We study the nonexistence of global solutions to the Cauchy problem for systems of time fractional parabolic-hyperbolic and time fractional hyperbolic thermo-elasticity equations in
. For certain nonlinearities, we present ‘threshold’ exponents depending on the space dimension *d*. Our proof rests on the test function method. Copyright © 2017 John Wiley & Sons, Ltd.

Modeling the cardiac conduction system is a challenging problem in the context of computational cardiac electrophysiology. Its ventricular section, the Purkinje system, is responsible for triggering tissue electrical activation at discrete terminal locations, which subsequently spreads throughout the ventricles. In this paper, we present an algorithm that is capable of estimating the location of the Purkinje system triggering points from a set of random measurements on tissue. We present the properties and the performance of the algorithm under controlled synthetic scenarios. Results show that the method is capable of locating most of the triggering points in scenarios with a fair ratio between terminals and measurements. When the ratio is low, the method can locate the terminals with major impact in the overall activation map. Mean absolute errors obtained indicate that solutions provided by the algorithm are useful to accurately simulate a complete patient ventricular activation map. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, the diffusion is introduced to an immunosuppressive infection model with delayed antiviral immune response. The direction and stability of Hopf bifurcation are effected by time delay, in the absence of which the positive equilibrium is locally asymptotically stable by means of analyzing eigenvalue spectrum; however, when the time delay increases beyond a threshold, the positive equilibrium loses its stability via the Hopf bifurcation. The stability and direction of the Hopf bifurcation is investigated with the norm form and the center manifold theory. The stability of the Hopf bifurcation leads to the emergence of spatial spiral patterns. Numerical calculations are performed to illustrate our theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Starting from the representation of the (*n* − 1) + *n* − dimensional Lorentz pseudo-sphere on the projective space
, we propose a method to derive a class of solutions underlying to a Dirac–Kähler type equation on the lattice. We make use of the Cayley transform
to show that the resulting group representation arises from the same mathematical framework as the conformal group representation in terms of the *general linear group*
. That allows us to describe such class of solutions as a commutative *n* − ary product, involving the quasi-monomials
) with membership in the paravector space
. Copyright © 2016 John Wiley & Sons, Ltd.

For graph domains without cycles, we show how unknown coefficients and source terms for a parabolic equation can be recovered from the dynamical Neumann-to-Dirichlet map associated with the boundary vertices. Through use of a companion wave equation problem, the topology of the tree graph, degree of the vertices, and edge lengths can also be recovered. The motivation for this work comes from a neuronal cable equation defined on the neuron's dendritic tree, and the inverse problem concerns parameter identification of *k* unknown distributed conductance parameters. Copyright © 2017 John Wiley & Sons, Ltd.

We prove the existence of global weak solution of the two-dimensional dissipative quasi-geostrophic equations with small initial data in and local well-posedness with the large initial data in the same space. Our proof is based on constructing a commutator related to the problem, as well as its estimate. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Based on the extended extragradient-like method and the linesearch technique, we propose three projection methods for finding a common solution of a finite family of equilibrium problems. The linesearch used in the proposed algorithms has allowed to reduce some conditions imposed on equilibrium bifunctions. The strongly convergent theorems are established without the Lipschitz-type condition of bifunctions. The paper also helps in the design and analysis of practical algorithms and gives us a generalization of some previously known problems. Copyright © 2017 John Wiley & Sons, Ltd.

]]>Time-dependent PDEs with fractional Laplacian ( − Δ)^{α} play a fundamental role in many fields and approximating ( − Δ)^{α} usually leads to ODEs' system like **u**^{′}(*t*) + *A***u**(*t*) = **g**(*t*) with *A* = *Q*^{α}, where
is a sparse symmetric positive definite matrix and *α* > 0 denotes the fractional order. The parareal algorithm is an ideal solver for this kind of problems, which is iterative and is characterized by two propagators
and
. The propagators
and
are respectively associated with large step size Δ*T* and small step size Δ*t*, where Δ*T* = *J*Δ*t* and *J*⩾2 is an integer. If we fix the
-propagator to the Implicit-Euler method and choose for
some proper Runge–Kutta (RK) methods, such as the second-order and third-order singly diagonally implicit RK methods, previous studies show that the convergence factors of the corresponding parareal solvers can satisfy
and
, where *σ*(*A*) is the spectrum of the matrix *A*. In this paper, we show that by choosing these two RK methods as the
-propagator, the convergence factors can reach
, provided the one-stage complex Rosenbrock method is used as the
-propagator. If we choose for both
and
, the complex Rosenbrock method, we show that the convergence factor of the resulting parareal solver can also reach
. Numerical results are given to support our theoretical conclusions. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, we consider a Riesz–Feller space-fractional backward diffusion problem with a time-dependent coefficient

