boundary conditions

By using Krasnoselskii fixed point theorem, the existence results of positive solutions for the boundary value problem in three cases and , are obtained, respectively. We also give out two corollaries as applications of the existence theorems and some examples to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we investigate a stochastic non-autonomous SIRS (susceptible-infected-recovered-susceptible) model. The extinction and the prevalence of the disease are discussed, and so, the threshold is given. Especially, we show there is a positive nontrivial periodic solution. At last, some examples and simulations are provided to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The Jacobi system on a full-line lattice is considered when it contains additional weight factors. A factorization formula is derived expressing the scattering from such a generalized Jacobi system in terms of the scattering from its fragments. This is performed by writing the transition matrix for the generalized Jacobi system as an ordered matrix product of the transition matrices corresponding to its fragments. The resulting factorization formula resembles the factorization formula for the Schrödinger equation on the full line. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, numerical solutions of fractional Fokker–Planck equations with Riesz space fractional derivatives have been developed. Here, the fractional Fokker–Planck equations have been considered in a finite domain. In order to deal with the Riesz fractional derivative operator, shifted Grünwald approximation and fractional centred difference approaches have been used. The explicit finite difference method and Crank–Nicolson implicit method have been applied to obtain the numerical solutions of fractional diffusion equation and fractional Fokker–Planck equations, respectively. Numerical results are presented to demonstrate the accuracy and effectiveness of the proposed numerical solution techniques. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Group classification of a class of systems of diffusion equations is carried out. Arbitrary elements that appear in the system depend on two variables. All forms of the arbitrary elements that provide additional Lie symmetries are determined. Equivalence transformations are used to simplify the analysis. Examples of similarity reductions are presented. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We establish the existence and multiplicity of weak solutions for a class of nonlocal equations involving the fractional Laplacian operator, nonlinearities with critical exponential growth, and potentials that may change sign. The proofs of our existence results rely on minimization methods and the mountain pass theorem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we develop an *h**p*-adaptive procedure for the numerical solution of general, semilinear elliptic boundary value problems in 1d, with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an *h**p*-version adaptive finite element discretization (based on a robust *a posteriori* residual analysis), thereby leading to a fully *h**p*-adaptive Newton–Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples. Copyright © 2016 John Wiley & Sons, Ltd.

In this research article, we present theoretical techniques that can be used to investigate and comprehend the convergence behavior patterns of single parent evolution strategies. In the process, we determine instances of divergence or prove log-linear convergence and estimate the related speed, for a single parent evolution strategies class. The tools and results provided herein entertain a wide range of evolution strategies variants of interest and can be readily adapted for global convergence studies of a multitude of continuous optimization methods, evolving a single solution. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study the magnetic Bénard problem in two-dimensional space with generalized dissipative and diffusive terms, namely, fractional Laplacians and logarithmic supercriticality. Firstly, we show that when the diffusive term for the magnetic field is a full Laplacian, the solution initiated from data sufficiently smooth preserves its regularity as long as the power of the fractional Laplacians for the dissipative term of the velocity field and the diffusive term of the temperature field adds up to 1. Secondly, we show that with zero dissipation for the velocity field and a full Laplacian for the diffusive term of the temperature field, the global regularity result also holds when the diffusive term for the magnetic field consists of the fractional Laplacian with its power strictly bigger than 1. Finally, we show that with no diffusion from the magnetic and the temperature fields, the global regularity result remains valid as long as the dissipation term for the velocity field has its strength at least at the logarithmically supercritical level. These results represent various extensions of previous work on both Boussinesq and magnetohydrodynamics systems. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The paper deals with the three-dimensional Dirichlet boundary value problem (BVP) for a second-order strongly elliptic self-adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary-domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary-domain integral equation system is studied. We establish that the obtained localized boundary-domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. © 2016 The Authors Mathematical Methods in the Applied Sciences Math. Meth. Appl. Sci. 2016 Published by John Wiley & Sons, Ltd.

