We prove the existence of weak solutions to a one-dimensional initial-boundary value problem for a model system of partial differential equations, which consists of a sub-system of linear elasticity and a nonlinear non-uniformly parabolic equation of second order. To simplify the existence proof of weak solutions in the 2006 paper of Alber and Zhu, we replace the function in that work by . The model is formulated by using a sharp interface model for phase transformations that are driven by material forces. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider the variable-coefficient fractional diffusion equations with two-sided fractional derivative. By introducing an intermediate variable, we propose a mixed-type Galerkin variational formulation and prove the existence and uniqueness of the variational solution over
. On the basis of the formulation, we develop a mixed-type finite element procedure on commonly used finite element spaces and derive the solvability of the finite element solution and the error bounds for the unknown and the intermediate variable. For the Toeplitz-like linear system generated by discretization, we design a fast conjugate gradient normal residual method to reduce the storage from *O*(*N*^{2}) to *O*(*N*) and the computing cost from *O*(*N*^{3}) to *O*(*N*log*N*). Numerical experiments are included to verify our theoretical findings. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we prove the existence and uniqueness of a solution for a class of backward stochastic differential equations driven by *G*-Brownian motion with subdifferential operator by means of the Moreau–Yosida approximation method. Moreover, we give a probabilistic interpretation for the viscosity solutions of a kind of nonlinear variational inequalities. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider the non-autonomous Navier–Stokes equations with discontinuous initial data. We prove the global existence of solutions, the decay rate of density, and the equilibrium state of solutions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider interactions of smooth and discontinuous germs as generalized integrations over non-rectifiable paths with applications in theory of boundary value problems of complex analysis. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider the initial-boundary value problem for a model of motion of aqueous polymer solutions in a bounded three-dimensional domain subject to the Navier slip boundary condition. We construct a global (in time) weak solution to this problem. Moreover, we establish some uniqueness results, assuming additional regularity for weak solutions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider a model of infinite dimensional differential variational inequalities formulated by a parabolic differential inclusion and an elliptic variational inequality. The existence of global solution and global attractor for the semiflow governed by our system is proved by using measure of noncompactness. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study the nonexistence result for the weighted Lane–Emden equation:

- (0.1)

and the weighted Lane–Emden equation with nonlinear Neumann boundary condition:

- (0.2)

where *f*(|*x*|) and *g*(|*x*|) are the radial and continuously differential functions,
is an upper half space in
, and
. Using the method of energy estimation and the Pohozaev identity of solution, we prove the nonexistence of the nontrivial solutions to problems (0.1) and (0.2) under appropriate assumptions on *f*(|*x*|) and *g*(|*x*|). Copyright © 2017 John Wiley & Sons, Ltd.

Synchronization of complex networks with time-varying coupling matrices is studied in this paper. Two kinds of time-varying coupling are taken into account. One is the time-varying inner coupling in the node state space and the other is the time-varying outer coupling in the network topology space. By respectively setting linear controllers and adaptive controllers, time-varying complex networks can be synchronized to a desired state. Meanwhile, different influences of the control parameters of linear controllers and adaptive controllers on the synchronization have also been investigated. Based on the Lyapunov function theory, we construct appropriate positive-definite functions, and several sufficient synchronization criteria are obtained. Numerical simulations further illustrate the effectiveness of conclusions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>A discrete multi-group SVIR epidemic model with general nonlinear incidence rate and vaccination is investigated by utilizing Mickens' nonstandard finite difference scheme to a corresponding continuous model. Mathematical analysis shows that the global asymptotic stability of the equilibria is fully determined by the basic reproduction number by constructing Lyapunov functions. The results imply that the discretization scheme can efficiently preserves the global asymptotic stability of the equilibria for corresponding continuous model, and numerical simulations are carried out to illustrate the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The existence of one non-trivial solution for a second-order impulsive differential inclusion is established. More precisely, a recent critical point result is exploited, in order to prove the existence of a determined open interval of positive eigenvalues for which the considered problem admits at least one non-trivial anti-periodic solution. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The seasonality of conception for populations of the past using no contraception has remained a *terra incognita*. First, the influence of that of marriages on the seasonality of births is highlighted, taking into account the different stages in women's reproductive lives and the presence of successive cohorts of unequal size. Second, the age-dependent and time-dependent monthly distribution of conception is disentangled from monthly marriage and birth time series by means of stochastic optimization under a Leslie recursion with time-varying and age-varying probability of conception. The application to Armenian-Gregorians in the Don Army Territory (South Russia) from 1889 to 1912 reveals strong consistency between reconstructed conception, mean age at marriage, and fertility time series. Copyright © 2017 John Wiley & Sons, Ltd.

We present an analytic approach to solve a degenerate parabolic problem associated with the Heston model, which is widely used in mathematical finance to derive the price of an European option on an risky asset with stochastic volatility. We give a variational formulation, involving weighted Sobolev spaces, of the second-order degenerate elliptic operator of the parabolic PDE. We use this approach to prove, under appropriate assumptions on some involved unknown parameters, the existence and uniqueness of weak solutions to the parabolic problem on unbounded subdomains of the half-plane. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The equivalence group is determined for systems of linear ordinary differential equations in both the standard form and the normal form. It is then shown that the normal form of linear systems reducible by an invertible point transformation to the canonical form **y**^{(n)}=0 consists of copies of the same iterative scalar equation. It is also shown that contrary to the scalar case, an iterative vector equation need not be reducible to the canonical form by an invertible point transformation. Other properties of iterative linear systems are also derived, as well as a simple algebraic formula for their general solution. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we are concerned with the existence of ground state solution for the following fractional differential equations with tempered fractional derivative:

- (FD)

where *α*∈(1/2,1), *λ*>0,
are the left and right tempered fractional derivatives,
is the fractional Sobolev spaces, and
. Assuming that *f* satisfies the Ambrosetti–Rabinowitz condition and another suitable conditions, by using mountain pass theorem and minimization argument over Nehari manifold, we show that (FD) has a ground state solution. Furthermore, we show that this solution is a radially symmetric solution. Copyright © 2017 John Wiley & Sons, Ltd.

