In this article, the sub-equation method is presented for finding the exact solutions of a nonlinear fractional partial differential equations. For this, the fractional complex transformation method has been used to convert fractional-order partial differential equation to ordinary differential equation. The fractional derivatives are described in Jumarie's the modified Riemann–Liouville sense. We apply to this method for the nonlinear time fractional differential equations. With the aid of symbolic computation, a variety of exact solutions for them are obtained. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The short-time Fourier transform has been shown to be a powerful tool for non-stationary signals and time-varying systems. This paper investigates the signal moments in the Hardy–Sobolev space that do not usually have classical derivatives. That is, signal moments become valid for non-smooth signals if we replace the classical derivatives by the Hardy–Sobolev derivatives. Our work is based on the extension of Cohen's contributions to the local and global behaviors of the signal. The relationship of the moments and spreads of the signal in the time, frequency and short-time Fourier domain are established in the Hardy–Sobolev space. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The concept of mono-component is widely used in nonstationary signal processing and time-frequency analysis. In this paper, we construct several classes of complete rational function systems in the Hardy space, whose boundary values are mono-components. Then, we propose a best approximation algorithm (BAA) based on optimal selections of two parameters in the orthonormal bases according to the approximation error. Effectiveness of BAA is evaluated by comparison experiments with the classical Fourier decomposition algorithm. It is also shown that BAA has promising results for filtering out noises and dealing with real-world signals. Copyright © 2014 John Wiley & Sons, Ltd.

]]>We consider the two-dimensional convection–diffusion equation with a fractional Laplacian, supplemented with step-like initial conditions. We show that the large time behavior of solutions to this IVP is described either by rarefaction waves, or diffusion waves, or suitable self-similar solutions, depending on the order of the fractional dissipation and on a direction of a convective nonlinearity. Copyright © 2014 John Wiley & Sons, Ltd.

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

where 0 < *α*,*β*≤1, 1 < *α* + *β*≤2, *λ* and *ρ* are constants, *γ* > 0, , is a continuous function, and *E*_{β}*x*(*t*) = *x*(*t* + *β* − 1). The existence and uniqueness of solutions are proved by using Banach's fixed point theorem. An illustrative example is also presented. Copyright © 2014 John Wiley & Sons, Ltd.

By means of a non-exact controllability result, we show the necessity of the conditions of compatibility for the exact synchronization by two groups for a coupled system of wave equations with Dirichlet boundary controls. Copyright © 2014 John Wiley & Sons, Ltd.

]]>The boundary function method [A. B. Vasil'eva, V. F. Butuzov, and L. V. Kalachev, *The boundary function method for singular perturbation problems*, SIAM Studies in Applied Mathematics, Philadelphia, 1995] is used to build an asymptotic expansion at any order of accuracy of a scalar time-harmonic wave scattered by a perfectly reflecting doubly periodic surface with oscillations at small and large scales. Error bounds are rigorously established, in particular in an optimal way on the relevant part of the field. It is also shown how the maximum principle can be used to design a homogenized surface whose reflected wave yields a first-order approximation of the actual one. The theoretical derivations are illustrated by some numerical experiments, which in particular show that using the homogenized surface outperforms the usual approach consisting in setting an effective boundary condition on a flat boundary. Copyright © 2014 John Wiley & Sons, Ltd.

The FFT-based homogenization method of Moulinec–Suquet has recently attracted attention because of its wide range of applicability and short computational time. In this article, we deduce an optimal *a priori* error estimate for the homogenization method of Moulinec–Suquet, which can be interpreted as a spectral collocation method. Such methods are well-known to converge for sufficiently smooth coefficients. We extend this result to rough coefficients. More precisely, we prove convergence of the fields involved for Riemann-integrable coercive coefficients without the need for an *a priori* regularization.

We show that our *L*^{2} estimates are optimal and extend to mildly nonlinear situations and *L*^{p} estimates for *p* in the vicinity of 2. The results carry over to the case of scalar elliptic and curl − curl-type equations, encountered, for instance, in stationary electromagnetism. Copyright © 2014 John Wiley & Sons, Ltd.