]]>In this paper, we introduce the Schatten class and the Berezin transform of quaternionic operators. The first topic is of great importance in operator theory, but it is also necessary to study the second one, which requires the notion of trace class operators, a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the *S*-spectrum of quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic *S*-functional calculus. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider the initial value problem for the Rosenau equation with damped term. The decay structure of the equation is of the regularity-loss type, which causes the difficulty in high-frequency region. Under small assumption on the initial value, we obtain the decay estimates of global solutions for *n*≥1. The proof also shows that the global solutions may be approximated by the solutions to the corresponding linear problem for *n*≥2. We prove that the global solutions may be approximated by the superposition of nonlinear diffusion wave for *n* = 1. Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents a direct method based on Legendre–Radau pseudospectral method for efficient and accurate solution of a class of singular optimal control problems. In this scheme, based on *a priori* knowledge of control, the problem is transformed to a multidomain formulation, in which the switching points appear as unknown parameters. Then, by utilizing Legendre-Radau pseudospectral method, a nonlinear programming problem is derived which can be solved by the well-developed parameter optimization algorithms. The main advantages of the present method are its superior accuracy and ability to capture the switching times. Accuracy and performance of the proposed method are examined by means of some numerical experiments. Copyright © 2016 John Wiley & Sons, Ltd.

This paper deals with a certain condenser capacity in an anisotropic environment. More precisely, we are going to investigate a free boundary problem for a class of anisotropic equations on a ring domain
*N*≥2. Our aim is to show that if the problem admits a solution in a suitable weak sense, then the underlying domain Ω is a Wulff-shaped ring. The proof makes use of a maximum principle for an appropriate P-function, in the sense of L. E. Payne, a Rellich type identity and some geometric arguments involving the anisotropic mean curvature of the free boundary. Copyright © 2016 John Wiley & Sons, Ltd.

This paper deals with the following Schrödinger–Poisson systems

where *λ*, *ν* are positive parameters and *V*(*x*) is sign-changing and may vanish at infinity. Under some suitable assumptions, the existence of positive ground state solutions is obtained by using variational methods. Our main results unify and improve the recent ones in the literatures. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a new model for the simulation of textiles with frictional contact between fibers and no bending resistance. In the model, one-dimensional hyperelasticity and the Capstan equation are combined, and its connection with conventional hyperelasticity and Coulomb friction models is shown. Then, the model is formulated as a problem with the rate-independent dissipation, and we prove that the problem possesses proper convexity and continuity properties. The article concludes with a numerical algorithm and provides numerical experiments along with a comparison of the results with a real measurement. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is mainly considered whether the mean-square stability of neutral stochastic delay differential equations (NSDDEs) with jumps is shared with that of the backward Euler–Maruyama method. Under the one-sided Lipschitz condition and the linear growth condition, the trivial solution of NSDDEs with jumps is proved to be mean-square stable by using the functional comparison principle and the Barbalat's lemma. It is shown that the backward Euler–Maruyama method can reproduce the mean-square stability of the trivial solution under the same conditions. The implicit backward Euler–Maruyama method shows better characteristic than the explicit Euler–Maruyama method for the reason that it works without the linear growth condition on the drift coefficient. Compared with some existing results, our results do not need to add extra condition on the neutral part. The conclusions can be applied to NSDDEs and SDDEs with jumps. The effectiveness of the theoretical results is illustrated by an example. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a robust mathematical method is proposed to study a new hybrid synchronization type, which is a combining generalized synchronization and inverse generalized synchronization. The method is based on Laplace transformation, Lyapunov stability theory of integer-order systems and stability theory of linear fractional systems. Sufficient conditions are derived to demonstrate the coexistence of generalized synchronization and inverse generalized synchronization between different dimensional incommensurate fractional chaotic systems. Numerical test of the method is used. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity-loss type, which causes difficulty in high-frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we first propose a concept of weighted pseudo-almost periodic functions on time scales and study some basic properties of weighted pseudo-almost periodic functions on time scales. Then, we establish some results about the existence of weighted pseudo-almost periodic solutions to linear dynamic equations on time scales. Finally, as an application of our results, we study the existence and global exponential stability of weighted pseudo-almost periodic solutions for a class of cellular neural networks with discrete delays on time scales. The results of this paper are completely new. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, first, we consider the existence of a positive solution for the nonlinear fractional differential equation boundary value problem