This paper shows the existence and the uniqueness of the nonnegative viscosity solution of the singular boundary value problem
for *t*>0,
, where *f* is a continuous non-decreasing function such that *f*(0)⩾0, and *h* is a nonnegative function satisfying the Keller–Osserman condition. Moreover, when *h*(*u*)=*u*^{p} with *p*>3, we obtain the global estimates for the classic solution *u*(*t*) and the exact blow-up rate of it at *t*=0. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we present a novel strategy to face the problem of dimensionality within datasets involved in conversational and feature selection systems. We base our work on a sound and complete logic along with an efficient attribute closure method to manage implications. All of them together allow us to reduce the overload of information we encounter when dealing with these kind of systems. An experiment carried out over a dataset containing real information comes to expose the benefits of our design. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The group analysis method is applied to the two-dimensional nonlinear Klein–Gordon equation with time-varying delay. Determining equations for equations with a time-varying delay are derived. A complete group classification of the studied equation with respect to the function involved into the equation is obtained. All admitted Lie algebras are classified. By using the classifications, representations of all invariant solutions are found. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, a second-order fast explicit operator splitting method is proposed to solve the mass-conserving Allen–Cahn equation with a space–time-dependent Lagrange multiplier. The space–time-dependent Lagrange multiplier can preserve the volume of the system and keep small features. Moreover, we analyze the discrete maximum principle and the convergence rate of the fast explicit operator splitting method. The proposed numerical scheme is of spectral accuracy in space and of second-order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, efficiency, mass conservation, and stability of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the asymptotic stability of a composite wave consisting of two traveling waves to a Keller–Segel chemotaxis model with logarithmic sensitivity and nonzero chemical diffusion. We show that the composite wave is asymptotically stable under general initial perturbation, which only be needed small in *H*^{1}-norm. This improves previous results. Copyright © 2017 John Wiley & Sons, Ltd.

We demonstrate how the problem of finding the effective property of quasiperiodic constitutive relations can be simplified to the periodic homogenization setting by transforming the original quasiperiodic material structure to a periodic heterogeneous material in a higher dimensional space. The characterization of two-scale cut-and-projection convergence limits of partial differential operators is presented. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Green's function technique serves as a powerful tool to find the particle displacements due to SH-wave propagation in layer of a shape different from the space between two parallel planes. Therefore, the present paper undertook to study the propagation of SH-wave in a transversely isotropic piezoelectric layer under the influence of a point source and overlying a heterogeneous substrate using Green's function technique. The coupled electromechanical field equations are solved with the aid of Green's function technique. Expression for displacements in both layer and substrate, scalar potential and finally the dispersion relation is obtained analytically for the case when wave propagates along the direction of layering. Numerical computations are carried out and demonstrated with the aid of graphs for six different piezoelectric materials namely PZT-5H ceramics, Barium titanate (BaTiO_{3}) ceramics, Silicon dioxide (SiO_{2}) glass, Borosilicate glass, Cobalt Iron Oxide (CoFe_{2}*O*_{4}), and Aluminum Nitride (AlN). The effects of heterogeneity, piezoelectric and dielectric constants on the dispersion curve are highlighted. Moreover, comparative study is carried out taking the phase velocity for different piezoelectric materials on one hand and isotropic case on the other. Dispersion relation is reduced to well-known classical Love wave equation with a view to illuminate the authenticity of problem. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider the following fractional Schrödinger–Poisson problem:

where *s*,*t*∈(0,1],4*s*+2*t*>3,*V*(*x*),*K*(*x*), and *f*(*x*,*u*) are periodic or asymptotically periodic in *x*. We use the non-Nehari manifold approach to establish the existence of the Nehari-type ground state solutions in two cases: the periodic one and the asymptotically periodic case, by introducing weaker conditions
uniformly in
with
and

with constant *θ*_{0}∈(0,1), instead of
uniformly in
and the usual Nehari-type monotonic condition on *f*(*x*,*τ*)/|*τ*|^{3}. Our results unify both asymptotically cubic or super-cubic nonlinearities, which are new even for *s*=*t*=1. Copyright © 2017 John Wiley & Sons, Ltd.

We consider the long time behavior of solutions for the non-autonomous stochastic *p*-Laplacian equation with additive noise on an unbounded domain. First, we show the existence of a unique
-pullback attractor, where *q* is related to the order of the nonlinearity. The main difficulty existed here is to prove the asymptotic compactness of systems in both spaces, because the Laplacian operator is nonlinear and additive noise is considered. We overcome these obstacles by applying the compactness of solutions inside a ball, a truncation method and some new techniques of estimates involving the Laplacian operator. Next, we establish the upper semi-continuity of attractors at any intensity of noise under the topology of
. Finally, we prove this continuity of attractors from domains in the norm of
, which improves an early result by Bates *et al.*(2001) who studied such continuity when the deterministic lattice equations were approached by finite-dimensional systems, and also complements Li *et al.* (2015) who discussed this approximation when the nonlinearity *f*(·,0) had a compact support. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider the shape inverse problem of a body immersed in the incompressible fluid governed by thermodynamic equations. By applying the domain derivative method, we obtain the explicit representation of the derivative of solution with respect to the boundary, which plays an important role in the inverse design framework. Moreover, according to the boundary parametrization technique, we present a regularized Gauss–Newton algorithm for the shape reconstruction problem. Finally, numerical examples indicate the proposed algorithm is feasible and effective for the low Reynolds numbers. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper deals with the following chemotaxis system:

in a bounded domain
with smooth boundary under no-flux boundary conditions, where
satisfies
for all
with *l*⩾2 and some nondecreasing function
on [0,*∞*). Here, *f*(*v*)∈*C*^{1}([0,*∞*)) is nonnegative for all *v*⩾0. It is proved that when
, the system possesses at least one global bounded weak solution for any sufficiently smooth nonnegative initial data. This extends a recent result by Wang (Math. Methods Appl. Sci. 2016 **39**: 1159–1175) which shows global existence and boundedness of weak solutions under the condition
. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we deal with discrete-time linear control systems in which the state is constrained to lie in the positive orthant independently of the inputs involved, that is, the inputs can take negative values. Such (positive state) systems appear, for example, in ecology models where the removal of individuals from a population is described. Controllability and reachability are fundamental properties of a system that show its ability to move in space, which are analyzed from an algebraic point of view throughout the text, paying special attention to the single-input case. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study the existence of periodic solutions for the Newtonian equation of motion with *p*-Laplacian operator by asymptotic behavior of potential function, establish some new sufficient criteria of existence of periodic solutions for the differential system under the frame of Fuc̆ik spectrum, generalize and improve some known works, and give an example to illustrate the application of the theorems. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we prove that if the initial data is small enough, we obtain an explicit *L*^{∞}(*Q*_{T})-estimate for a two-dimensional mathematical model of cancer invasion, proving an explicit bound with respect to time *T* for the estimate of solutions. Copyright © 2017 John Wiley & Sons, Ltd.

This paper deals with Lasota–Wazewska red blood cell model with perturbation on time scales. By applying the fixed point theorem of decreasing operator, we establish sufficient conditions for the existence of unique almost periodic positive solution. Particularly, we give iterative sequence which converges to the almost periodic positive solution. Moreover, we investigate exponential stability of the almost periodic positive solution by means of Gronwall inequality. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we use the arrowhead matrices as a tool to study graph theory. More precisely, we deal with an interesting class of directed multigraphs, the hub-directed multigraphs. We associate the arrowhead matrices with the adjacency matrices of a class of directed multigraph, and we obtain new properties of the second objects by using properties of the first ones. The hub-directed multigraphs with potential use in applications are also defined. As main result, we show that a hub-directed multigraph *G*(*H*) with adjacency matrix *H*^{∗} is a dominant hub-directed multigraph if and only if *H*^{∗}=*C**E*, where *C* is the adjacency matrix of another directed multigraph and *E* is the adjacency matrix of a particular elementary dominant hub-directed pseudo-graph. Another decomposition of its Gram (arrowhead) matrix
is also given. Copyright © 2017 John Wiley & Sons, Ltd.

Through solving the problem step by step and by applying the method of a *C*_{0} semigroup of operators combined with the Banach contraction theorem, we investigate the existence and uniqueness of a mild solution of semilinear impulsive integro-differential evolution equation in Banach spaces. In addition, an explicit iterative approximation sequence of the mild solution is derived. The assumed conditions in the present theorems are weaker and more general, and the results obtained are the generalizations and improvements of some known results. Examples are also given to illustrate our main results. Copyright © 2017 John Wiley & Sons, Ltd.

Many works on hybrid projective synchronization (or simply ‘HPS’ for short) of nonlinear real dynamic systems have been performed, while the HPS of chaotic complex systems and its application have not been extensively studied. In this paper, the HPS of complex Duffing–Holmes oscillators with known and unknown parameters is separately investigated via nonlinear control. The adaptive control methods and explicit expressions are derived for controllers and parameters estimation law, which are respectively used to achieve HPS. These expressions on controllers are tested numerically, which are in excellent agreement with theory analysis. The proposed synchronization scheme is applied to image encryption with exclusive or (or simply ‘XOR’ for short). The related security analysis shows the high security of the encryption scheme. Concerning the complex Duffing–Holmes oscillator, we also discuss its chaotic properties via the maximum Lyapunov exponent. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this work, we study the integrability aspects of the Schamel–Korteweg–de Vries equation that play an important role in studying the effect of electron trapping on the nonlinear interaction of ion-acoustic waves by including a quasi-potential. Lie symmetry analysis together with the simplest equation method and Kudryashov method is used to obtain exact traveling wave solutions for this equation. In addition, conservation laws are constructed using two different techniques, namely, the multiplier method and the new conservation theorem. Using the conservation laws and symmetries of the underlying equation, double reduction and exact solution were also constructed. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study the following Kirchhoff-type equations

where *a*>0,*b*⩾0,4<*p*<2^{∗}=6, and
. Under some suitable conditions, we prove that the equation has three solutions of mountain pass type: one positive, one negative, and sign-changing. Furthermore, this problem has infinitely many sign-changing solutions. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we are concerned with the general decay result of the quasi-linear wave equation with a time-varying delay in the boundary feedback and acoustic boundary conditions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this article, a new approach for pseudo almost periodic solution under the measure theory, under Acquistpace-Terreni conditions. We make extensive use of interpolation spaces and exponential dichotomy techniques to obtain the existence of *μ*-pseudo almost periodic solutions to some classes of nonautonomous partial evolution equations. For illustration, we propose some application to a nonautonomous heat equation. Copyright © 2017 John Wiley & Sons, Ltd.