This work deals with the existence and uniqueness of a nontrivial solution for the third-order *p*-Laplacian *m*-point eigenvalue problems on time scales. We find several sufficient conditions of the existence and uniqueness of nontrivial solution of eigenvalue problems when *λ* is in some interval. The proofs are based on the nonlinear alternative of Leray–Schauder. To illustrate the results, some examples are included. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, sufficient conditions are established for the existence and uniqueness of global solutions to stochastic impulsive systems with expectations in the nonlinear terms. The maximal interval and the estimate of mild solutions are also discussed. These results are obtained by using the fixed point theorem, interval partition, and Lyapunov-like technique. Finally, examples are given to illustrate the theory. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, by incorporating latencies for both human beings and female mosquitoes to the mosquito-borne diseases model, we investigate a class of multi-group dengue disease model and study the impacts of heterogeneity and latencies on the spread of infectious disease. Dynamical properties of the multi-group model with distributed delays are established. The results showthat the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium depends only on the basic reproduction number. Our proofs for global stability of equilibria use the classical method of Lyapunov functions and the graph-theoretic approach for large-scale delay systems. Copyright © 2014 John Wiley & Sons, Ltd.

]]>A new variety of (3 + 1)-dimensional Burgers equations is presented. The recursion operator of the Burgers equation is employed to establish these higher-dimensional integrable models. A generalized dispersion relation and a generalized form for the one kink solutions is developed. The new equations generate distinct solitons structures and distinct dispersion relations as well. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Solvability of Cauchy's problem in for fractional Hamilton–Jacobi equation (1.1) with subcritical nonlinearity is studied here both in the classical Sobolev spaces and in the locally uniform spaces. The first part of the paper is devoted to the global in time solvability of subcritical equation (1.1) in *locally uniform phase space*, a generalization of the standard Sobolev spaces. Subcritical growth of the nonlinear term with respect to the gradient is considered. We prove next the global in time solvability in classical Sobolev spaces, in Hilbert case. Regularization effect is used there to guarantee global in time extendibility of the local solution. Copyright © 2014 John Wiley & Sons, Ltd.

By the means of a differential inequality technique, we obtain a lower bound for blow-up time if p and the initial value satisfy some conditions. Also, we establish a blow-up criterion and an upper bound for blow-up time under some conditions as well as a nonblow-up and exponential decay under some other conditions. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this work, we deal with the existence of the fractional integrable equations involving two generalized symmetries compatible with nonlinear systems. The method used is based on the Bä cklund transformation or B-transformation. Furthermore, we shall factorize the fractional heat operator in order to yield the fractional Riccati equation. This is done by utilizing matrix transform Miura type and matrix operators, that is, matrices whose entries are differential operators of fractional order. The fractional differential operator is taken in the sense of Riemann–Liouville calculus. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we are concerned with the Cauchy problem for the density-dependent incompressible flow of liquid crystals in thewhole space (N ≥ 2).We prove the localwell-posedness for large initial velocity field and director field of the system in critical Besov spaces if the initial density is close to a positive constant. We show also the global well-posedness for this system under a smallness assumption on initial data. In particular, this result allows us to work in Besov space with negative regularity indices, where the initial velocity becomes small in the presence of the strong oscillations. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we study the existence of solutions with prescribed *L*^{2}-norm for the following Kirchhoff type equation:

where *N*≤3,*a*,*b* > 0 are constants, *p*∈(2,2*),2*=6 if *N* = 3, and 2*=+*∞* if *N* = 1,2. We obtain the sharp existence of global constraint minimizers for . In the case , the existence of solutions with prescribed *L*^{2}-norm is obtained for all *L*^{2}-norm. The key point is the analysis of excluding the dichotomy of the minimizing sequences for the related constrained minimization problem. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper we construct two distinct generalized holomorphic orthogonal function systems over infinite cylinders in . Explicit representation formulae and properties of the obtained basis functions are given. As an application, we present some plot simulations that illustrate the results of this work. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we consider a one-dimensional linear Bresse system with infinite memories acting in the three equations of the system. We establish well-posedness and asymptotic stability results for the system under some conditions imposed into the relaxation functions regardless to the speeds of wave propagations. Copyright © 2014 John Wiley & Sons, Ltd.