where 0≤*λ* < 1,^{C}D^{α} is the Caputo's differential operator of order *α*, and *f*:[0,1] × [0,*∞*)[0,*∞*) is a continuous function. Using some cone theoretic techniques, we deduce a general existence theorem for this problem. Then, we consider two following more general problems for arbitrary *α*, 1≤*n* < *α*≤*n* + 1:

Problem 1:

where
, 0≤*λ* < *k* + 1;

Problem 2:

where 0≤*λ*≤*α* and *D*^{α} is the Riemann–Liouville fractional derivative of order *α*.

For these problems, we give existence results, which improve recent results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.

An infection-age virus dynamics model for human immunodeficiency virus (or hepatitis B virus) infections with saturation effects of infection rate and immune response is investigated in this paper. It is shown that the global dynamics of the model is completely determined by two critical values *R*_{0}, the basic reproductive number for viral infection, and *R*_{1}, the viral reproductive number at the immune-free infection steady state (*R*_{1}<*R*_{0}). If *R*_{0}<1, the uninfected steady state *E*^{0} is globally asymptotically stable; if *R*_{0}>1 > *R*_{1}, the immune-free infected steady state *E*^{∗} is globally asymptotically stable; while if *R*_{1}>1, the antibody immune infected steady state
is globally asymptotically stable. Moreover, our results show that ignoring the saturation effects of antibody immune response or infection rate will result in an overestimate of the antibody immune reproductive number. Copyright © 2016 John Wiley & Sons, Ltd.

Shifted and modulated Gaussian functions play a vital role in the representation of signals. We extend the theory into a quaternionic setting, using two exponential kernels with two complex numbers. As a final result, we show that every continuous and quaternion-valued signal *f* in the Wiener space can be expanded into a unique *ℓ*^{2} series on a lattice at critical density 1, provided one more point is added in the middle of a cell. We call that a *relaxed Gabor expansion*. Copyright © 2016 John Wiley & Sons, Ltd.

It is well known that the time fractional equation
where
is the fractional time derivative in the sense of Caputo of *u* does not generate a dynamical system in the standard sense.

In this paper, we study the algebraic properties of the solution operator *T*(*t*,*s*,*τ*) for that equation with *u*(*s*) = *v*. We apply this theory to linear time fractional PDEs with constant coefficients. These equations are solved by the Fourier multiplier techniques. It appears that their solution exhibits some singularity, which leads us to introduce a new kind of solution for abstract time fractional problems. Copyright © 2016 John Wiley & Sons, Ltd.

We prove in this paper a generalized result with a unified proof of global existence in time of classical solutions to a class of a reaction diffusion system with triangular diffusion matrix on a bounded domain in
. The system in question is *u*_{t}=*a*Δ*u* − *f*(*x*,*t*,*u*,*v*), *v*_{t}=*c*Δ*u* + *d*Δ*v* + *ρ**f*(*x*,*t*,*u*,*v*),
, *t* > 0 with *f*(*x*,*t*,0,*η*) = 0 and *f*(*x*,*t*,*ξ*,*η*)≤*K**φ*(*ξ*)*e*^{ση}, for all *x*∈Ω, *t* > 0, *ξ*≥0, *η*≥0; where *ρ*, *K* and *σ* are real positive constants. Copyright © 2016 John Wiley & Sons, Ltd.