On the basis of the ideas of non-traditional biomanipulation control in fresh water body, a kind of nutrient–algae fish model is presented to investigate the effects of constant releasing fish on the nutrient and the algae. The threshold conditions for the extinction of the algae are obtained by discussing the stability of boundary equilibrium. The conditions for the coexistence of the algae and the fish are obtained by discussing the existence and stability of positive equilibrium. Besides, Hopf bifurcation is also analyzed by considering the parameter about the amount of the released fish. Furthermore, a kind of optimal problem is presented, and the necessary condition for the existence of the optimal solution is given by Pontryagin maximum principle. Finally, the mathematical results are verified by numerical simulations. The mathematical results show that there is a threshold amount of the released fish, above which the activity of releasing fish can control the growth of the algae and further reduce the probability of the algae bloom, but can not decrease the eutrophication level of the water body. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider the nonstationary 3-D flow of a compressible viscous heat-conducting micropolar fluid in the domain to be the subset of **R**^{3} bounded with two coaxial cylinders that present the solid thermoinsulated walls. In the thermodynamical sense, the fluid is perfect and polytropic. We assume that the initial density and temperature are bounded from below with a positive constant, and that the initial data are sufficiently smooth cylindrically symmetric functions. The starting problem is transformed into the Lagrangian description on the spatial domain ]0,*L*[. In this work, we prove that our problem has a generalized solution for any time interval [0,*T*], *T*∈**R**^{+}. The proof is based on the local existence theorem and the extension principle. Copyright © 2017 John Wiley & Sons, Ltd.

We consider a non-stationary Stokes system in a thin porous medium of thickness *ε* that is perforated by periodically distributed solid cylinders of size *ε*, and containing a fissure of width *η*_{ε}. Passing to the limit when *ε* goes to zero, we find a critical size
in which the flow is described by a 2D quasi-stationary Darcy law coupled with a 1D quasi-stationary Reynolds problem. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we apply the family of potential wells to the initial boundary value problem of semilinear hyperbolic equations on the cone Sobolev spaces. We not only give some results of global existence and nonexistence of solutions but also obtain the vacuum isolating of solutions. Finally, we show blow-up in finite time of solutions on a manifold with conical singularities. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we establish exact solutions of the Cauchy problem for the 3D cylindrically symmetric incompressible Navier–Stokes equations and further study the global existence and asymptotic behavior of solutions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>The aim of this paper is to deal with the existence of mild solutions and exact controllability for a class of fractional evolution inclusions with damping (FEID, for short) in Banach spaces. Firstly, we provide the representation of mild solutions for FEID by applying the method of Laplace transform and the theory of (*α*,*κ*)-regularized families of operators. Next, we are concerned with the existence and exact controllability of FEID under some suitable sufficient conditions by using the method of measure of noncompactness and an appropraite fixed point theorem. Finally, an application to nonlinear partial differential equations with temporal fractional derivatives is presented to illustrate the effectiveness of our main results. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we prove the existence and uniqueness for the global solutions of Cauchy problem for coupled nonlinear Schrödinger equations and obtain the continuous dependence result on the initial data and the stronger decay estimate of global solutions. In particular, we show the existence and uniqueness of self-similar solutions. Also, we build some asymptotically self-similar solutions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We investigate the propagation of infinitesimal harmonic mechanical waves emitted from a boundary with variable velocity and arriving at a stationary observer. In the classical Doppler effect, *X*_{s}(*t*)=*v**t* is the location of the source with constant velocity *v*. In the present work, however, we consider a source co-located with a moving boundary *x*=*X*_{s}(*t*), where *X*_{s}(*t*) can have an arbitrary functional form. For ‘slowly moving’ boundaries (i.e., ones for which the timescale set by the mechanical motion is large in comparison to the inverse of the frequency of the emitted wave), we present a multiple-scale asymptotic analysis of the moving boundary problem for the linear wave equation. We obtain a closed-form leading-order (with respect to the latter small parameter) solution and show that the variable velocity of the boundary results not only in frequency modulation but also in amplitude modulation of the received signal. Consequently, our results extend the applicability of two basic tenets of the theory of a moving source on a stationary domain, specifically that (i)
for non-uniform boundary motion can be inserted in place of the constant velocity *v* in the classical Doppler formula and (ii) that the non-uniform boundary motion introduces variability in the amplitude of the wave. The specific examples of decelerating and oscillatory boundary motion are worked out and illustrated. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider the 2D incompressible Boussinesq system with fractional Laplacian dissipation and thermal diffusion. On the basis of the previous works and some new observations, we show that the condition with suffices in order for the solution pair of velocity and temperature to remain smooth for all time. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This work is devoted to the estimation of the effective energy density of porous nonlinear materials by means of variational bounds. Because of the unavailability of an improved upper bound, attention is turned to the improved bound obtained by considering constant nonzero polarization fields in the formal lower bound, which follows from the generalization to nonlinear behavior of the Hashin–Shtrikman variational principles. A particular class of nonlinearity is considered which is relevant in different physical situations. Also, 11 different microstructures are considered. Several computational experiments performed show that the elementary upper bound and the improved lower bound are indistinguishable, suggesting that nonlinearity dominates over the microstructural effects. In other words, at least for the nonlinearity considered here, the influence of microstructure on the effective behavior is negligible. So, in this case, there is no need to search for an improved upper bound, as the elementary one provides a simple but accurate estimate of the effective energy density. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, the approximate analytical solutions of Lotka–Volterra model with fractional derivative have been obtained by using hybrid analytic approach. This approach is amalgamation of homotopy analysis method, Laplace transform, and homotopy polynomials. First, we present an alternative framework of the method that can be used simply and effectively to handle nonlinear problems arising in several physical phenomena. Then, existence and uniqueness of solutions for the fractional Lotka–Volterra equations are discussed. We also carry out a detailed analysis on the stability of equilibrium. Further, we have derived the approximate solutions of predator and prey populations for different particular cases by using initial values. The numerical simulations of the result are depicted through different graphical representations showing that this hybrid analytic method is reliable and powerful method to solve linear and nonlinear fractional models arising in science and engineering. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Starting form basic principles, we obtain mathematical models that describe the traffic of material objects in a network represented by a graph. We analyze existence, uniqueness, and positivity of solutions for some implicit models. Also, some linear models and their equilibria are analyzed. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, an adaptive method for sampling and reconstructing high-dimensional shift-invariant signals is proposed. First, the integrate-and-fire sampling scheme and an approximate reconstruction algorithm for one-dimensional bandlimited signals are generalized to shift-invariant signals. Then, a high-dimensional shift-invariant signal is reduced to be a sequence of one-dimensional shift-invariant signals along the trajectories parallel to some coordinate axis, which can be approximately reconstructed by the generalized integrate-and-fire sampling scheme. Finally, an approximate reconstruction for the high-dimensional shift-invariant signal is obtained by solving a series of stable linear systems of equations. The main result shows that the final reconstructed error is completely determined by the initial threshold in integrate-and-fire sampling scheme, which is generally very small. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for the density. We study Galerkin finite elements method for the initial boundary value problem. The existence and uniqueness of the approximation are proved. A prior estimates for the solutions in
, time derivative in
and gradient in
, with *a*∈(0,1) are established. Error estimates for the density variable are derived in several norms for both continuous and discrete time procedures. Numerical experiments using backward Euler scheme confirm the theoretical analysis regarding convergence rates. Copyright © 2017 John Wiley & Sons, Ltd.

