In this paper, we introduced geometric algebra to develop a new multisource multisink optimal evacuation route planning method. A dynamically updatable data structure and a matrix-based greedy searching algorithm were developed to support the dynamic evacuation route searching for multiple evacuees. Unlike most existing methods, which iteratively search the optimal path for each evacuee, our method can search all the possible evacuation routes synchronously. The dynamic updating of the network topography, weights, and constraints during the route searching is direct and flexible, thus can support the evacuation in dynamic cases. The method is demonstrated and tested by an evacuation case in the city of Changzhou, China. The simulation experiments suggest that the method can well support the dynamic route searching in large scales with dynamic weight changes.

]]>In this paper, we investigate the dynamics of a time-delay ratio-dependent predator-prey model with stage structure for the predator. This predator-prey system conforms to the realistically biological environment. The existence and stability of the positive equilibrium are thoroughly analyzed, and the sufficient and necessary conditions for the stability and instability of the positive equilibrium are obtained for the case without delay. Then, the influence of delay on the dynamics of the system is investigated using the geometric criterion developed by Beretta and Kuang. We show that the positive steady state can be destabilized through a Hopf bifurcation and there exist stability switches under some conditions. The formulas determining the direction and the stability of Hopf bifurcations are explicitly derived by using the center manifold reduction and normal form theory. Finally, some numerical simulations are performed to illustrate and expand our theoretical results.

]]>We consider, in this paper, the following nonlinear equation with variable exponents:

where *a*,*b*>0 are constants and the exponents of nonlinearity *m*,*p*, and *r* are given functions. We prove a finite-time blow-up result for the solutions with negative initial energy and for certain solutions with positive energy.

In this paper, we develop the main ideas of the quantized version of affinely rigid (homogeneously deformable) motion. We base our consideration on the usual Schrödinger formulation of quantum mechanics in the configuration manifold, which is given, in our case, by the affine group or equivalently by the semi-direct product of the linear group
and the space of translations
, where *n* equals the dimension of the “physical space.” In particular, we discuss the problem of dynamical invariance of the kinetic energy under the action of the whole affine group, not only under the isometry subgroup. Technically, the treatment is based on the 2-polar decomposition of the matrix of the internal configuration and on the Peter-Weyl theory of generalized Fourier series on Lie groups. One can hope that our results may be applied in quantum problems of elastic media and microstructured continua.

We introduce a new type of linear and cyclic codes. These codes are defined over a direct product of 2 finite chain rings. The definition of these codes as certain submodules of the direct product of copies of these rings is given, and the cyclic property is defined. Cyclic codes can be seen as submodules of the direct product of polynomial rings. Generator matrices for linear codes and generator polynomials for cyclic codes are determined. Further, we study the concept of duality.

]]>Using the existence of integrable bi–almost-periodic Green functions of linear homogeneous differential equations and the contraction fixed point, we are able to prove the existence of almost and pseudo–almost-periodic mild solutions under quite general hypotheses for the differential equation with constant delay

in a Banach space *X*, where *τ*>0 is a fixed constant. The results extend the corresponding ones in the case of exponential dichotomy. Some examples illustrate the importance of the concepts.

Inversion of the scalar and vector ray transforms is performed in domain , ie, with the presence of an obstacle or singularity in the origin. Initially, the ray transforms of the basis functions for the scalar and vector fields are evaluated in an analytical form, and next, the inversion procedure is reduced to a linear system of equations by the use of the least squares method.

]]>In this work, we obtain the fundamental solution (FS) of the multidimensional time-fractional telegraph Dirac operator where the 2 time-fractional derivatives of orders *α*∈]0,1] and *β*∈]1,2] are in the Caputo sense. Explicit integral and series representation of the FS are obtained for any dimension. We present and discuss some plots of the FS for some particular values of the dimension and of the fractional parameters *α* and *β*. Finally, using the FS, we study some Poisson and Cauchy problems.