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

Let *u*_{ϵ} be the solution of the Poisson equation in a domain perforated by thin tubes with a nonlinear Robin-type boundary condition on the boundary of the tubes (the flux here being *β*(*ϵ*)*σ*(*x*,*u*_{ϵ})), and with a Dirichlet condition on the rest of the boundary of Ω. *ϵ* is a small parameter that we shall make to go to zero; it denotes the period of a grid on a plane where the tubes/cylinders have their bases; the size of the transversal section of the tubes is *O*(*a*_{ϵ}) with *a*_{ϵ}≪*ϵ*. A certain nonperiodicity is allowed for the distribution of the thin tubes, although the perimeter is a fixed number *a*. Here, is a strictly monotonic function of the second argument, and the adsorption parameter *β*(*ϵ*) > 0 can converge toward infinity. Depending on the relations between the three parameters *ϵ*, *a*_{ϵ}, and *β*(*ϵ*), the effective equations in volume are obtained. Among the multiple possible relations, we provide *critical relations*, which imply different averages of the process ranging from linear to nonlinear. All this allows us to derive spectral convergence as *ϵ*0 for the associated spectral problems in the case of *σ* a linear function of *u*_{ϵ}. Copyright © 2014 John Wiley & Sons, Ltd.

The main objective of this research note is to provide an interesting result for the reducibility of the Kampé de Fériet function. The result is derived with the help of two results for the terminating _{3}*F*_{2} series very recently obtained by Rakha *et al*. A few interesting special cases have also been given. Copyright © 2014 John Wiley & Sons, Ltd.

Applying three critical point theorems, we prove the existence of at least three weak solutions for a class of differential equations with *p*(*x*)-Laplacian and subject to small perturbations of nonhomogeneous Neumann conditions. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with asymptotic stability of a Bresse system with two frictional dissipations. Under mathematical condition of equal speed of wave propagation, we prove that the system is exponentially stable. Otherwise, we show that Bresse system is not exponentially stable. Then, in the latter case, by using a recent result in linear operator theory, we prove the solution decays polynomially to zero with optimal decay rate. Better rates of polynomial decay depending on the regularity of initial data are also achieved. Copyright © 2014 John Wiley & Sons, Ltd.

We obtain an explicit formula for the diagonal singularities of the scattering amplitude for the Dirac equation with short-range electromagnetic potentials. Using this expansion we uniquely reconstruct an electric potential and magnetic field from the high-energy limit of the scattering amplitude. Moreover, supposing that the electric potential and magnetic field are asymptotic sums of homogeneous terms we give the unique reconstruction procedure for these asymptotics from the scattering amplitude, known for some energy *E*. Furthermore, we prove that the set of the averaged scattering solutions to the Dirac equation is dense in the set of all solutions to the Dirac equation that are in *L*^{2}(Ω), where Ω is any connected bounded open set in with smooth boundary, and we show that if we know an electric potential and a magnetic field for , then the scattering amplitude, given for some energy *E*, uniquely determines these electric potential and magnetic field everywhere in . Combining this uniqueness result with the reconstruction procedure for the asymptotics of the electric potential and the magnetic field we show that the scattering amplitude, known for some *E*, uniquely determines a electric potential and a magnetic field, that are asymptotic sums of homogeneous terms, which converges to the electric potential and the magnetic field respectively. Moreover, we discuss the symmetries of the kernel of the scattering matrix, which follow from the parity, charge-conjugation and time-reversal transformations for the Dirac operator. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we are interested in looking for multiple solutions for the following system of nonhomogenous Kirchhoff-type equations:

- (1.1)

where constants *a*,*c* > 0;*b*,*d*,*λ*≥0, *N* = 1,2 or 3, *f*,*g*∈*L*^{2}(*R*^{N}) and *f*,*g*≢0, *F*∈*C*^{1}(*R*^{N}×*R*^{2},*R*), , *V*∈*C*(*R*^{N},*R*) satisfy some appropriate conditions. Under more relaxed assumptions on the nonlinear term *F*, the existence of one negative energy solution and one positive energy solution for the nonhomogenous system 2.1 is obtained by Ekeland's variational principle and Mountain Pass Theorem, respectively. Copyright © 2014 John Wiley & Sons, Ltd.