In this work, we analyze the existence, uniqueness, and asymptotic behavior of solution to the model of a thermoelastic mixture of type III. We establish sufficient conditions to guarantee the exponential decay of solutions. When the decay is not of exponential type, we prove that the solutions decay polynomially and we find the optimal polynomial decay rate. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the existence of positive solutions for a nonlinear *m*-point boundary value problem for the *p*-Laplacian dynamic equations on time scales, by applying a Krasnosel'skii's fixed point theorem. As an application, an example is included to demonstrate the main results. Copyright © 2017 John Wiley & Sons, Ltd.

This study considers the steady flow of a viscous, incompressible and electrically conducting fluid in a lid-driven square cavity under the effect of a uniform horizontally applied magnetic field. The governing equations are obtained from the Navier-Stokes equations including buoyancy and Lorentz force terms and the energy equation including Joule heating and viscous dissipation terms. These equations are solved iteratively in terms of velocity components, stream function, vorticity, temperature, and pressure by using radial basis function approximation. Particular solution, which is approximated by radial basis functions to satisfy both differential equation and boundary conditions, becomes the solution of the differential equation itself. Vorticity boundary conditions are obtained from stream function equation using finite difference scheme. Normal derivative of pressure is taken as zero on the boundary. The numerical results are obtained for several values of Hartmann number and Grashof number for the Stokes approximation (*R**e* << 1). The results show that when the viscous dissipation is present, the flow and isolines concentrate through the cold wall forming boundary layers as Grashof number increases. An increase in the magnetic field intensity retards the effect of buoyancy force in the square cavity, whereas the movement of the upper lid causes buoyancy force to be dominant. The solution is obtained in a considerably low computational expense through the use of radial basis function approximations for the MHD equations. Copyright © 2017 John Wiley & Sons, Ltd.

The Shilnikov-type single-pulse homoclinic orbits and chaotic dynamics of a simply supported truss core sandwich plate subjected to the transverse and the in-plane excitations are investigated in detail. The resonant case considered here is principal parametric resonance and 1:2 internal resonance. Based on the normal form theory, the desired form for the global perturbation method is obtained. By using the global perturbation method developed by Kovacic and Wiggins, explicit sufficient conditions for the existence of a Shilnikov-type homoclinic orbit are obtained, which implies that chaotic motions may occur for this class of truss core sandwich plate in the sense of Smale horseshoes. Numerical results obtained by using the fourth-order Runge–Kutta method agree with theoretical analysis at least qualitatively. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this article, we propose a mathematical model that describes the dynamics of a population divided into susceptible drinkers, moderate drinkers, and heavy drinkers subject to an external influence. The external influence is modelled using a supplementary dynamical variable that is not a group of individuals but that enters the equations affecting the choices of the population classes.

The system we define can be investigated using two simplified systems (one of which is a real subsystem), which model the populations of susceptible and moderate drinkers or susceptible and heavy drinkers independently. The dynamics of these two subsystems can be described exhaustively. The full system is too rich in possible scenarios, but its qualitative behaviour is connected to that of the two simplified systems. We make a complete description only in one particular case by means of numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.

Three-compartment mathematical models of non-toxic phytoplankton (NTP), toxin producing phytoplankton (TPP), and zooplankton are proposed to explore the role of TPP in algal blooms. The mutual interference between predator zooplankton and avoidance of TPP by zooplankton are incorporated into the model. The NTP and TPP engage in exploit competition and the toxin produced by TPP has no effect on NTP. Using the concept of uniform persistence, we establish coexistence of NTP, TPP, and zooplankton in certain parameter regimes. We study the effects of mutual interference and avoidance by zooplankton upon the population interactions. In addition to the toxin producing mechanism, it is concluded that mutual interference of zooplankton is an important factor for diminishing harmful blooms. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we consider the compressible bipolar Navier–Stokes–Poisson equations with a non-flat doping profile in three-dimensional space. The existence and uniqueness of the non-constant stationary solutions are established when the doping profile is a small perturbation of a positive constant state. Then under the smallness assumption of the initial perturbation, we show the global existence of smooth solutions to the Cauchy problem near the stationary state. Finally, the convergence rates are obtained by combining the energy estimates for the nonlinear system and the *L*^{2}-decay estimates for the linearized equations. Copyright © 2017 John Wiley & Sons, Ltd.

By using the unfolding operators for periodic homogenization, we give a general compactness result for a class of functions defined on bounded domains presenting perforations of two different size. Then we apply this result to the homogenization of the flow of a Bingham fluid in a porous medium with solid obstacles of different size. Next, we give the interpretation of the limit problem in terms of a nonlinear Darcy law. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we derive the explicit formulas for computing the zeros of certain cubic quaternionic polynomial. From these, we obtain a necessary and sufficient condition to quaternionic cubic polynomial have a spherical zero, and some examples are also provided. Moreover, we will discuss some applications of the cubic quaternionic formulas. Copyright © 2017 John Wiley & Sons, Ltd.