We show that this problem is ill-posed; therefore, we propose a convolution regularization method to solve it. New error estimates for the regularized solution are given under *a priori* and *a posteriori* parameter choice rules, respectively. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is in continuation of the work performed by Kajla *et al*. (Applied Mathematics and Computation 2016; 275 : 372–385.) wherein the authors introduced a bivariate extension of *q*-Bernstein–Schurer–Durrmeyer operators and studied the rate of convergence with the aid of the Lipschitz class function and the modulus of continuity. Here, we estimate the rate of convergence of these operators by means of Peetre's *K*-functional. Then, the associated generalized Boolean sum operator of the *q*-Bernstein–Schurer–Durrmeyer type is defined and discussed. The smoothness properties of these operators are improved with the help of mixed *K*-functional. Furthermore, we show the convergence of the bivariate Durrmeyer-type operators and the associated generalized Boolean sum operators to certain functions by illustrative graphics using Maple algorithm. Copyright © 2017 John Wiley & Sons, Ltd.

This paper is concerned with the exponential stability for the discrete-time bidirectional associative memory neural networks with time-varying delays. Based on Lyapunov stability theory, some novel delay-dependent sufficient conditions are obtained to guarantee the globally exponential stability of the addressed neural networks. In order to obtain less conservative results, an improved Lyapunov–Krasovskii functional is constructed and the reciprocally convex approach and free-weighting matrix method are employed to give the upper bound of the difference of the Lyapunov–Krasovskii functional. Several numerical examples are provided to illustrate the effectiveness of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.

]]>We investigate a linear, fully coupled thermoelasticity problem for a highly heterogeneous, two-phase medium. The medium in question consists of a connected matrix with disconnected, initially periodically distributed inclusions separated by a sharp interface undergoing an *a priori* known interface movement because of phase transformations. After transforming the moving geometry to an *ϵ*-periodic, fixed reference domain, we establish the well-posedness of the model and derive a number of *ϵ*-independent *a priori* estimates. Via a two-scale convergence argument, we then show that the *ϵ*-dependent solutions converge to solutions of a corresponding upscaled model with distributed time-dependent microstructures. Copyright © 2017 John Wiley & Sons, Ltd.

A matrix formulation of the generalised finite difference method is introduced. A necessary and sufficient condition for the uniqueness of the solution is demonstrated, and important practical consequences are obtained. A generalised finite differences scheme for SH wave is obtained, the stability of the scheme is analysed and the formula for the velocity of the wave due to the scheme is obtained in order to deal with the numerical dispersion. The method is applied to seismic waves propagation problems, specifically to the problem of reflection and transmission of plane waves in heterogeneous media. A heterogeneous approach without nodes at the interface is chosen to solve the problem in heterogeneous media. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, a numerical solution of fractional partial differential equations (FPDEs) for electromagnetic waves in dielectric media will be discussed. For the solution of FPDEs, we developed a numerical collocation method using an algorithm based on two-dimensional shifted Legendre polynomials approximation, which is proposed for electromagnetic waves in dielectric media. By implementing the partial Riemann–Liouville fractional derivative operators, two-dimensional shifted Legendre polynomials approximation and its operational matrix along with collocation method are used to convert FPDEs first into weakly singular fractional partial integro-differential equations and then converted weakly singular fractional partial integro-differential equations into system of algebraic equation. Some results concerning the convergence analysis and error analysis are obtained. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we study the *n*th-order nonlinear dynamic equation with Laplacians and a deviating argument

on an above-unbounded time scale, where *n*⩾2,

New oscillation criteria are established for the cases when *n* is even and odd and when *α* > *γ*,*α* = *γ*, and *α* < *γ*, respectively, with *α* = *α*_{1}⋯*α*_{n − 1}. Copyright © 2017 John Wiley & Sons, Ltd.

This paper concerns new continuum phenomenological model for epitaxial thin-film growth with three different forms of the Ehrlich–Schwoebel current. Two of these forms were first proposed by Politi and Villain 1996 and then studied by Evans, Thiel, and Bartelt 2006. The other one is completely new. Energy structure and properties of the new model are studied. Following the techniques used in Li and Liu 2003, we present rigorous analysis of the well-posedness, regularity, and time stability for the new model. We also studied both the global and the local behavior of the surface roughness in the growth process. By using a convex–concave time-splitting scheme, one can naturally build unconditionally stable semi-implicit numerical discretizations with linear implicit parts, which is much easier to implement than conventional models requiring nonlinear implicit parts. Despite this fundamental difference in the model, numerical experiments show that the nonlinear morphological instability of the new model agrees well with results of other models published before which indicates that the new model correctly captures the essential morphological states in the thin-film growth process. Copyright © 2017 John Wiley & Sons, Ltd.

]]>This paper contains results on well-posedness, stability, and long-time behavior of solutions to a class of plate models subject to damping and source terms given by the product of two nonlinear components [EQUATION1] where Ω is a bounded open set of **R**^{n} with smooth boundary, *γ*,*ρ*⩾0 and
are nonlocal functions. The main result states that the dynamical system {*S*(*t*)}_{t⩾0} associated with this problem has a compact global attractor. In addition, in the limit case *γ* = 0, it is also shown that {*S*(*t*)}_{t⩾0} has a finite dimensional global attractor by using an approach on quasi-stability because of Chueshov–Lasiecka (2010). Copyright © 2017 John Wiley & Sons, Ltd.