This paper presents 2 new classes of the Bessel functions on a compact domain [0,*T*] as generalized-tempered Bessel functions of the first- and second-kind which are denoted by GTBFs-1 and GTBFs-2. Two special cases corresponding to the GTBFs-1 and GTBFs-2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self-adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders.

The well posedness of the evolutive problem for visco-plastic materials represented by two different fractional constitutive equations is proved. We show that, for these materials, we can observe permanent deformations. So that, as it is usual in plasticity, when the stress goes to zero, then the strain assumes a constant nonzero behavior. Moreover, we prove the compatibility of our models with the classical laws of thermodynamics. For the second model, described through a fractional derivative with an exponential kernel, we obtain the exponential decay of the solutions by means of the semigroup theory.

]]>In this paper, we devote ourselves to establishing the unconditionally stable and absolutely convergent classical finite difference Crank-Nicholson (CN) implicit (CFDCNI) scheme and optimized finite difference CN-extrapolated implicit (OFDCNEI) scheme containing very few degrees of freedom but holding fully second-order accuracy for the two-dimensional viscoelastic wave via the proper orthogonal decomposition technique, analyzing the existence, stability, and convergence of the CFDCNI and OFDCNEI solutions, and using the numerical simulations to verify that the OFDCNEI scheme is far more superior than the CFDCNI scheme.

]]>The asymptotic behavior of the sequence {*u*_{n}} of positive first eigenfunctions for a class of eigenvalue problems is studied in a bounded domain
with smooth boundary *∂*Ω. We prove
, where *δ* is the distance function to *∂*Ω. Our study complements some earlier results by Payne and Philippin, Bhattacharya, DiBenedetto, and Manfredi, and Kawohl obtained in relation with the “*torsional creep problem*.”

We establish a local well-posedness and a blow-up criterion of strong solutions for the compressible Navier-Stokes-Fourier-*P*1 approximate model arising in radiation hydrodynamics. For the local well-posedness result, we do not need the assumption on the positivity of the initial density and it may vanish in an open subset of the domain.

We investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) *L*^{1}-spaces. We deal with both the cases of hard and soft potentials (with angular cut-off). For hard potentials, we provide a new proof of the fact that, in weighted *L*^{1}-spaces with exponential or algebraic weights, the solutions converge exponentially fast towards equilibrium. Our approach uses weak-compactness arguments combined with recent results of the second author on positive semigroups in *L*^{1}-spaces. For soft potentials, in *L*^{1}-spaces, we exploit the convergence to ergodic projection for perturbed substochastic semigroup to show that, for very general initial datum, solutions to the linear Boltzmann equation converges to equilibrium in large time. Moreover, for a large class of initial data, we also prove that the convergence rate is at least algebraic. Notice that, for soft potentials, no exponential rate of convergence is expected because of the absence of spectral gap.

In this paper, the notions of annulets and normal filters are introduced in Stone lattices and their properties are studied. A set of equivalent conditions is obtained to characterize normal filters of a Stone lattice. The extensions of the Glivenko-type congruences on a Stone lattice are investigated via annulets and normal filters. A description of the lattice of all extensions of the Glivenko-type congruences on a Stone lattice is given. A one-to-one correspondence between the class of all extensions and the class of all normal filters of a Stone lattice is obtained. Finally, we observe that every 2 extensions of the Glivenko-type congruences are permutable.

]]>We study nonlinear Volterra-type evolution integral equations of the form:
in a *C*^{∗}-algebra
or in a Hilbert algebra of Dixmier-Segal type, acting on a Hilbert space tensor product
, where
denotes a Hilbert space and
is the Boson-Einstein (Fermion-Dirac) Fock space, over a complex Hilbert space
. Under suitable Carathéodory-type conditions on the corresponding Nemytskii operator Φ of *f* and assuming that *k* is a quantum dynamical-type semigroup, we obtain exactly one classical global solution in the space
of bounded continuous (operator-valued) quantum stochastic processes. Moreover, we prove the existence of exactly one positive (respectively completely positive) classical global solution in
(respectively in
, applying a positivity (respectively completely positivity preserving) quantum stochastic integration process and assuming that *k* is a quantum dynamical semigroup acting on
, where Φ defines a positive (respectively completely positive) quantum stochastic process.