In this note, we study the application of generalized fractional operators to a particular class of nonstandard Lagrangians. These are typical of dissipative systems, and the corresponding Euler–Lagrange and Hamilton equations are analyzed. The dependence of the equation of motion on the generalized kernel permits to obtain a wide range of different configurations of motion. Some examples are discussed and analyzed.Copyright © 2014 John Wiley & Sons, Ltd.

This paper considers the one parameter families of extrinsic differential geometries of Lorentzian hypersurfaces on pseudo *n*-spheres. We give a continuous relationship among the three types Gauss indicatrices by one parameter map. Meanwhile, we also give the singularity analysis of the one parameter Gauss indicatrices of Lorentzian hypersurfaces on pseudo *n*-spheres by the Legendrian singularity theory. Copyright © 2014 John Wiley & Sons, Ltd.

We develop a numerical predictive tool for multiphase fluid mixtures consisting of biofilms grown in a viscous fluid matrix by implementing a second-order finite difference discretization of the multiphase biofilm model developed recently on a general purpose graphic processing unit. With this numerical tool, we study a 3-D biomass–flow interaction resulting in biomass growth, structure formation, deformation, and detachment phenomena in biofilms grown in a water channel in quiescent state and subject to a shear flow condition, respectively. The numerical investigation is limited in the viscous regime of the biofilm–solvent mixture. In quiescent flows, the model predicts growth patterns consistent with experimental findings for single or multiple adjacent biofilm colonies, the so-called mushroom shape growth pattern. The simulated biomass growth both in density and thickness matches very well with the experimentally grown biofilm in a water channel. When shear is imposed at a boundary, our numerical studies reproduce wavy patterns, pinching, and streaming phenomena observed in biofilms grown in a water channel. Copyright © 2013 John Wiley & Sons, Ltd.

]]>This paper deals with a fully parabolic attraction–repulsion chemotaxis system in two-dimensional smoothly bounded domains. It is shown that the system admits global *bounded* classical solutions whenever the repulsion is dominated. The proof is based on an entropy-like inequality and coupled estimate techniques. Copyright © 2014 John Wiley & Sons, Ltd.

This paper studies the construction and approximation of quasi-interpolation for spherical scattered data. First of all, a kind of quasi-interpolation operator with Gaussian kernel is constructed to approximate the spherical function, and two Jackson type theorems are established. Second, the classical Shepard operator is extended from Euclidean space to the unit sphere, and the error of approximation by the spherical Shepard operator is estimated. Finally, the compact supported kernel is used to construct quasi-interpolation operator for fitting spherical scattered data, where the spherical modulus of continuity and separation distance of scattered sampling points are employed as the measurements of approximation error. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we consider three-dimensional incompressible magnetohydrodynamics equations. By using interpolation inequalities in anisotropic Lebesgue space, we provide regularity criteria involving the velocity or alternatively involving the fractional derivative of velocity in one direction, which generalize some known results. Copyright © 2014 John Wiley & Sons, Ltd.

]]>We give sufficient conditions for blows up of positive mild solutions for the weakly coupled system:

where is a fractional Laplacian, 0 < *α*_{i}≤2,*β*_{i}≥1,*ρ*_{i}>0,*σ*_{i}>−1 are constants, and the initial condition *ϕ*_{i} are positive, bounded and integrable functions. We also discuss the critical dimension. Copyright © 2014 John Wiley & Sons, Ltd.

The goal of this work is to present a new nonlinear inequality which is used in a study of the Lyapunov uniform stability and uniform asymptotic stability of solutions to time-varying perturbed differential equations. New sufficient conditions for global uniform asymptotic stability and/or practical stability in terms of Lyapunov-like functions for nonlinear time-varying systems is obtained. Our conditions are expressed as relation between the Lyapunov function and the existence of specific function which appear in our analysis through the solution of a scalar differential equation. Moreover, an example in dimensional two is given to illustrate the applicability of the main result. Copyright © 2014 John Wiley & Sons, Ltd.