In this article, we want to solve a free boundary problem which models tumor growth with drug application. This problem includes five time dependent partial differential equations. The tumor considered in this model consists of three kinds of cells, proliferative cells, quiescent cells, and dead cells. Three different first-order hyperbolic equations are given that describe the evolution of cells and other two second-order parabolic equations describe the diffusion of nutrient and drug concentration. We solve the problem using the collocation method. Then, we prove stability and convergence of method. Also, some examples are considered to show the efficiency of method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, the truncated Painlevé analysis and the consistent *tanh* expansion method are developed for the modified Boussinesq system, and new exact solutions such as the single-soliton, the two-soliton, the rational solutions, and the explicit interaction solutions among a soliton and the cnoidal periodic waves are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

Two-dimensional time-fractional diffusion equations with given initial condition and homogeneous Dirichlet boundary conditions in a bounded domain are considered. A semidiscrete approximation scheme based on the pseudospectral method to the time-fractional diffusion equation leads to a system of ordinary fractional differential equations. To preserve the high accuracy of the spectral approximation, an approach based on the evaluation of the Mittag-Leffler function on matrix arguments is used for the integration along the time variable. Some examples along with numerical experiments illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, an infinite integral concerning numerical computation in crystallography is investigated, which was studied in two recent articles, and integration by parts is employed for calculating this typical integral. A variable transformation and a single integration by parts lead to a new formula for this integral, and at this time, it becomes a completely definite integral. Using integration by parts iteratively, the singularity at the points near three points *a* = 0,1,2 can be eliminated in terms containing obtained integrals, and the factors of amplifying round-off error are released into two simple fractions independent of the integral. Series expansions for this integral are obtained, and estimations of its remainders are given, which show that accuracy 2^{−n} is achieved in about 2*n* operations for every value in a given domain. Finally, numerical results are given to verify error analysis, which coincide well with the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

In this article, we discuss the modelling of elastic and electromagnetic wave propagation through one-dimensional and two-dimensional structured piezoelectric solids. Dispersion and the effect of piezoelectricity on the group velocity and positions of stop bands are studied in detail. We will also analyse the reflection and transmission associated with the problem of scattering of an elastic wave by a heterogeneous piezoelectric stack. Special attention is given to the occurrence of transmission resonances in finite stacks and their dependence on a piezoelectric effect. A two-dimensional doubly periodic piezoelectric checkerboard structure is subsequently introduced, for which the dispersion surfaces for Bloch waves have been constructed and analysed, with the emphasis on the dynamic anisotropy and special features of standing waves within the piezoelectric structure. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the following quasilinear Schrödinger equations:

where Ω is a bounded smooth domain of
,
. Under some suitable conditions, we prove that this equation has three solutions of mountain pass type: one positive, one negative, and sign-changing. Furthermore, if *g* is odd with respect to its second variable, this problem has infinitely many sign-changing solutions. Copyright © 2016 John Wiley & Sons, Ltd.

We propose a practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard system with a logarithmic free energy. We use Eyre's convex splitting scheme for the time discretization and a Fourier spectral method for the space variables. Given an absolute temperature, we find composition values that make the total free energy be minimum. Then, we find the splitting parameter value that makes the two split homogeneous free energies be convex on the neighborhood of the local minimum concentrations. For general use, we also propose a sixth-order polynomial approximation of the minimum concentration and derive a useful formula for the practical estimation of the splitting parameter in terms of the absolute temperature. The numerical tests are phase separation and total energy decrease with different temperature values. The linear stability analysis shows a good agreement between the exact and numerical solutions with an optimal value *s*. Various computational experiments confirm that the proposed splitting parameter estimation gives stable numerical results. Copyright © 2016 John Wiley & Sons, Ltd.

We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhedral domain, where Ventcel boundary conditions are imposed on one face of the polyhedron and Dirichlet boundary conditions are imposed on the complement of that face in the boundary. We establish improved regularity estimates for the trace of the variational solution on the Ventcel face and use them to derive a decomposition of the solution into a regular and a singular part that belongs to suitable weighted Sobolev spaces. This decomposition, in turn, via interpolation estimates both in the interior as well as on the Ventcel face, allows us to perform an *a priori* error analysis for the finite element approximation of the solution on anisotropic graded meshes. Numerical tests support the theoretical analysis. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we obtain conservation laws of (2+1) dimensional Calogero–Bogoyavlenskii–Schiff equation by non-local conservation theorem method. Besides, exact solutions are obtained by the aid of the symmetries associated with conservation laws. Double reduction is used to obtain these exact solution of Calogero–Bogoyavlenskii–Schiff equation. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The problem of model aggregation from various information sources of unknown validity is addressed in terms of a variational problem in the space of probability measures. A weight allocation scheme to the various sources is proposed, which is designed to lead to the best aggregate model compatible with the available data and the set of prior measures provided by the information sources. Copyright © 2016 John Wiley & Sons, Ltd.