In this paper, we construct a more general Besov spaces and consider the global well-posedness of incompressible Navier-Stokes equations with small data in for . In particular, we show that for any and , the solution with initial data in belong to , which, as far as we know, has not been discussed in other papers. Moreover, the smoothing effect of the solution to Navier-Stokes equations is proved, which may have its own interest.

]]>In this paper, we apply wavelets to study the Triebel-Lizorkin type oscillation spaces and identify them with the well-known Triebel-Lizorkin-Morrey spaces. Further, we prove that Calderón-Zygmund operators are bounded on .

]]>The purpose of this paper is to study the traveling wave solutions of a nonlocal reaction-diffusion system with delay arising from the spread of an epidemic by oral-faecal transmission. Under monostable and quasimonotone it is well known that the system has a minimal wave speed c* of traveling wave fronts. In this paper, we first prove the monotonicity and uniqueness of traveling waves with speed *c*⩾*c*_{∗}. Then we show that the traveling wave fronts with speed *c*>*c*_{∗} are exponentially asymptotically stable.

Birkhoff interpolation is the most general interpolation scheme. We study the Lagrange-type basis for uniform integrable tensor-product Birkhoff interpolation. We prove that the Lagrange-type basis of multivariate uniform tensor-product Birkhoff interpolation can be obtained by multiplying corresponding univariate Lagrange-type basis when the integrable condition is satisfied. This leads to less computational complexity, which drops to from .

]]>In this paper, we introduce the notion of partial affine system that is a subset of an affine system. It has potential applications in signal analysis. A general affine system has been extensively studied; however, the partial one has not. The main focus of this paper is on partial affine system–based frames and dual frames. We obtain a necessary condition and a sufficient condition for a partial affine system to be a frame and present a characterization of partial affine system–based dual frames. Some examples are also provided.

]]>This paper describes the geometric meaning of the inner product of 2 circles based on *Compass Ruler Algebra*, the Conformal Geometric Algebra in 2D. It analyzes in detail the cases of intersecting and tangent circles as well as when one circle is completely inside or completely outside the other circle. The paper reflects the systematic way of deriving the results based on the Geometric Algebra software tool Gaalop. The results are easily conferrable to arbitrary dimensions.

In this note, we point out two errors in the article “On the Neumann function and the method of images in spherical and ellipsoidal geometry” by Dassios and Sten. Two corrections are then proposed.

]]>In this paper, we study a general discrete-time model representing the dynamics of a contest competition species with constant effort exploitation. In particular, we consider the difference equation *x*_{n+1}=*x*_{n}*f*(*x*_{n−k})−*h**x*_{n} where *h*>0, *k*∈{0,1}, and the density dependent function *f* satisfies certain conditions that are typical of a contest competition. The harvesting parameter *h* is considered as the main parameter, and its effect on the general dynamics of the model is investigated. In the absence of delay in the recruitment (*k*=0), we show the effect of *h* on the stability, the maximum sustainable yield, the persistence of solutions, and how the intraspecific competition change from contest to scramble competition. When the delay in recruitment is 1 (*k*=1), we show that a Neimark-Sacker bifurcation occurs, and the obtained invariant curve is supercritical. Furthermore, we give a characterization of the persistent set.

Although foraging patterns have long been predicted to autonomously adapt to environmental conditions, empirical evidence has been found in recent years. This evidence suggests that the search strategy of animals is open to change so that animals can flexibly respond to their environment. In this study, we began with a simple computational model that possesses the principal features of an intermittent strategy, ie, careful local searches separated by longer steps, as a mechanism for relocation, where an agent in the model follows a rule to switch between two phases, but it could misunderstand this rule, ie, the agent follows an ambiguous switching rule. Thanks to this ambiguity, the agent's foraging strategy can continuously change. First, we demonstrate that our model can exhibit an autonomous change of strategy from Brownian-type to Lévy type depending on the prey density, and we investigate the distribution of time intervals for switching between the phases. Moreover, we show that the model can display higher search efficiency than a correlated random walk.