]]>We deal with the generalized Emden–Fowler equation *f*^{″}(*x*) + *g*(*x*)*f*^{−θ}(*x*) = 0, where belongs to *L*^{p}((*a*,*b*)). We obtain a priori estimates for the solutions, as well as information about their asymptotic behavior near boundary points. As a tool, we derive new nonlinear variants of first-order and second-order Poincaré inequalities, which are based on strongly nonlinear multiplicative inequalities obtained recently by first author and Peszek. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we employed the bifurcation method and qualitative theory of dynamical systems to study the peakons and periodic cusp waves of the generalization of the Camassa-Holm equation, which may be viewed as an extension of peaked waves of the same equation. Through the bifurcation phase portraits of traveling wave system, we obtained the explicit peakons and periodic cusp wave solutions. Further, we exploited the numerical simulation to confirmthe qualitative analysis, and indeed, the simulation results are in accord with the qualitative analysis. Compared with the previous works, several new nonlinear wave solutions are obtained. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the existence and multiplicity of solutions to the following second-order impulsive Hamiltonian systems:

where is a continuousmap form the interval [0, *T*] to the set of N-order symmetric matrices. Our methods are based on critical point theory for nondifferentiable functionals. Copyright © 2014 John Wiley & Sons, Ltd.

The nonstationary Stokes system with slip boundary conditions is considered in a bounded domain . We prove the existence and uniqueness of solutions to the problem in anisotropic Sobolev spaces . Thanks to the slip boundary conditions, the Stokes problem is transformed to the Poisson and the heat equation. In this way, difficult calculations that must be performed in considerations of boundary value problems for the Stokes system are avoided. This approach does not work for the Dirichlet and the Neumann boundary conditions. Because solvability of the Poisson and the heat equation is carried out by the regularizer technique, we have that *σ* > 3,*α* > 0. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, the Darboux transformation of the Kundu–nonlinear Schrödinger equation is derived and generalized to the matrix of *n*-fold Darboux transformation. From known solution *Q*, the determinant representation of *n*-th new solutions of *Q*^{[n]} are obtained by the *n*-fold Darboux transformation. Then soliton solutions and positon solutions are generated from trivial seed solutions, breather solutions and rogue wave solutions that are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu–nonlinear Schrödinger equation are given. We show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves. Meanwhile, the third-order rogue waves are given explicitly. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is devoted to the investigation of the global dynamics of a SEIR model with information dependent vaccination. The basic reproduction number is derived for the model, and it is shown that gives the threshold dynamics in the sense that the disease-free equilibrium is globally asymptotically stable and the disease dies out if , while there exists at least one positive periodic solution and the disease is uniformly persistent when . Further, we give the approximation formula of . This answers the concerns presented in [B. Buonomo, A. d'Onofrio, D. Lacitignola, Modeling of pseudo-rational exemption to vaccination for SEIR diseases, J. Math. Anal. Appl. 404 (2013) 385–398]. Copyright © 2014 John Wiley & Sons, Ltd.

]]>Dedicated to Professor Dr. Martin Costabel on the occasion of his 65th birthday.Martin Costabel showed in his celebrated paper on boundary integral operators on Lipschitz domains also that the trace theorem for functions in *H*^{s}(Ω) holds for and not only for . Here, we show that his approach can be extended to *C*^{k − 1,1}-domains with . Copyright © 2014 John Wiley & Sons, Ltd.

The weighted *L*^{r}-asymptotic behavior of the strong solution and its first-order spacial derivatives to the incompressible magnetohydrodynamic (MHD) equations is established in a half-space. Further, the *L*^{∞}-decay rates of the second-order spatial derivatives of the strong solution are derived by using the Stokes solution formula and employing a decomposition for the nonlinear terms in MHD equations. Copyright © 2014 John Wiley & Sons, Ltd.

We consider a weakly dissipative modified two-component Dullin–Gottwald–Holm system. The existence of global weak solutions to the system is established. We first give the well-posedness result of viscous approximate problem and obtain the basic energy estimates. Then, we show that the limit of the viscous approximation solutions is a global weak solution to the system. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In the present study, we propose and analyze a predator–prey system with disease in the predator population. To understand the role of cannibalism, we modify the model considering predator population is of cannibalistic type. Local and global stability around the biologically feasible equilibria are studied. The conditions for the persistence of the system are worked out. We also analyze and compare the community structure of the model systems with the help of ecological and disease basic reproduction numbers. Finally, through numerical simulation, we observe that inclusion of cannibalism in predator population may control the disease transmission in the susceptible predator population. Copyright © 2014 John Wiley & Sons, Ltd.