We establish a two-wave mode equation for the integrable Kadomtsev–Petviashvili equation, which describes the propagation of two different wave modes in the same direction simultaneously. We determine the necessary conditions that make multiple soliton solutions exist for this new equation. The simplified Hirota's method will be used to conduct this work. We also use other techniques to obtain other set of periodic and singular solutions for the two-mode Kadomtsev-Petviashvili equation. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, we introduce a generalized form of Cole-Hopf transformation and apply it to find new closed-form (analytic) solutions to Painleve III equation. The same transformation is used then to find analytic solutions for the van der Pol and other nonlinear convective equations. These solutions provide analytic insights to some practical problems and might be used also to test the accuracy of numerical solutions. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, our main aim is to establish some new fractional integral inequalities involving Hadamard-type *k*-fractional integral operators recently given by Mubeen *et al.* Furthermore, the paper discusses some of their relevance with known results. Copyright © 2016 John Wiley & Sons, Ltd.

The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot both be sharply localized. It plays an important role in signal processing and physics. This paper generalizes the uncertainty principle for measurable sets from complex domain to hypercomplex domain using quaternion algebras, associated with the quaternion Fourier transform. The performance is then evaluated in signal recovery problems where there is an interplay of missing and time-limiting data. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we aim to investigate the exponential stability of general hybrid stochastic functional differential systems with delayed impulses. By using the average impulsive interval and the Lyapunov function method, we derive some sufficient conditions for exponential stability, which are less conservative than those existing results based on the supremum or infimum of impulsive interval and more convenient to be applied than those Razumikhin-type conditions in the literature. Meanwhile, we show that unstable hybrid stochastic delay differential systems, both linear and nonlinear, can be stabilized by suitably impulsive sequence. Finally, two examples are discussed to illustrate the effectiveness and advantages of the obtained results. Copyright © 2017 John Wiley & Sons, Ltd.

]]>In this paper, stability and bifurcation of a two-dimensional ratio-dependence predator–prey model has been studied in the close first quadrant
. It is proved that the model undergoes a period-doubling bifurcation in a small neighborhood of a boundary equilibrium and moreover, Neimark–Sacker bifurcation occurs at a unique positive equilibrium. We study the Neimark–Sacker bifurcation at unique positive equilibrium by choosing *b* as a bifurcation parameter. Some numerical simulations are presented to illustrate theocratical results. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, blow-up property to a system of nonlinear stochastic PDEs driven by two-dimensional Brownian motions is investigated. The lower and upper bounds for blow-up times are obtained. When the system parameters satisfy certain conditions, the explicit solutions of a related system of random PDEs are deduced, which allows us to use Yor's formula to obtain the distribution functions of several blow-up times. Particularly, the impact of noises on the life span of solutions is studied as the system parameters satisfy different conditions. 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.

]]>We consider a Euler–Bernoulli beam equation with a boundary control condition of fractional derivative type. We study stability of the system using the semigroup theory of linear operators and a result obtained by Borichev and Tomilov. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The paper is devoted to study the stability of nonlinear fractional order difference systems by their linear approximation. Additionally, we show the relation between the stability of linear fractional order differential systems and their discretizations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>By equivalently replacing the dynamical boundary condition by a kind of nonlocal boundary conditions, and noting a hidden regularity of solution on the boundary with a dynamical boundary condition, a constructive method with modular structure is used to get the local exact boundary controllability for 1-D quasilinear wave equations with dynamical boundary conditions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>A new proposal for group key exchange is introduced which proves to be both efficient and secure and compares favorably with state of the art protocols. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper presents a method for nested fluid simulation based on smoothed particle hydrodynamics. Given suitable background flow information of an “external flow” as it evolves in time, our method simulates the motion of particles only within a local material region. In order to perform the simulation, the background physical quantities need to be transferred to the local fluid particles. We employ ghost particles to carry the given physical quantities to the nested fluid. We also solve the problem of density computation appropriately for the ghost particles. Our numerical tests show that accurate local fluid motion can be obtained in such a nested volume of fluid particles. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper presents an a posteriori error analysis for the stationary Stokes–Darcy coupled problem approximated by finite element methods on anisotropic meshes in
or 3. Korn's inequality for piecewise linear vector fields on anisotropic meshes is established and is applied to non-conforming finite element method. Then the existence and uniqueness of the approximation solution are deduced for non-conforming case. With the obtained finite element solutions, the error estimators are constructed and based on the residual of model equations plus the stabilization terms. *The lower error bound* is proved by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the *upper error bound*, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so-called *matching function* is defined, and its discussion shows it to be useful tool. With its help, the *upper error bound* is shown by means of the corresponding anisotropic interpolation estimates and a special Helmholtz decomposition in both media. Copyright © 2016 John Wiley & Sons, Ltd.