]]>The aim through this work is to suppress the transverse vibrations of an axially moving viscoelastic strip. A controller mechanism (dynamic actuator) is attached at the right boundary to control the undesirable vibrations. The moving strip is modeled as a moving beam pulled at a constant speed through 2 eyelets. The left eyelet is fixed in the sense that there is no transverse displacement (see Figure ). The mathematical model of this system consists of an integro-partial differential equation describing the dynamic of the strip and an integro-differential equation describing the dynamic of the actuator. The multiplier method is used to design a boundary control law ensuring an exponential stabilization result.

]]>We propose a deterministic model to study the impact of environmental pollution on the dynamics of cholera. We consider both human to human and human-environment-human transmission modes in our model. We obtain the expression for the basic reproduction number of the proposed model. The study of our model reveals that environmental pollution plays a significant role in the spread of cholera and should not be ignored. Although various dimensions of cholera has been studied using mathematical models but scanty efforts have been made to understand impact of environmental pollution on this disease. Through this study, we try to bridge this gap.

]]>In the paper, we investigate the mixed type transmission problem arising in the model of fluid–solid acoustic interaction when a piezoceramic elastic body (Ω^{+}) is embedded in an unbounded fluid domain (Ω^{−}). The corresponding physical process is described by the boundary-transmission problem for second-order partial differential equations. In particular, in the bounded domain Ω^{+}, we have a 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 based on the Wiener–Hopf factorization method, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. We derive asymptotic expansion of solutions, and on the basis of asymptotic analysis, we establish optimal Hölder smoothness results for solutions. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, the geometric structure for normal distribution manifold, von Mises distribution manifold and their joint distribution manifold are firstly given by the metric, curvature, and divergence, respectively. Furthermore, the active detection with sensor networks is presented by a classical measurement model based on metric manifold, and the information resolution is presented for the range and angle measurements sensor networks. The preliminary analysis results introduced in this paper indicate that our approach is able to offer consistent and more comprehensive means to understand and solve sensor network problems containing sensors management and target detection, which are not easy to be handled by conventional analysis methods.

]]>We show the advantages and disadvantages of specific geometric algebras and propose practical implementations in colorimetry. The colour space *C**I**E**L*^{∗}*a*^{∗}*b*^{∗} is endowed by an Euclidean metric; the neighbourhood of a point is therefore a sphere, and the choice of a conformal geometric algebra is thus obvious. For the colour space *C**M**C*(*l*:*c*), the neighbourhood is an ellipsoid and thus we choose the quadric geometric algebra to linearize the metric by means of the scalar product. We discuss the distance problems in colour spaces with these particular geometric algebras applied.

The Bogdanov-Takens bifurcations of a Leslie-Gower predator-prey model with Michaelis-Menten–type prey harvesting were studied. In the paper “Diff. Equ. Dyn. Syst. 20(2012), 339-366,” Gupta et al proved that the Leslie-Gower predator-prey model with Michaelis-Menten–type prey harvesting has rich dynamics. Some equilibria of codimension 1 and their bifurcations were discussed. In this paper, we find that the model has an equilibrium of codimensions 2 and 3. We also prove analytically that the model undergoes Bogdanov-Takens bifurcations (cusp cases) of codimensions 2 and 3. Hence, the model can have 2 limit cycles, coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1 as the values of parameters vary. Moreover, several numerical simulations are conducted to illustrate the validity of our results.

]]>In this paper, we present a general phase transition model that describes the evolution of vehicular traffic along a one-lane road. Two different phases are taken into account, according to whether the traffic is low or heavy. The model is given by a scalar conservation law in the *free-flow* phase and by a system of 2 conservation laws in the *congested* phase. The free-flow phase is described by a one-dimensional fundamental diagram corresponding to a Newell-Daganzo type flux. The congestion phase is described by a two-dimensional fundamental diagram obtained by perturbing a general fundamental flux. In particular, we study the resulting Riemann problems in the case a local point constraint on the flow of the solutions is enforced.