]]>This paper revisits and complement in different directions the classical work by W. T. Reid on symmetrizable completely continuous transformations in Hilbert spaces and a more recent paper by one of the authors. More precisely, we deal with spectral properties of *% non-compact* operators *G* on a complex Hilbert space *H* such that *SG* is self-adjoint where *S* is a (*not* necessarily injective) nonnegative operator. We study the isolated eigenvalues of *G* outside its essential spectral interval and provide variational characterization of them as well as stability estimates. We compare them also to spectral objects of *S**G*. Finally, we characterize the Schechter essential spectrum of strongly symmetrizable operators in terms singular Weyl sequences; in particular, we complement J. I. Nieto's paper on the essential spectrum of symmetrizable. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we present a collection of *a priori* estimates of the electromagnetic field scattered by a general bounded domain. The constitutive relations of the scatterer are in general anisotropic. Surface averages are investigated, and several results on the decay of these averages are presented. The norm of the exterior Calderón operator for a sphere is investigated and depicted as a function of the frequency. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we investigate the computability of the solution operator of the generalized KdV-Burgers equation with initial-boundary value problem. Here, the solution operator is a nonlinear map *H*^{3m − 1}(*R*^{+}) × *H*^{m}(0,*T*)*C*([0,*T*];*H*^{3m − 1}(*R*^{+})) from the initial-boundary value data to the solution of the equation. By a technique that is widely used for the study of nonlinear dispersive equation, and using the type 2 theory of effectivity as computable model, we prove that the solution map is Turing computable, for any integer *m* ≥ 2, and computable real number *T* > 0. Copyright © 2014 John Wiley & Sons, Ltd.

This paper is concerned with the controllability and stabilizability problem for control systems described by a time-varying linear abstract differential equation with distributed delay in the state variables. An approximate controllability property is established, and for periodic systems, the stabilization problem is studied. Assuming that the semigroup of operators associated with the uncontrolled and non delayed equation is compact, and using the characterization of the asymptotic stability in terms of the spectrum of the monodromy operator of the uncontrolled system, it is shown that the approximate controllability property is a sufficient condition for the existence of a periodic feedback control law that stabilizes the system. The result is extended to include some systems which are asymptotically periodic. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, we constructed the split-step *θ *(SS*θ*)-method for stochastic age-dependent population equations. The main aim of this paper is to investigate the convergence of the SS *θ*-method for stochastic age-dependent population equations. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from the theory, and comparative analysis with Euler method is given, the results show the higher accuracy of the SS *θ*-method. Copyright © 2014 John Wiley & Sons, Ltd.

We systematically studied the optical properties (narrows peaks position) of the transmission spectra for microspheres coated by a multilayered stack. Three different sequences of spherical stack—periodic, quasiperiodic, and disordered—are studied by the transfer matrix approach. Dependence of the number of resonances in the transmission spectrum as function of number of layers in the stack is numerically investigated with details. It is shown that characteristic shape of the recursive return map forms well-defined ordered spectrum in the state space. The shape of such structures is different for different frequency ranges and various spherical quantum numbers. For quasiperiodic case, the latter leads to a specific signature of studied sequences and generates the self-similarity in the transmittance spectra. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we are concerned with stroboscopic averaging for highly oscillatory evolution equations posed in a Banach space. Using Taylor expansion, we construct a non-oscillatory high-order system whose solution remains exponentially close to the exact one over a long time. We then apply this result to the nonlinear wave equation in one dimension. We present the stroboscopic averaging method, which is a numerical method introduced by Chartier, Murua and Sanz-Serna, and apply it to our problem. Finally, we conclude by presenting the qualitative and quantitative efficiency of this numerical method for some nonlinear wave problem. Copyright © 2014 John Wiley & Sons, Ltd.