Fractional cosine transform (FRCT) and fractional sine transform (FRST), which are closely related to the fractional Fourier transform (FRFT), are useful mathematical and optical tool for signal processing. Many properties for these transforms are well investigated, but the convolution theorems are still to be determined. In this paper, we derive convolution theorems for the fractional cosine transform (FRCT) and fractional sine transform (FRST) based on the four novel convolution operations. And then, a potential application for these two transforms on designing multiplicative filter is presented. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The implementation of countermeasures to avoid licence abuse is now obligatory, especially with the burgeoning of the Internet. The protocol proposed here is implemented within the session initiation protocol (SIP); this has been selected as the official end-to-end signalling protocol for establishing multimedia sessions in the Universal Mobile Telecommunication Systems network. This paper introduces blind signatures, enforced with user-specific and unique data, modelled from CCD sensors to trace users of these online services, thus avoiding licence sharing that gives access to them. Blind signatures are useful in providing anonymity and establishing a way to tag users. The proposed protocol takes advantage of elliptic curve-based cryptosystems – smaller key sizes and lower computational resources, an interesting issue for session establishment in S-Universal Mobile Telecommunication Systems (satellite-linked networks), where fast and light authentication protocols are a requirement ideal. SIP is a powerful signalling protocol for transmitting media over Internet protocol. Authentication is a vital security requirement for SIP. Hitherto, many authentication schemes have been proposed to enhance SIP security; indeed, the problem of impersonation is one of the topics most discussed. Consequently, a novel authentication and key agreement scheme is proposed for SIP using an elliptic curve cryptosystem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this work, we utilize the existing Carleman estimates and propagation estimates of smallness from measurable sets for real analytic functions, together with the telescoping series method, to establish an observability inequality from measurable subsets in time-space variable for the parabolic equation with Grushin operator in some multidimension domains. We can apply this observability inequality to show the bang–bang property for both time optimal and norm optimal control problems for this kind of singular parabolic equation. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Under the shrinking curvature flow with inner normal velocity *V* = *k*^{α}(*α* > 1), it is shown that highly symmetric, locally convex initial curves evolve into a point asymptotically like an multi-circles. The proof relies on a crucial use of Bonnensen inequality for highly symmetric, locally convex curves. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, combining with a new generalized ansätz and the fractional Jacobi elliptic equation, an improved fractional Jacobi elliptic equation method is proposed for seeking exact solutions of space-time fractional partial differential equations. The fractional derivative used here is the modified Riemann-Liouville derivative. For illustrating the validity of this method, we apply it to solve the space-time fractional Fokas equation and the the space-time fractional BBM equation. As a result, some new general exact solutions expressed in various forms including the solitary wave solutions, the periodic wave solutions, and Jacobi elliptic functions solutions for the two equations are found with the aid of mathematical software Maple. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the one and one-half dimensional multi-species relativistic Vlasov–Maxwell system with non-decaying (in space) initial data. We prove its well-posedness and nonrelativistic limit as the speed of light . These results mainly rely on a delicate analysis of energy structure and application of estimates along the characteristic lines. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper deals with the non-uniform dependence and persistence properties for a coupled Camassa–Holm equations. Using the method of approximate solutions in conjunction with well-posedness estimate, it is proved that the solution map of the Cauchy problem for this coupled Camassa–Holm equation is not uniformly continuous in Sobolev spaces *H*^{s} with *s* > 3/2. On the other hand, the persistence properties in weighted *L*^{p} spaces for the solution of this coupled Camassa–Holm system are considered. Copyright © 2016 John Wiley & Sons, Ltd.

The solution form of the system of nonlinear difference equations

where the coefficients *a*,*b*,*α*,*β* and the initial values *x*_{ − i},*y*_{ − i},*i*∈{0,1,…,*k*} are non-zero real numbers, is obtained. Furthermore, the behavior of solutions of the aforementioned system when *p* = 1 is examined. Copyright © 2016 John Wiley & Sons, Ltd.

This work proposes a general class of estimators for the population total of a sensitive variable using auxiliary information. Under a general randomized response model, the optimal estimator in this class is derived. Design-based properties of proposed estimators are obtained. A simulation study reflects the potential gains from the use of the proposed estimators instead of the customary estimators. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider a linear parabolic problem in a thick junction domain which is the union of a fixed domain and a collection of periodic branched trees of height of order 1 and small width connected on a part of the boundary. We consider a three-branched structure, but the analysis can be extended to n-branched structures. We use unfolding operator to study the asymptotic behavior of the solution of the problem. In the limit problem, we get a multi-sheeted function in which each sheet is the limit of restriction of the solution to various branches of the domain. Homogenization of an optimal control problem posed on the above setting is also investigated. One of the novelty of the paper is the characterization of the optimal control via the appropriately defined unfolding operators. Finally, we obtain the limit of the optimal control problem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we introduce a new generalization of the Helgason–Fourier transform using the angular Dirac operator on both the hyperboloid and unit ball models. The explicit integral kernels of even dimension are derived. Furthermore, we establish the formal generating function of the even dimensional kernels. In the computations, fractional integration plays a key unifying role. Copyright © 2016 John Wiley & Sons, Ltd.