In this paper, the boundary output feedback stabilization problem is addressed for a class of coupled nonlinear parabolic systems. An output feedback controller is presented by introducing a Luenberger-type observer based on the measured outputs. To determine observer gains, a backstepping transform is introduced by choosing a suitable target system with nonlinearity. Furthermore, based on the state observer, a backstepping boundary control scheme is presented. With rigorous analysis, it is proved that the states of nonlinear closed-loop system including state estimation and estimation error of plant system are locally exponentially stable in the *L*^{2}norm. Finally, a numerical example is proposed to illustrate the effectiveness of the presented scheme.

For 1-D first order quasilinear hyperbolic systems without zero eigenvalues, based on the theory of exact boundary controllability of nodal profile, using an extension method, the exact controllability of nodal profile can be realized in a shorter time by means of additional internal controls acting on suitably small space-time domains. On the other hand, using a perturbation method, the exact controllability of nodal profile for 1-D first order quasilinear hyperbolic systems with zero eigenvalues can be realized by additional internal controls to the part of equations corresponding to zero eigenvalues. Furthermore, by adding suitable internal controls to all the equations on suitable domains, the exact controllability of nodal profile for systems with zero eigenvalues can be realized in a shorter time.

]]>By introducing a trigonal curve , which constructed from the characteristic polynomial of Lax matrix for the Hirota-Satsuma hierarchy, we present the associated Baker-Akhiezer function and algebraic functions carrying the data of the divisor. Then the Hirota-Satsuma equations are decomposed into the system of Dubrovin-type ordinary differential equations. Based on the theory of algebraic geometry, we obtain the explicit Riemann theta function representations of the Baker-Akhiezer function, the meromorphic function, and solutions for the Hirota-Satsuma hierarchy.

]]>This paper investigates the properties of the *p*-mean Stepanov-like doubly weighted pseudo almost automorphic (*S*^{p}DWPAA) processes and its application to Sobolev-type stochastic differential equations driven by *G*-Brownian motion. We firstly prove the equivalent relation between the *S*^{p}DWPAA and Stepanov-like asymptotically almost automorphic stochastic processes based on ergodic zero set. We further establish the completeness of the space and the composition theorem for *S*^{p}DWPAA processes. These results obtained improve and extend previous related conclusions. As an application, we show the existence and uniqueness of the *S*^{p} DWPAA solution for a class of nonlinear Sobolev-type stochastic differential equations driven by *G*-Brownian motion and present a decomposition of this unique solution. Moreover, an example is given to illustrate the effectiveness of our results.

In this paper, we obtain optimal decay estimates for the solutions to an evolution equation with critical, structural, dissipation, and absorbing power nonlinearity:

with *μ*>0, *θ* is a positive integer, and *p*>1+4*θ*/*n*, in space dimension *n*∈(2*θ*,4*θ*). We use these estimates to find the self-similar asymptotic profile of the solutions, when *μ*≥1.

In this paper, we derive an asymptotic expansion for the semi-infinite sum of Dirac-*δ* functions centered at discrete equidistant points defined by the set
. The method relies on the Laplace transform of the semi-infinite sum of Dirac-*δ* functions. The derived series distribution takes the form of the Euler-Maclaurin summation when the distributions are defined for complex or real-valued continuous functions over the interval
. For *n*=1, the series expansion contributes with a term equal to *δ*(*x*)/2, which survives in the limit when *a*0^{+}. This term represents a correction term, which is in general omitted in calculations of the density of states of quantum confined systems by finite-size effects.

As far as the numerical solution of boundary value problems defined on an infinite interval is concerned, in this paper, we present a test problem for which the exact solution is known. Then we study an a posteriori estimator for the global error of a nonstandard finite difference scheme previously introduced by the authors. In particular, we show how Richardson extrapolation can be used to improve the numerical solution using the order of accuracy and numerical solutions from 2 nested quasi-uniform grids. We observe that if the grids are sufficiently fine, the Richardson error estimate gives an upper bound of the global error.