]]>This work deals with a mathematical model of an age-cycle length structured cell population. Each cell is distinguished by its age and its cycle length. The cellular mitosis is mathematically described by non-compact boundary conditions. We prove then that this mathematical model is governed by a positive *C*_{0}-semigroup. Copyright © 2014 John Wiley & Sons, Ltd.

In the present review, we deal with the recently introduced method of spectral parameter power series (SPPS) and show how its application leads to an explicit form of the characteristic equation for different eigenvalue problems involving Sturm–Liouville equations with variable coefficients. We consider Sturm–Liouville problems on finite intervals; problems with periodic potentials involving the construction of Hill's discriminant and Floquet–Bloch solutions; quantum-mechanical spectral and transmission problems as well as the eigenvalue problems for the Zakharov–Shabat system. In all these cases, we obtain a characteristic equation of the problem, which in fact reduces to finding zeros of an analytic function given by its Taylor series. We illustrate the application of the method with several numerical examples, which show that at present, the SPPS method is the easiest in the implementation, the most accurate, and efficient. We emphasize that the SPPS method is not a purely numerical technique. It gives an analytical representation both for the solution and for the characteristic equation of the problem. This representation can be approximated by different numerical techniques and leads to a powerful numerical method, but most important, it offers a different insight into the spectral and transmission problems. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this study, we investigate the existence of mild solutions for a class of impulsive neutral stochastic integro-differential equations with infinite delays, using the Krasnoselskii–Schaefer type fixed point theorem combined with theories of resolvent operators. As an application, an example is provided to illustrate the obtained result. Copyright © 2014 John Wiley & Sons, Ltd.

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

In this paper, we prove the -boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter , where , and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 < *ϵ* < *π* ∕ 2 and *γ*_{0} > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large *γ*_{0} > 0 for given 0 < *ϵ* < *π* ∕ 2. We also prove the maximal *L*_{p} − *L*_{q} regularity theorem of the nonstationary Stokes problem as an application of the -boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable . Copyright © 2014 John Wiley & Sons, Ltd.

In this article, we are concerned with the interactions of delta shock waves with contact discontinuities for the relativistic Euler equations for Chaplygin gas by using split delta functions method. The solutions are obtained constructively and globally when the initial data consists of three piecewise constant states. The global structure and large time-asymptotic behaviors of the solutions are analyzed case by case. During the process of the interaction, the strengths of delta shock waves are computed completely. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with special initial data by letting perturbed parameter *ε* tends to zero. Copyright © 2014 John Wiley & Sons, Ltd.

We consider standing waves for 4-superlinear Schrödinger–Kirchhoff equations in with potential indefinite in sign. The nonlinearity considered in this study satisfies a condition that is much weaker than the classical Ambrosetti–Rabinowitz condition. We obtain a nontrivial solution and, in the case of odd nonlinearity, an unbounded sequence of solutions via the Morse theory and the fountain theorem, respectively. Copyright © 2014 John Wiley & Sons, Ltd.

]]>In this paper, a SIR model with two delays and general nonlinear incidence rate is considered. The local and global asymptotical stabilities of the disease-free equilibrium are given. The local asymptotical stability and the existence of Hopf bifurcations at the endemic equilibrium are also established by analyzing the distribution of the characteristic values. Furthermore, the sufficient conditions for the permanence of the system are given. Some numerical simulations to support the analytical conclusions are carried out. At last, some conclusions are given. Copyright © 2014 John Wiley & Sons, Ltd.

In the paper, we investigate the basic transmission problems arising in the model of fluid-solid acoustic interaction when a piezo-ceramic elastic body ( Ω^{ + }) is embedded in an unbounded fluid domain ( Ω^{ − }). The corresponding physical process is described by boundary-transmission problems for second order partial differential equations. In particular, in the bounded domain Ω^{ + }, we have 4 × 4 dimensional matrix strongly elliptic second order partial differential equation, while in the unbounded complement domain Ω^{ − }, we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. Copyright © 2014 John Wiley & Sons, Ltd.

In this paper, we are concerned with the behavior of shock waves in a 2 × 2 balance law with discontinuous source terms. We obtain the existence of a local shock wave solution of this problem and deduce that the discontinuous source terms create a weak discontinuity in this solution. Copyright © 2013 John Wiley & Sons, Ltd.