]]>An isogeometric discontinuous Galerkin method for Euler equations is proposed. It integrates the idea of isogeometric analysis with the discontinuous Galerkin framework by constructing each element through the knots insertion and degree elevation techniques in non-uniform rational B-splines. This leads to the solution inherently shares the same function space as the non-uniform rational B-splines representation, and results in that the curved boundaries as well as the interfaces between neighboring elements are naturally and exactly resolved. Additionally, the computational cost is reduced in contrast to that of structured grid generation. Numerical tests demonstrate that the presented method can be high order of accuracy and flexible in handling curved geometry. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study existence of invariant curves of an iterative equation

which is from dissipative standard map. By constructing an invertible analytic solution *g*(*x*) of an auxiliary equation of the form

invertible analytic solutions of the form *g*(*λ**g*^{ − 1}(*x*)) for the original iterative functional equation are obtained. Besides the hyperbolic case 0 < |*λ*| < 1, we focus on those *λ* on the unit circle *S*^{1}, that is, |*λ*| = 1. We discuss not only those *λ* at resonance, that is, at a root of the unity, but also those *λ* near resonance under the Brjuno condition. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a generalized Kirchhoff equation in a bounded domain under the effect of a sublinear nonlinearity. Under suitable assumptions on the data of the problem, we show that, with a simple change of variable, the equation can be reduced to a classical semilinear equation and then studied with standard tools. Copyright © 2016 John Wiley & Sons, Ltd.

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

under homogeneous Neumann boundary conditions in a bounded domain
with smooth boundary. Here, *δ* and *χ* are some positive constants and *f* is a smooth function that satisfies

with some constants *a*⩾0,*b* > 0, and *γ* > 1. We prove that the classical solutions to the preceding system are global and bounded provided that

Copyright © 2016 John Wiley & Sons, Ltd.

We consider a fourth-order variational model, to solve the image inpainting problem, with the emphasis on the recovery of low-dimensional sets (edges and corners) and the curvature of the edges. The model permits also to perform simultaneously the restoration (filtering) of the initial image where this one is available. The multiscale character of the model follows from an adaptive selection of the diffusion parameters that allows us to optimize the regularization effects in the neighborhoods of the small features that we aim to preserve. In addition, because the model is based on the high-order derivatives, it favors naturally the accurate capture of the curvature of the edges, hence to balance the tasks of obtaining long curved edges and the obtention of short edges, tip points, and corners. We analyze the method in the framework of the calculus of variations and the Γ-convergence to show that it leads to a convergent algorithm. In particular, we obtain a simple discrete numerical method based on a standard mixed-finite elements with well-established approximation properties. We compare the method to the Cahn–Hilliard model for the inpainting, and we present several numerical examples to show its performances. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the initial boundary value problem for generalized logarithmic improved Boussinesq equation. By using the Galerkin method, logarithmic Sobolev inequality, logarithmic Gronwall inequality, and compactness theorem, we show the existence of global weak solution to the problem. By potential well theory, we show the norm of the solution will grow up as an exponential function as time goes to infinity under some suitable conditions. Furthermore, for the generalized logarithmic improved Boussinesq equation with damped term, we obtain the decay estimate of the energy. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we discuss two inverse problems for differential pencils with boundary conditions dependent on the spectral parameter. We will prove the Hochstadt–Lieberman type theorem of – except for arbitrary one eigenvalue and the Borg type theorem of – except for at most arbitrary two eigenvalues, respectively. The new results are generalizations of the related results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the Maxwell equations for a composite material consisting of two phases and enjoying a periodical structure in the presence of a time-harmonic current source. We perform the two-scale homogenization taking into account both the interfacial layer thickness and the complex conductivity of the interfacial layer. We introduce a variational formulation of the differential system equipped with boundary and interfacial conditions. We show the unique solvability of the variational problem. Then, we analyze the low frequency case, high and very high frequency cases, with different strength of the interfacial currents. We find the macroscopic equations and determine the effective constant matrices such as the magnetic permeability, dielectric permittivity, and electric conductivity. The effective matrices depend strongly on the frequency of the current source; the dielectric permittivity and electric conductivity also depend on the strength of the interfacial currents. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper deals with an inverse problem of determining the diffusion coefficient, spacewise dependent source term, and the initial value simultaneously for a one-dimensional heat equation based on the boundary control, boundary measurement, and temperature distribution at a given single instant in time. By a Dirichlet series representation for the boundary observation, the identification of the diffusion coefficient and initial value can be transformed into a spectral estimation problem of an exponential series with measurement error, which is solved by the matrix pencil method. For the identification of the source term, a finite difference approximation method in conjunction with the truncated singular value decomposition is adopted, where the regularization parameter is determined by the generalized cross-validation criterion. Numerical simulations are performed to verify the result of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Ductal carcinoma *in situ* – a special cancer – is confined within the breast ductal only. We derive the mathematical ductal carcinoma *in situ* model in a form of a nonlinear parabolic equation with initial, boundary, and free boundary conditions. Existence, uniqueness, and stability of problem are proved. Algorithm and illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2016 John Wiley & Sons, Ltd.