]]>In the paper, a necessary and sufficient criterion it provided such that any local optimal solution is also global in a not necessarily differentiable constrained optimization problem. This criterion is compared to others earlier appeared in the literature, which are sufficient but not necessary for a local optimal solution to be global. The importance of the established criterion is illustrated by suitable examples of nonconvex optimization problems presented in the paper.

]]>In this paper, we investigate a Vector-Borne disease model with nonlinear incidence rate and 2 delays: One is the incubation period in the vectors and the other is the incubation period in the host. Under the biologically motivated assumptions, we show that the global dynamics are completely determined by the basic reproduction number *R*_{0}. The disease-free equilibrium is globally asymptotically stable if *R*_{0}≤1; when *R*_{0}>1, the system is uniformly persistent, and there exists a unique endemic equilibrium that is globally asymptotically. Numerical simulations are conducted to illustrate the theoretical results.

For a fissured medium with uncertainty in the knowledge of fractures' geometry, a conservative tangential flow field is constructed, which is consistent with the physics of stationary fluid flow in porous media and an interpolated geometry of the cracks. The flow field permits computing preferential fluid flow directions of the medium, rates of mechanical energy dissipations, and a stochastic matrix modeling stream lines and fluid mass transportation, for the analysis of solute/contaminant mass advection-diffusion as well as drainage times.

]]>This paper is concerned with the following nonlinear fractional Schrödinger equation

where *ε*>0 is a small parameter, *V*(*x*) is a positive function, 0<*s*<1, and
. Under some suitable conditions, we prove that for any positive integer *k*, one can construct a nonradial sign-changing (nodal) solutions with exactly *k* maximum points and *k* minimum points near the local minimum point of *V*(*x*).

This paper studies the dynamics of the generalized Lengyel-Epstein reaction-diffusion model proposed in a recent study by Abdelmalek and Bendoukha. Two main results are shown in this paper. The first of which is sufficient conditions that guarantee the nonexistence of Turing patterns, ie, nonconstant solutions. Second, more relaxed conditions are derived for the stability of the system's unique steady-state solution.

]]>A class of inverse problems for restoring the right-hand side of a fractional heat equation with involution is considered. The results on existence and uniqueness of solutions of these problems are presented.

]]>A deterministic model for the transmission dynamics of measles in a population with fraction of vaccinated individuals is designed and rigorously analyzed. The model with standard incidence exhibits the phenomenon of backward bifurcation, where a stable disease-free equilibrium coexists with a stable endemic equilibrium whenever the associated reproduction number is less than unity. This phenomenon can be removed if either measles vaccine is assumed to be perfect or disease related mortality rates are negligible. In the latter case, the disease-free equilibrium is shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, the model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. This equilibrium is shown, using a nonlinear Lyapunov function of Goh-Volterra type, to be globally asymptotically stable for a special case.

]]>Malaria is one of the most common mosquito-borne diseases widespread in the tropical and subtropical regions. Few models coupling the within-host malaria dynamics with the between-host mosquito-human dynamics have been developed. In this paper, by adopting the nested approach, a malaria transmission model with immune response of the host is formulated. Applying age-structured partial differential equations for the between-host dynamics, we describe the asymptomatic and symptomatic infectious host population for malaria transmission. The basic reproduction numbers for the within-host model and for the coupled system are derived, respectively. The existence and stability of the equilibria of the coupled model are analyzed. We show numerically that the within-host model can exhibit complex dynamical behavior, possibly even chaos. In contrast, equilibria in the immuno-epidemiological model are globally stable and their stabilities are determined by the reproduction number. Increasing the activation rate of the within-host immune response “dampens” the sensitivity of the population level reproduction number and prevalence to the increase of the within-host reproduction of the pathogen. From public health perspective this means that treatment in a population with higher immunity has less impact on the population-level reproduction number and prevalence than in a population with less immunity.

]]>We provide a non-linear image transformation by means of conformal geometric algebra (CGA) elements and operations. We show the correspondence to the fisheye correction algorithms; more precisely, we prove the proportionality of classical models with CGA algorithm and provide exact formula in terms of CGA. Consequently, we show that the geometric construction allows to determine the inverse model quite universally.