]]>In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space was constructed recently, including a higher dimensional analogue of the logarithmic function in the complex plane. Their distributional limits at the boundary were also determined. In this paper, the potentials and their distributional boundary values are calculated in dimensions 3 and 4, dimensions for which the expressions in general dimension break down. Copyright © 2013 John Wiley & Sons, Ltd.

]]>Using the theory of fixed point index, we discuss the existence and multiplicity of nonnegative solutions of a wide class of boundary value problems with coupled nonlinear boundary conditions. Our approach is fairly general and covers a variety of situations. We illustrate in an example that all the constants that occur in our theory can be computed. Copyright © 2013 John Wiley & Sons, Ltd.

]]>The topic of the present paper is the reflection of electromagnetic plane waves by rough surfaces, that is, by smooth and bounded perturbations of planar faces. Moreover, the contrast between the cover material and the substrate beneath the rough surface is supposed to be low. In this case, a modification of Stearns’ formula based on Born approximation and Fourier techniques is derived for a special class of surfaces. This class contains the graphs of functions where the interface function is a radially modulated almost periodic function. For the Born formula to converge, a sufficient and almost necessary condition is given. A further technical condition is defined, which guarantees the existence of the corresponding far field of the Born approximation. This far field contains plane waves, far-field terms such as those for bounded scatterers, and, additionally, a new type of terms. The derived formulas can be used for the fast numerical computations of far fields and for the statistics of random rough surfaces. Copyright © 2013 John Wiley & Sons, Ltd.

The main purpose of this work is to provide a numerical approach for the delay partial differential equations based on a spectral collocation approach. In this research, a rigorous error analysis for the proposed method is provided. The effectiveness of this approach is illustrated by numerical experiments on two delay partial differential equations. Copyright © 2013 John Wiley & Sons, Ltd.

]]>In this paper, an eco-epidemiological model with Holling type-III functional response and a time delay representing the gestation period of the predators is investigated. In the model, it is assumed that the predator population suffers a transmissible disease. The disease basic reproduction number is obtained. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free equilibrium and the endemic-coexistence equilibrium are established, respectively. By using the persistence theory on infinite dimensional systems, it is proved that if the disease basic reproduction number is greater than unity, the system is permanent. By means of Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are obtained for the global stability of the endemic-coexistence equilibrium, the disease-free equilibrium and the predator-extinction equilibrium of the system, respectively. Copyright © 2013 John Wiley & Sons, Ltd.

]]>In this paper, we deal with the M-essential spectra of unbounded linear operators in Banach spaces where some generalizations of earlier work are given. Furthermore, we give an application from transport theory. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, a stage-structured SI epidemic model with time delay and nonlinear incidence rate is investigated. By analyzing the corresponding characteristic equations, the local stability of an endemic equilibrium and a disease-free equilibrium, and the existence of Hopf bifurcations are established. By comparison arguments, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Copyright © 2013 John Wiley & Sons, Ltd.

]]>In the present article, the authors have proposed a modified projective adaptive synchronization technique for fractional-order chaotic systems. The adaptive projective synchronization controller and identification parameters law are developed on the basis of Lyapunov direct stability theory. The proposed method is successfully applied for the projective synchronization between fractional-order hyperchaotic Lü system as drive system and fractional-order hyperchaotic Lorenz chaotic system as response system. A comparison between the effects on synchronization time due to the presence of fractional-order time derivatives for modified projective synchronization method and proposed modified adaptive projective synchronization technique is the key feature of the present article. Numerical simulation results, which are carried out using Adams–Boshforth–Moulton method show that the proposed technique is effective, convenient and also faster for projective synchronization of fractional-order nonlinear dynamical systems. Copyright © 2013 John Wiley & Sons, Ltd.

]]>In this paper, we show the existence and uniqueness of the mild solution for a class of time-dependent stochastic evolution equations with finite delay driven by a standard cylindrical Wiener process and an independent cylindrical fractional Brownian motion with Hurst parameter *H* ∈ (1 / 2,1). An example is provided to illustrate the theory. Copyright © 2013 John Wiley & Sons, Ltd.