The aim of this paper is twofold. We first establish the Besov spaces on metric spaces endowed with a doubling measure, via the remarkable orthonormal wavelet basis constructed recently by T. Hytönen and O. Tapiola, and characterize the dual spaces of these Besov spaces. Second, we prove the *T*1 type theorem for the boundedness of Calderón–Zygmund operators on these Besov spaces. Finally, we introduce a new class of Lipschitz spaces and characterize these spaces via the Littlewood–Paley theory. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we investigate the influence of boundary dissipation on the decay property of solutions for a transmission problem of Kirchhoff-type wave equations with a memory condition on one part of the boundary. Without the condition *u*_{0} = 0 on Γ_{0}, we establish a general decay of energy depending on the behavior of relaxation function by introducing suitable energy and Lyapunov functionals. This result allows a wider class of relaxation functions. Copyright © 2016 John Wiley & Sons, Ltd.

We analyze a highly nonlinear system of partial differential equations related to a model solidification and/or melting of thermoviscoelastic isochoric materials with the possibility of motion of the material during the process. This system consists of an internal energy balance equation governing the evolution of temperature, coupled with an evolution equation for a phase field whose values describe the state of material and a balance equation for the linear moments governing the material displacements. For this system, under suitable dissipation conditions, we prove global existence and uniqueness of weak solutions. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we investigate the existence of homoclinic solutions for a class of fourth-order nonautonomous differential equations

where *w* is a constant,
and
. By using variational methods and the mountain pass theorem, some new results on the existence of homoclinic solutions are obtained under some suitable assumptions. The interesting is that *a*(*x*) and *f*(*x*,*u*) are nonperiodic in *x*,*a* does not fulfil the coercive condition, and *f* does not satisfy the well-known (*A**R*)-condition. Furthermore, the main result partly answers the open problem proposed by Zhang and Yuan in the paper titled with Homoclinic solutions for a nonperiodic fourth-order differential equations without coercive conditions (see Appl. Math. Comput. 2015; 250:280–286). Copyright © 2016 John Wiley & Sons, Ltd.

We consider the Lamé system for an elastic medium consisting of an inclusion embedded in a homogeneous background medium. Based on the field expansion method and layer potential techniques, we rigorously derived the asymptotic expansion of the perturbed displacement field because of small perturbations in the interface of the inclusion. We extend these techniques to determine a relationship between traction-displacement measurements and the shape of the object and derive an asymptotic expansion for the perturbation in the elastic moment tensors because of the presence of small changes in the interface of the inclusion. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, a class of fourth-order Sturm-Liouville problems with transmission conditions is considered. The eigenvalues depend not only continuously but also smoothly on the problem. An expression for the derivative of the eigenvalues with respect to a given parameter: an endpoint, a boundary condition, a transmission condition, a coefficient, or the weight function, is found. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The process of transporting nanoparticles at the blood vessels level stumbles upon various physical and physiological obstacles; therefore, a Mathematical modeling will provide a valuable means through which to understand better this complexity. In this paper, we consider the motion of nanoparticles in capillaries having cylindrical shapes (i.e., tubes of finite size). Under the assumption that these particles have spherical shapes, the motion of these particles reduces to the motion of their centers. Under these conditions, we derive the mathematical model, to describe the motion of these centers, from the equilibrium of the gravitational force, the hemodynamic force and the van der Waals interaction forces. We distinguish between the interaction between the particles and the interaction between each particle and the walls of the tube. Assuming that the minimum distance between the particles is large compared with the maximum radius *R* of the particles and hence neglecting the interactions between the particles, we derive simpler models for each particle taking into account the particles-to-wall interactions. At an error of order *O*(*R*) or *O*(*R*^{3})(depending if the particles are 'near' or 'very near' to the walls), we show that the horizontal component of each particle's displacement is solution of a nonlinear integral equation that we can solve via the fixed point theory. The vertical components of the displacement are computable in a straightforward manner as soon as the horizontal components are estimated. Finally, we support this theory with several numerical tests. Copyright © 2016 John Wiley & Sons, Ltd.

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term. Based on a low-frequency and high-frequency decomposition, Green's function method and the classical energy method, we not only obtain *L*^{2} time-decay estimates but also establish the global existence of solutions to Cauchy problem when the initial data *u*_{0}(*x*) satisfies the smallness condition on
, but not on
. Furthermore, by taking a time-frequency decomposition, we obtain the optimal decay estimates of solutions. Copyright © 2016 John Wiley & Sons, Ltd.