]]>No abstract is available for this article.

]]>We prove the existence of pullback and uniform attractors for the process associated to a non-autonomous SIR model, with several types of non-autonomous features. The Hausdorff dimension of the pullback attractor is also estimated. We illustrate some examples of pullback attractors by numerical simulations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we provide a sufficient condition, in terms of the horizontal gradient of two horizontal velocity components and the gradient of liquid crystal molecular orientation field, for the breakdown of local in time strong solutions to the three-dimensional incompressible nematic liquid crystal flows. More precisely, let *T*_{∗} be the maximal existence time of the local strong solution (*u*,*d*), then *T*_{∗}<+*∞* if and only if

where *u*^{h}=(*u*^{1},*u*^{2}), ∇_{h}=(*∂*_{1},*∂*_{2}). This result can be regarded as the generalization of the well-known Beale-Kato-Majda (BKM) type criterion and is even new for the three-dimensional incompressible Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a plate equation with infinite memory in the presence of delay and source term. Under suitable conditions on the delay and source term, we establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity. Our result allows a wider class of relaxation functions and improves earlier results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The main purpose of this paper is to construct sign-changing solution for the following Neumann problem:

where *n*≥3 and *K* is a bounded and continuous function on
, which concentrate around two critical points satisfying some conditions. Copyright © 2016 John Wiley & Sons, Ltd.

An inverse problem for the wave equation outside an obstacle with a *dissipative boundary condition* is considered. The observed data are given by a single solution of the wave equation generated by an initial data supported on an open ball. An explicit analytical formula for the computation of the coefficient at a point on the surface of the obstacle, which is nearest to the center of the support of the initial data, is given. Copyright © 2016 John Wiley & Sons, Ltd.

Backward heat equation with time dependent variable coefficient is severely ill-posed in the sense of Hadamard, so we need regularization. In this paper, we consider Backward heat equation with time dependent variable coefficient, and by small perturbing, we obtain an approximation problem. We show this approximation problem is well-posed with small parameter. Also, we show this approximation system converges to the original problem when parameter goes to zero. Here, we use modified-quasi boundary value method to regularize this problem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is devoted to the time-fractional gas dynamics equation with Caputo derivative. Fractional operators are very natural tools to model memory-dependent phenomena. Modified iteration method is proposed to obtain the approximate and analytical solution of the fractional gas dynamics equation. This method is a combined form of the new iteration method and Laplace transform. Modified iteration method really is powerful and simple method compared with other methods. Existence and uniqueness of solution are proven. Numerical results for different cases of the equation are obtained. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the well-posedness of a one-dimensional transport equation with nonlocal velocity in the Lei–Lin space . We first modify the product estimate and then establish the global existence of solutions to the Cauchy problem with small enough initial data. Finally, we discuss the stability of the global solution. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper investigates an inverse problem for parabolic equations backward in time, which is solved by total-variation-like (TV-like, in abbreviation) regularization method with cost function ∥*u*_{x}∥^{2}. The existence, uniqueness and stability estimate for the regularization problem are deduced in the linear case. For numerical illustration, the variational adjoint method, which presents a simple method to derive the gradient of the optimization functional, is introduced to reconstruct the unknown initial condition for both linear and nonlinear parabolic equations. The conjugate gradient method is used to iteratively search for the optimal approximation. Numerical results validate the feasibility and effectiveness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

The aim of this paper is to study the existence and the asymptotic behavior of solutions for some reaction–diffusion equations arising in epidemic biology phenomena. We will show that for a rather broad class of nonlinearities, the solutions are global and uniformly bounded, and under suitable assumptions on the parameters of the system, these solutions converge as time goes to infinity to a disease-free equilibrium point. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the existence of infinitely many solutions for the indefinite quasilinear Schrödinger equations

where *α*≥2,
. When *g*(*x*,*u*) is only of locally superlinear growth at infinity in *u* and *h*(*x*,*u*) is not odd in *u*, the existence of infinitely many solutions is proved in spite of the lack of the symmetry of this problem by using dual approach and Bolle's perturbation method. Our results generalize some known results and are new even in the symmetric situation. Copyright © 2016 John Wiley & Sons, Ltd.

We prove that on a smooth metric measure space with *m*−Bakry–Émery curvature bounded from below by −(*m* − 1)*K* for some constant *K*≥0 (i.e., Ric_{f,m}≥−(*m* − 1)*K*), the following degenerate elliptic equation

- (0.1)

has no nonconstant positive solution when *p* > 1 and constant *λ*_{f,p} satisfies

Our approach is based on the local Sobolev inequality and the Moser's iterative technique and is different from Cheng-Yau's method, which was used by Wang-Zhu in 2012 to derive a same Liouville theorem when 1 < *p*≤2, Ric_{f,m}≥−(*m* − 1)*K* and the sectional curvature is bounded from below. Copyright © 2016 John Wiley & Sons, Ltd.

The Schrödinger equation is solved for some of the *q*-deformed physical potentials within the framework of supersymmetric approach and supersymmetric Wentzel–Kramers–Brillouin approximation method. The energy levels are obtained with the corresponding normalized wave functions in terms of hypergeometric functions. The energy equations for some special cases of these potentials are discussed, and some of the thermodynamic quantities of the canonical system and the optical quantities of the two-level system are calculated for one of the potentials. Some of the numerical results are shown, too. Copyright © 2016 John Wiley & Sons, Ltd.