]]>No abstract is available for this article.

]]>Based on a general isospectral problem of fractional order, a fractional bilinear form variational identity, the new integrable coupling of fractional L-hierarchy and the Hamiltonian structures of the integrable coupling of fractional L-hierarchy are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper presents a new numerical approach for the static and dynamic stabilities analysis of viscoelastic columns. By expanding all variables in a discretized time interval, a time and space coupled integral-differential equation system is converted into a series of recursive spatial problems that are solved via FEM. A temporally piecewised adaptive algorithm is developed to maintain computing accuracy. Consequently, the descriptions of static stability of imperfect viscoelastic columns and dynamic stability of perfect viscoelastic columns are realized, respectively. Numerical examples are presented to verify the proposed approach. Additionally, two analytical expressions are derived for the dynamic stability analysis of linear and Burgers models, respectively. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider continuous random dynamical systems with jumps. We estimate the dimension of the invariant measures and apply the results to a model of stochastic gene expression. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Over the last years, considerable attention has been paid to the role of the prolate spheroidal wave functions (PSWFs) introduced in the early sixties by D. Slepian and H.O. Pollak to many practical signal and image processing problems. The PSWFs and their applications to wave phenomena modeling, fluid dynamics, and filter design played a key role in this development. In this paper, we introduce the prolate spheroidal quaternion wave functions (PSQWFs), which refine and extend the PSWFs. The PSQWFs are ideally suited to study certain questions regarding the relationship between quaternionic functions and their Fourier transforms. We show that the PSQWFs are orthogonal and complete over two different intervals: the space of square integrable functions over a finite interval and the three-dimensional Paley–Wiener space of bandlimited functions. No other system of classical generalized orthogonal functions is known to possess this unique property. We illustrate how to apply the PSQWFs for the quaternionic Fourier transform to analyze Slepian's energy concentration problem. We address all of the aforementioned and explore some basic facts of the arising quaternionic function theory. We conclude the paper by computing the PSQWFs restricted in frequency to the unit sphere. The representation of these functions in terms of generalized spherical harmonics is explicitly given, from which several fundamental properties can be derived. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we apply the dual reciprocity boundary elements method for the numerical solution of two-dimensional linear and nonlinear time-fractional modified anomalous subdiffusion equations and time-fractional convection–diffusion equation. The fractional derivative of problems is described in the Riemann–Liouville and Caputo senses. We employ the linear radial basis function for interpolation of the nonlinear, inhomogeneous and time derivative terms. This method is improved by using a predictor–corrector scheme to overcome the nonlinearity which appears in the nonlinear problems under consideration. The accuracy and efficiency of the proposed schemes are checked by five test problems. The proposed method is employed for solving some examples in two dimensions on unit square and also in complex regions to demonstrate the efficiency of the new technique. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We are interested in the coexistence of three species forming a tritrophic food chain model. Considering a linear grow for the lowest trophic species, Holling III and Holling II functional response for the predator and the top-predator, respectively. We prove that this model has stable periodic orbits for adequate values of its parameters. Copyright © 2016 John Wiley & Sons, Ltd.