No abstract is available for this article.

]]>Optical coherence tomography (OCT) and photoacoustic tomography are emerging non-invasive biological and medical imaging techniques. It is a recent trend in experimental science to design experiments that perform photoacoustic tomography and OCT imaging at once. In this paper, we present a mathematical model describing the dual experiment. Because OCT is mathematically modelled by Maxwell's equations or some simplifications of it, whereas the light propagation in quantitative photoacoustics is modelled by (simplifications of) the radiative transfer equation, the first step in the derivation of a mathematical model of the dual experiment is to obtain a unified mathematical description, which in our case are Maxwell's equations. As a by-product, we therefore derive a new mathematical model of photoacoustic tomography based on Maxwell's equations. It is well known by now that without additional assumptions on the medium, it is not possible to uniquely reconstruct all optical parameters from either one of these modalities alone. We show that in the combined approach, one has additional information, compared with a single modality, and the inverse problem of reconstruction of the optical parameters becomes feasible. © 2016 The Authors. *Mathematical Methods in the Applied Sciences* Published by John Wiley & Sons Ltd.

Solution of any engineering problem starts with a modelling process, which typically involves a choice among different kinds of models. To create a realistic model, one has to think carefully about the modelling process. Particularly in the case of coupled problems when several models are coupled together to represent a given physical phenomenon. This paper presents an approach based on the category theory that allows to describe this modelling process on a more abstract level. Using the advantages of abstract level, one can describe the coupling process in a concise way and introduce certain criteria to check consistency of a coupled model. The main idea of the proposed approach is to introduce a structure in the modelling process, which allows to see how different models interact without a precise look into them. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The main objective of this paper was to study the global stability of the positive solutions and the periodic character of the difference equation

where the parameters *a*, *b*, *c*, *d*, and *e* are positive real numbers and the initial conditions *x*_{−t},*x*_{−t + 1},...,*x*_{−1}, *x*_{0} are positive real numbers where *t* = *m**a**x*{*l*,*k*,*s*}. Some numerical examples will be given to explicate our results. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is concerned with the short time strong solutions for Cauchy problem to a simplified Ericksen–Leslie system of compressible nematic liquid crystals in two dimensions with vacuum as far field density. We establish a blow-up criterion for possible breakdown of such solutions at a finite time, which is analogous to the well-known Serrin's blow-up criterion for the incompressible Navier–Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>It is well known that the damping term will give more smooth effect to obtain global solutions. In this paper, we consider the effect of damping term on the solutions to system of inhomogeneous wave equation with damping term. We can obtain the singularity that will be formed in finite time for some large initial data. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with the mathematical analysis of the electromagnetic wave scattering by an unbounded dielectric medium, which is mounted on a perfectly conducting infinite plane. By introducing a transparent boundary condition on a plane surface confining the medium, the scattering problem is modeled as a boundary value problem of Maxwell's equations. Based on a variational formulation, the problem is shown to have a unique weak solution for a wide class of dielectric permittivity and magnetic permeability by using the generalized Lax–Milgram theorem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study the stability problems and *L*_{2}-gain analysis for switched singular linear systems with jumps. Based on the concept of average impulsive interval, some novel sufficient conditions on the stability and *L*_{2}-gain for switched singular linear systems with jumps are developed. Two examples are given to illustrate the effectiveness of the results. Copyright © 2016 John Wiley & Sons, Ltd.

Real traffic data are very versatile and are hard to explain with the so-called standard fundamental diagram. A simple microscopic model can show that the heterogeneity of traffic results in a reduced mean flow and that the reduction is proportional to the density variance. Standard averaging techniques allow us to evaluate this reduction without having to describe the complex microscopic interactions.

Using a second equation for the variance results in a two-dimensional hyperbolic system that can be put in conservative form. The Riemann problem is completely solved in the case of a parabolic fundamental diagram, and the solutions are compared with the famous second-order Aw–Rascle–Zhang model in a simulation of lane reduction. Adding a diffusion term results in entropy production, and the diffusive model is studied as well. Finally, a numerical scheme is used and converges to the analytical solution. Copyright © 2016 John Wiley & Sons, Ltd.