This article discusses the analyticity and the long-time asymptotic behavior of solutions to space-time fractional diffusion-reaction equations in
. By a Laplace transform argument, we prove that the decay rate of the solution as *t**∞* is dominated by the order of the time-fractional derivative. We consider the decay rate also in a bounded domain. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider the following coupled Schrödinger system with critical exponent:

where
is a smooth bounded domain, *λ* > 0,*μ*≥0, and
. Under certain conditions on *λ* and *μ*, we show that this problem has at least one positive least energy solution. Copyright © 2016 John Wiley & Sons, Ltd.

We consider a singularly perturbed Dirichlet spectral problem for an elliptic operator of second order. The coefficients of the operator are assumed to be locally periodic and oscillating in the scale *ϵ*. We describe the leading terms of the asymptotics of the eigenvalues and the eigenfunctions to the problem, as the parameter *ϵ* tends to zero, under structural assumptions on the potential. More precisely, we assume that the local average of the potential has a unique global minimum point in the interior of the domain and its Hessian is non-degenerate at this point. Copyright © 2016 John Wiley & Sons, Ltd.

We consider a two-dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and interfacial tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate *x* as a time-like coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the (*β*,*α*)-plane are obtained, where *α* and *β* are two parameters. The curves depend on two additional parameters *ρ* and *h*, where *ρ* is the ratio of the densities and *h* is the ratio of the fluid depths. However, the bifurcation diagram is found to be qualitatively the same as for surface waves. In particular, we find that a Hamiltonian-Hopf bifurcation, Hamiltonian real 1:1 resonance, and a Hamiltonian 0^{2}-resonance occur for certain values of (*β*,*α*). Of particular interest are solitary wave solutions of the Euler equations. Such solutions correspond to homoclinic solutions of the Hamiltonian system. We investigate the parameter regimes where the Hamiltonian-Hopf bifurcation and the Hamiltonian real 1:1 resonance occur. In both these cases, we perform a center manifold reduction of the Hamiltonian system and show that homoclinic solutions of the reduced system exist. In contrast to the case of surface waves, we find parameter values *ρ* and *h* for which the leading order nonlinear term in the reduced system vanishes. We make a detailed analysis of this phenomenon in the case of the real 1:1 resonance. We also briefly consider the Hamiltonian 0^{2}-resonance and recover the results found by Kirrmann. Copyright © 2016 John Wiley & Sons, Ltd.

Water bodies located nearby cities are much prone to pollution, especially in the developing countries, where effluents treatment facilities are generally lacking. The main reason for this phenomenon is the increasing population in the cities, and the large number of industries located near them. This leads to generation of huge amounts of domestic and industrial sewage that is discharged into the water bodies, increasing their organic pollutant load and resulting in the depletion of dissolved oxygen. In this paper, we propose a mathematical model for this situation, focusing especially on the resulting quality of the water, determined by the level of dissolved oxygen. The model also accounts for resources needed for the population survival and for the industrial operations. In addition, we describe also the decomposition of organic pollutants by bacteria in the aquatic medium. Feasibility conditions and stability criteria of the system's equilibria are determined analytically. The results show that human population and industries are relevant influential factors responsible for the increase in organic pollutants and the decrease in dissolved oxygen in the water body, in the sense that they may exert a destabilizing effect on the system. The numerical simulations confirm the analytical results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this work, we define the notions of ‘impulsive non-autonomous dynamical systems’ and ‘impulsive cocycle attractors’. Such notions generalize (we will see that not in the most direct way) the notions of autonomous dynamical systems and impulsive global attractors in the current published literature. We also establish conditions to ensure the existence of an impulsive cocycle attractor for a given impulsive non-autonomous dynamical system, which are analogous to the continuous case. Moreover, we prove the existence of such attractor for a non-autonomous 2D Navier–Stokes equation with impulses, using energy estimates. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper presents a numerical method for shape optimization of a body immersed in an incompressible viscous flow governed by Stokes–Oseen equations. The purpose of this work is to optimize the shape that minimizes a given cost functional. Based on the continuous adjoint method, the shape gradient of the cost functional is derived by involving a Lagrangian functional with the function space parametrization technique. Then, a gradient-type algorithm is applied to the shape optimization problem. The numerical examples indicate the proposed algorithm is feasible and effective in low Reynolds number flow. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the approximate controllability of Hilfer fractional differential inclusions with nonlocal conditions. The main techniques rely on the fixed point theorem combined with the semigroup theory, fractional calculus, and multivalued analysis. An interesting example is provided to illustrate the obtained results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a small-time large deviation principle for the stochastic non-Newtonian fluids driven by multiplicative noise is proved. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Modern ground-based telescopes rely on a technology called adaptive optics in order to compensate for the loss of angular resolution caused by atmospheric turbulence. Next-generation adaptive optics systems designed for a wide field of view require a stable and high-resolution reconstruction of the turbulent atmosphere. By introducing a novel Bayesian method, we address the problem via reconstructing the atmospheric turbulence strength profile and the turbulent layers simultaneously, where we only use wavefront measurements of incoming light from guide stars. Most importantly, we demonstrate how this method can be used for model optimization as well. We propose two different algorithms for solving the maximum a posteriori estimate: the first approach is based on alternating minimization and has the advantage of integrability into existing atmospheric tomography methods. In the second approach, we formulate a convex non-differentiable optimization problem, which is solved by an iterative thresholding method. This approach clearly illustrates the underlying sparsity-enforcing mechanism for the strength profile. By introducing a tuning/regularization parameter, an automated model reduction of the layer structure of the atmosphere is achieved. Using numerical simulations, we demonstrate the performance of our method in practice. 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.