]]>A two-grid variational multiscale method based on two local Gauss integrations for solving the stationary natural convection problem is presented in this article. A significant feature of the method is that we solve the natural convection problem on a coarse mesh using finite element variational multiscale method based on two local Gauss integrations firstly, and then find a fine grid solution by solving a linearized problem on a fine grid. In the computation, we introduce two local Gauss integrations as a stabilizing term to replace the projection operator without adding other variables. The stability estimates and convergence analysis of the new method are derived. Ample numerical experiments are performed to validate the theoretical predictions and demonstrate the efficiency of the new method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The paper investigates an extension of the coupled integrable dispersionless equations, which describe the current-fed string within an external magnetic field. By using the relation among the coupled integrable dispersionless equations, the sine-Gordon equation and the two-dimensional Toda lattice equation, we propose a generalized coupled integrable dispersionless system. *N*-soliton solutions to the generalized system are presented in the Casorati determinant form with arbitrary parameters. By choosing real or complex parameters in the Casorati determinant, the properties of one-soliton and two-soliton solutions are investigated. It is shown that we can obtain solutions in soliton profile and breather profile. Copyright © 2016 John Wiley & Sons, Ltd.

In this article, the new exact travelling wave solutions of the nonlinear space-time fractional Burger's, the nonlinear space-time fractional Telegraph and the nonlinear space-time fractional Fisher equations have been found. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into ordinary differential equations of integer order in the sense of the Jumarie's modified Riemann–Liouville derivative. The -expansion method is effective for constructing solutions to the nonlinear fractional equations, and it appears to be easier and more convenient by means of a symbolic computation system. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we develop the energy argument in homogeneous Besov space framework to study the large time behavior of global-in-time strong solutions to the Cauchy problem of the three-dimensional incompressible nematic liquid crystal flows with low regularity assumptions on initial data. More precisely, if the small initial data
with 1 < *p* < *∞* and further assume that
with 1 < *q*≤*p* and
, then the global-in-time strong solution (*u*,*d*) to the nematic liquid crystal flows admits the following temporal decay rate:

Here, is a constant unit vector. The highlight of our argument is to show that the -norms (with ) of solution are preserved along time evolution. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The *p*-Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The *p*-Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the *p*-Laplace equation for 1 < *p* < 2 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the *p*-Laplace equation into the *p*-Dirac equation. This equation will be solved iteratively by using a fixed-point theorem. Applying operator-theoretical methods for the *p*-Dirac equation and *p*-Laplace equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. Copyright © 2016 John Wiley & Sons, Ltd.

An existence result of smooth solutions for a complex material flow problem is provided. The considered equations are of hyperbolic type including a nonlocal interaction term. The existence proof is based on a problem-adapted linear iteration scheme exploiting the structure conditions of the nonlocal term. 35Q70, 35L65 Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study the existence of ground states for the nonlinear Choquard equation driven by fractional Laplacian:

where the nonlinearity satisfies the general Berestycki–Lions-type assumptions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We introduce a class of tent-type spaces and establish a Poisson extension result of Triebel–Lizorkin spaces . As an application, we get the well-posedness of Navier–Stokes equations and magnetohydrodynamic equations with initial data in critical Triebel–Lizorkin spaces , . Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider semilinear elliptic equations of the form Δ*u* + *f*(u) = 0 over a quarter space with Dirichlet boundary conditions. Given a suitable positive root *z* of *f*, we show how to construct a non-negative bounded solution *u* converging to a one-dimensional limiting profile *V* with *V*
. This is established using Perron's method by constructing sub-solutions and super-solutions and employing a sliding argument. Copyright © 2016 John Wiley & Sons, Ltd.