]]>In this paper, we consider the integration of systems of second-order linear inhomogeneous initial value problems with constant coefficients. Hybrid Numerov methods are used that are constructed in the sense of Runge-Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. We present the order conditions taking advantage of the special structure of the problem at hand. These equations are solved using differential evolution technique, and we present a method with algebraic order eighth at a cost of only 5 function evaluations per step. Numerical results over some linear problems, especially arising from the semidiscretization of the wave equation indicate the superiority of the new method.

]]>In this paper, we investigate the dynamics of a multigroup disease propagation model with distributed delays and nonlinear incidence rates, which accounts for the relapse of recovered individuals. The main concern is the stability of the equilibria, sufficient conditions for global stability being obtained by applying Lyapunov-LaSalle invariance principle and using Lyapunov functionals, which are constructed using their single-group counterparts. The situation in which the deterministic model is subject to perturbations of white noise type is also investigated from a stability viewpoint.

]]>In this paper, we investigate a backward problem for a space-fractional partial differential equation. The main purpose is to propose a modified regularization method for the inverse problem. The existence and the uniqueness for the modified regularized solution are proved. To derive the gradient of the optimization functional, the variational adjoint method is introduced, and hence, the unknown initial value is reconstructed. Finally, numerical examples are provided to show the effectiveness of the proposed algorithm. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study a reaction-diffusion equation where nonlinear memory and concentration effects are considered at the same time. We are specially concerned with the local solvability of this problem for singular initial data in Lebesgue spaces. We also analyze the vanishing concentration problem and prove a blow-up alternative. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we present the approximate solution of damped Boussinesq equation using extended Raviart–Thomas mixed finite element method. In this method, the numerical solution of this equation is obtained using triangular meshes. Also, for discretization in time direction, we use an implicit finite difference scheme. In addition, error estimation and stability analysis of both methods are shown. Finally, some numerical examples are considered to confirm the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we consider the almost periodic dynamics of an impulsive multispecies Lotka–Volterra competition system with time delays on time scales. By establishing some comparison theorems of dynamic equations with impulses and delays on time scales, a permanence result for the model is obtained. Furthermore, based on the permanence result, by studying the Lyapunov stability theory of impulsive dynamic equations on time scales, we establish the existence and uniformly asymptotic stability of a unique positive almost periodic solution of the system. Finally, we give an example to show the feasibility of our main results, and our example also shows that the continuous time system and its corresponding discrete time system have the same dynamics. Our results of this paper are completely new even if for both the case of the time scale and the case of the time scale . Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we use the domain decomposition method to prove well-posedness and smoothness results in anisotropic weighted Sobolev spaces for a multidimensional high-order parabolic equation set in conical time-dependent domains of . Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study a model of hematopoiesis with time-varying delays and discontinuous harvesting, which is described by a nonsmooth dynamical system. Based on a newly developed method, nonsmooth analysis, and the generalized Lyapunov method, some new delay-dependent criteria are established to ensure the existence and global exponential stability of positive periodic solutions. Moreover, an example with numerical simulations is presented to demonstrate the effectiveness of theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Quaternion-valued signals along with quaternion Fourier transforms (QFT) provide an effective framework for vector-valued signal and image processing. However, the sampling theory of quaternion-valued signals has not been well developed. In this paper, we present the generalized sampling expansions associated with QFT by using the generalized translation and convolution. We show that a *σ*-bandlimited quaternion-valued signal in QFT sense can be reconstructed from the samples of output signals of *M* linear systems based on QFT. Quaternion linear canonical transform is a generalization of QFT with six parameters. Using the relationship between QFT, we derive the sampling formula for *σ*-bandlimited quaternion-valued signal in quaternion linear canonical transform sense. Examples are given to illustrate our results. Copyright © 2017 John Wiley & Sons, Ltd.