This paper focuses on a distributed optimization problem associated with a time-varying multi-agent network with quantized communication, where each agent has local access to its convex objective function, and cooperatively minimizes a sum of convex objective functions of the agents over the network. Based on subgradient methods, we propose a distributed algorithm to solve this problem under the additional constraint that agents can only communicate quantized information through the network. We consider two kinds of quantizers and analyze the quantization effects on the convergence of the algorithm. Furthermore, we provide explicit error bounds on the convergence rates that highlight the dependence on the quantization levels. Finally, some simulation results on a *l*_{1}-regression problem are presented to demonstrate the performance of the algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

The passivity theory is used to achieve projective synchronization in coupled partially linear complex-variable systems with known parameters. By using this theory, the control law is thus adopted to make state vectors asymptotically synchronized up to a desired scaling factor. This paper deals with sending different large messages which include image and voice signals. The theoretical foundation of the projective synchronization based on the passivity theory is exploited for application to secure communications. The numerical simulations of secure communication are used to send large message, an image and sound (voice) signal. The errors are controlled to zero that show the agreement between theoretical and numerical simulations results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the initial-boundary value problem of semilinear damped wave equation *u*_{tt}−Δ*u* + *u*_{t}=|*u*|^{p} with power
in an exterior domain. Blow-up result in a finite time will be established in higher dimensions (*n*≥3), no matter how small the initial data are. A special test function will be constructed, and then, we obtain the blow-up result by a contradiction argument. Copyright © 2016 John Wiley & Sons, Ltd.

We study the contributions of within-host (virus-to-cell) and synaptic (cell-to-cell) transmissions in a mathematical model for human immunodeficiency virus epidemics. The model also includes drug resistance. We prove the local and global stability of the disease-free equilibrium and the local stability of the endemic equilibrium. We analyse the effect of the cell-to-cell transmission rate on the value of the reproduction number, *R*_{0}. Moreover, we show evidence of a qualitative change in the models' dynamics, subjected to the value of the drug efficacy. In the end, important inferences are drawn. Copyright © 2016 John Wiley & Sons, Ltd.

Ebola virus disease (EVD) can rapidly cause death to animals and people, for less than 1month. In addition, EVD can emerge in one region and spread to its neighbors in unprecedented durations. Such cases were reported in Guinea, Sierra Leone, and Liberia. Thus, by blocking free travelers, traders, and transporters, EVD has had also impacts on economies of those countries. In order to find effective strategies that aim to increase public knowledge about EVD and access to possible treatment while restricting movements of people coming from regions at high risk of infection, we analyze three different optimal control approaches associated with awareness campaigns, treatment, and travel-blocking operations that health policy-makers could follow in the war on EVD. Our study is based on the application of Pontryagin's maximum principle, in a multi-regional epidemic model we devise here for controlling the spread of EVD. The model is in the form of multi-differential systems that describe dynamics of susceptible, infected, and removed populations belonging to *p* different geographical domains with three control functions incorporated. The forward–backward sweep method with integrated progressive-regressive Runge–Kutta fourth-order schemes is followed for resolving the multi-points boundary value problems obtained. Copyright © 2016 John Wiley & Sons, Ltd.

Sufficient conditions are obtained for the nonexistence of solutions to the nonlinear higher order pseudo-parabolic equation

where
is the Kohn-Laplace operator on the (2*N* + 1)-dimensional Heisenberg group
, *m*≥1,*p* > 1. Then, this result is extended to the case of a 2 × 2-system of the same type. Our technique of proof is based on judicious choices of the test functions in the weak formulation of the sought solutions. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, a cholera epidemic model with saturated recovery rate is studied. Backward bifurcation leading to bistability possibly occurs, and global dynamics are shown by compound matrices and geometric approaches. Numerical simulations are presented to illustrate the results. Our results suggest that the basic reproduction number itself is not enough to describe whether cholera will prevail or not when the resources for treatment of infectives are limited and suggest that we should pay more attention to the initial state of cholera. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid-saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin–Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdélyi–Kober integral. In this way, we obtain a nonlinear model in the framework of time-fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time-fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We propose a model-building framework unifying those continuum models of condensed matter accounting for second-neighbor interactions. A notion of material isomorphism justifies restrictions that we impose to changes in observers on the material manifold. In the presence of dissipation due to evolution of inhomogeneities, we extend the notion of relative power including hyperstresses and derive pertinent balance equations by exploiting an invariance axiom. The scheme presented here permits an extension of the multi-field model-building framework for complex materials to account at a gross scale for second-neighbor microstructural interactions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the *b*-family of equations on the torus *u*_{t}−*u*_{txx}+(*b* + 1)*u**u*_{x}=*b**u*_{x}*u*_{xx}+*u**u*_{xxx}, which for appropriate values of *b* reduces to well-known models, such as the Camassa–Holm equation or the Degasperis–Procesi equation. We establish a local-in-space blow-up criterion. Copyright © 2016 John Wiley & Sons, Ltd.

The Breakthrough Starshot Initiative is suggested to develop the concept of propelling a nanoscale spacecraft by the radiation pressure of an intense laser beam. In this project, the nanocraft is a gram-scale robotic spacecraft comprising two main parts: StarChip and Lightsail. To achieve the goal of the project, it is necessary to solve a number of scientific problems. One of these tasks is to make sure that the nanocraft position and orientation inside the intense laser beam column are stable. The nanocraft driven by intense laser beam pressure acting on its Lightsail is sensitive to the torques and lateral forces reacting on the surface of the sail. These forces influence the orientation and lateral displacement of the spacecraft, thus affecting its dynamics. If unstable, the nanocraft might be expelled from the area of laser beam. In choosing the models for nanocraft stability studies, we are using several assumptions: (i) configuration of nanocraft is treated as a rigid body; (ii) flat or concave shape of circular sail; and (iii) mirror reflection of laser beam from surface of the Lightsail. We found conditions of position stability for spherical and conical shapes of the sail. The simplest stable configurations require the StarChip to be removed from the sail to make the distance to the center of mass of the nanocraft bigger than the curvature radius of the sail. Stability criteria do not require the spinning of the nanocraft. A flat sail is never stable. Copyright © 2016 John Wiley & Sons, Ltd.

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