We review the theory of hypercomplex numbers and hypercomplex analysis with the ultimate goal of applying them to issues related to the integration of systems of ordinary differential equations (ODEs). We introduce the notion of *hypercomplexification,* which allows the lifting of some results known for scalar ODEs to systems of ODEs. In particular, we provide another approach to the construction of superposition laws for some Riccati-type systems, we obtain invariants of Abel-type systems, we derive integrable Ermakov systems through hypercomplexification, we address the problem of linearization by hypercomplexification, and we provide a solution to the inverse problem of the calculus of variations for some systems of ODEs. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, a reaction-diffusion predator–prey system that incorporates the Holling-type II and a modified Leslie-Gower functional responses is considered. For ODE, the local stability of the positive equilibrium is investigated and the specific conditions are obtained. For partial differential equation, we consider the dissipation and persistence of solutions, the Turing instability of the equilibrium solutions, and the Hopf bifurcation. By calculating the normal form, we derive the formulae, which can determine the direction and the stability of Hopf bifurcation according to the original parameters of the system. We also use some numerical simulations to illustrate our theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We start recalling the characterizing property of the ‘partial symmetries’ of a differential problem, that is, the property of transforming solutions into solutions only in a proper subset of the full solution set. This paper is devoted to analyze the role of partial symmetries in the special context of dynamical systems and also to compare this notion with other notions of ‘weak’ symmetries, namely, the *λ*-symmetries and the orbital symmetries. Particular attention is addressed to discuss the relevance of partial symmetries in dynamical systems admitting homoclinic (or heteroclinic) manifolds, which can be ‘broken’ by periodic perturbations, thus giving rise, according to the (suitably rewritten) Mel'nikov theorem, to the appearance of a chaotic behavior of Smale-horseshoes type. Many examples illustrate all the various aspects and situations. Copyright © 2016 John Wiley & Sons, Ltd.

We propose and analyze a recurrent epidemic model of cholera in the presence of bacteriophage. The model is extended by general periodic incidence functions for low-infectious bacterium and high-infectious bacterium, respectively. A general periodic shedding function for two infected class (phage-positive and phage-negative) and a generalized contact and intrinsic growth function for susceptible class are also considered. Under certain biological assumptions, we derive the basic reproduction number (*R*_{0}) in a periodic environment for the proposed model. We also observe the global stability of the disease-free equilibrium, existence, permanence, and global stability of the positive endemic periodic solution of our proposed model. Finally, we verify our results with specific functional form. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we establish some new interior regularity criteria for suitable weak solutions of the liquid crystals flow in terms of the smallness of the scaled *L*^{p,q}-norm of the velocity field or the vorticity, which extends the results by Scheffer in [*Communications in Mathematical Physics* 1980; **73**:1–42]. Copyright © 2016 John Wiley & Sons, Ltd.

High-order variational models are powerful methods for image processing and analysis, but they can lead to complicated high-order nonlinear partial differential equations that are difficult to discretise to solve computationally. In this paper, we present some representative high-order variational models and provide detailed descretisation of these models and numerical implementation of the split Bregman algorithm for solving these models using the fast Fourier transform. We demonstrate the advantages and disadvantages of these high-order models in the context of image denoising through extensive experiments. The methods and techniques can also be used for other applications, such as image decomposition, inpainting and segmentation. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The aim of this paper is to establish a continuation principle for strong solutions to the full compressible magnetohydrodynamic system without resistivity and heat conductivity. We prove that if the solution loses its regularity in finite time, the dominated part is due to the hyperbolic effect. More precisely, it is essentially shown that the strong solution exists globally if the density, temperature, and magnetic field are bounded from above, where vacuum is allowed to exist. This verifies a problem proposed by D.Serre. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We investigate the initial value problem for the three-dimensional incompressible magnetohydrodynamics flows. Global existence and uniqueness of flows are established in the function space , provided that the norm of the initial data is less than the minimal value of the viscosity coefficients of the flows. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this article, the local well-posedness of Cauchy's problem is explored for a system of quadratic nonlinear Schrödinger equations in the space *L*^{p}(**R**^{n}). In a special case of mass resonant 2 × 2 system, it is well known that this problem is well posed in *H*^{s}(*s*≥0) and ill posed in *H*^{s}(*s* < 0) in two-space dimensions. By translation on a linear semigroup, we show that the general system becomes locally well posed in *L*^{p}(**R**^{2}) for 1 < *p* < 2, for which *p* can arbitrarily be close to the scaling limit *p*_{c}=1. In one-dimensional case, we show that the problem is locally well posed in *L*^{1}(**R**); moreover, it has a measure valued solution if the initial data are a Dirac function. Copyright © 2016 John Wiley & Sons, Ltd.

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

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