We consider general virus dynamics model with virus-to-target and infected-to-target infections. The model is incorporated by intracellular discrete or distributed time delays. We assume that the virus-target and infected-target incidences, the production, and clearance rates of all compartments are modeled by general nonlinear functions that satisfy a set of reasonable conditions. The non-negativity and boundedness of the solutions are studied. The existence and stability of the equilibria are determined by a threshold parameter. We use suitable Lyapunov functionals and apply LaSalle's invariance principle to prove the global asymptotic stability of the all equilibria of the model. We confirm the theoretical results by numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.

]]>One of the fundamental problems in communications is finding the energy distribution of signals in time and frequency domains. It should therefore be of great interest to find the quaternionic signal whose time-frequency energy distribution is most concentrated in a given time-frequency domain. The present paper finds a new kind of quaternionic signals whose energy concentration is maximal in both time and frequency under the quaternionic Fourier transform. The new signals are a generalization of the classical prolate spheroidal wave functions to a quaternionic space, which are called the quaternionic prolate spheroidal wave functions. The purpose of this paper is to present the definition and fundamental properties of the quaternionic prolate spheroidal wave functions and to show that they can reach the extreme case within the energy concentration problem both from the theoretical and experimental description. The superiority of the proposed results can be widely applied to the application of 4D valued problems. In particular, these functions are shown as an effective method for bandlimited quaternionic signals relying on the extrapolation problem. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We consider the Calderón problem in an infinite cylindrical domain, whose cross section is a bounded domain of the plane. We prove log–log stability in the determination of the isotropic periodic conductivity coefficient from partial Dirichlet data and partial Neumann boundary observations of the solution. Copyright © 2017 John Wiley & Sons, Ltd.

]]>No abstract is available for this article.

]]>A mixed boundary value problem for the stationary heat transfer equation in a thin layer around a surface with the boundary is investigated. The main objective is to trace what happens in Γ-limit when the thickness of the layer converges to zero. The limit Dirichlet BVP for the Laplace–Beltrami equation on the surface is described explicitly, and we show how the Neumann boundary conditions in the initial BVP transform in the Γ-limit. For this, we apply the variational formulation and the calculus of Günter's tangential differential operators on a hypersurface and layers, which allow global representation of basic differential operators and of corresponding boundary value problems in terms of the standard Euclidean coordinates of the ambient space . 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, 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.

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 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.

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 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 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.

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.

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.

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.

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.

The current paper is devoted to stochastic Ginzburg–Landau–Newell equation with degenerate random forcing. The existence and pathwise uniqueness of strong solutions with *H*^{1}-initial data is established, and then the existence of an invariant measure for the Feller semigroup is shown by Krylov–Bogoliubov theorem. Because of the coupled items in the stochastic Ginzburg–Landau–Newell equations, the higher order momentum estimates can be only obtained in the *L*^{2}-norm. We show the ergodicity of invariant measure for the transition semigroup by asymptotically strong Feller property and the support property. 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.

In this paper, we study the global well-posedness and scattering theory of the solution to the Cauchy problem of a generalized fourth-order wave equation

where
if *d*⩽4, and
if *d*⩾5. The main strategy we use in this paper is concentration-compactness argument, which was first introduced by Kenig and Merle to handle the scattering problem vector so as to control the momentum. 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.

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.

]]>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 consider smooth compactly supported solution to the classical Vlasov–Poisson system in three space dimensions in the electrostatic case. An estimate on velocities is derived, showing a growth rate at most like the power 1/8 of the time variable. As a consequence, a better decay estimate is obtained for the electric field in the norm. 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 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 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.

]]>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.

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.

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 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.

]]>In this paper, a computational technique based on the pseudo-spectral method is presented for the solution of the optimal control problem constrained with elliptic variational inequality. In fact, our aim in this paper is to present a direct approach for this class of optimal control problems. By using the pseudo-spectral method, the infinite dimensional mathematical programming with equilibrium constraint, which can be an equivalent form of the considered problem, is converted to a finite dimensional mathematical programming with complementarity constraint. Then, the finite dimensional problem can be solved by the well-developed methods. Finally, numerical examples are presented to show the validity and efficiency of the technique. Copyright © 2017 John Wiley & Sons, Ltd.

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