We consider a version of the stationary phase method in one dimension of A. Erdélyi, allowing the phase to have stationary points of non-integer order and the amplitude to have integrable singularities. After having completed the original proof and improved the error estimate in the case of regular amplitude, we consider a modification of the method by replacing the smooth cut-off function employed in the source by a characteristic function, leading to more precise remainder estimates. We exploit this refinement to study the time-asymptotic behaviour of the solution of the free Schrödinger equation on the line, where the Fourier transform of the initial data is compactly supported and has a singularity. We obtain asymptotic expansions with respect to time in certain space-time cones as well as uniform and optimal estimates in curved regions, which are asymptotically larger than any space-time cone. These results show the influence of the frequency band and of the singularity on the propagation and on the decay of the wave packets. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we are concerned with the system of the non-isentropic compressible Navier–Stokes equations coupled with the Maxwell equations through the Lorentz force in three space dimensions. The global existence of solutions near constant steady states is established, and the time-decay rates of perturbed solutions are obtained. The proof for existence is due to the classical energy method, and the investigation of large-time behavior is based on the linearized analysis of the non-isentropic Navier–Stokes–Poisson equations and the electromagnetic part for the linearized isentropic Navier–Stokes–Maxwell equations. In the meantime, the time-decay rates obtained by Zhang, Li, and Zhu [*J. Differential Equations, 250(2011), 866-891*] for the linearized non-isentropic Navier–Stokes–Poisson equations are improved. Copyright © 2016 John Wiley & Sons, Ltd.

We present some new matrix spectral problems, based on the real special orthogonal Lie algebra , and construct corresponding soliton hierarchies by means of zero curvature equations associated with these spectral problems. With the aid of symbolic computation by Maple, new soliton hierarchies of Kaup–Newell type, Ablowitz–Kaup–Newell–Segur type and Wadati–Konno–Ichikawa type are obtained to illustrate the use of . Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we formulate a (*n* + 3)-dimensional nonlinear virus dynamics model that considers *n*-stages of the infected cells and *n* + 1 distributed time delays. The model incorporates humoral immune response and general nonlinear forms for the incidence rate of infection, the generation and removal rates of the cells and viruses. Under a set of conditions on the general functions, the basic infection reproduction number
and the humoral immune response activation number
are derived. Utilizing Lyapunov functionals and LaSalle's invariance principle, the global asymptotic stability of all steady states of the model are proven. Numerical simulations are carried out to confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is concerned with the existence of positive solutions to a class of nonlocal boundary value problem of the *p*-Kirchhoff type
where
is a bounded smooth domain and *M*,*f*, and *g* are continuous functions. The existence of a positive solution is stated through an iterative method based on mountain pass theorem. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is concerned with the following Schrödinger elliptic system

where the potential *V* is periodic and 0 lies in a gap of the spectrum of −Δ+*V*, *f*(*x*,*t*) and *g*(*x*,*t*) are superlinear in *t* at infinity. By using non-Nehari manifold method developed recently by Tang, we demonstrate the existence of the ground state solutions of Nehari-Pankov type for the above problem with periodic and asymptotically periodic nonlinearity. Copyright © 2016 John Wiley & Sons, Ltd.

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

]]>We establish the existence of local in time semi-strong solutions and global in time strong solutions for the system of equations describing flows of viscous and incompressible asymmetric fluids with variable density in general three-dimensional domains with boundary uniformly of class *C*^{3}. Under suitable assumptions, uniqueness of local semi-strong solutions is also proved. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is concerned with the following systems of Kirchhoff-type equations:

Under more relaxed assumptions on *V*(*x*) and *F*(*x*,*u*,*v*), we first prove the existence of at least two nontrivial solutions for the aforementioned system by using Morse theory in combination with local linking arguments. Then by using the Clark theorem, the existence results of at least 2*k* distinct pairs of solutions are obtained. Some recent results from the literature are extended. Copyright © 2016 John Wiley & Sons, Ltd.

In this study, by using the concepts and results on spherical curves in dual Lorentzian space, we give the criterions for ruled surfaces with non-lightlike ruling to be closed (periodic). Moreover, we obtain the necessary and sufficient conditions to guarantee that a timelike or a spacelike ruled surface is closed (periodic). Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with the existence and uniqueness of time periodic solutions in the whole-space
for a heat equation with nonlinear term. The nonlinear term we considered is of this type, |*u*|^{q − 1}*u* + *f*(*x*,*t*), with
, *N* > 2. We show that there exists a unique time periodic solution when the source term *f* is small. In fact,
is a critical exponent; when
, there is no time periodic solution. Copyright © 2016 John Wiley & Sons, Ltd.

Multi-species Boltzmann equations for gaseous mixtures, with analytic cross sections and under Grad's angular cutoff assumption, are considered under diffusive scaling. In the limit, we formally obtain an explicit expression for the binary diffusion coefficients in the Maxwell–Stefan equations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study a class of generalized differential neoclassical growth model with time-varying delays, new criteria for the existence, and global attractivity of almost positive periodic solutions are established by using the theory of dichotomy and differential inequality techniques, together with constructing a suitable Lyapunov function. Finally, we present an example with numerical simulations to support the theoretical results. The obtained results are essentially new and complement previously known results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we analyze a model presenting formation of microstructure depending on the parameters and the initial data. In particular, we investigate how the presence of stochastic perturbations affects this phenomenon in its asymptotic behavior. Two different sufficient conditions are provided in order to prevent the formation of microstructure: the first one for Stratonovich noise while the second for Itô noise. The main contribution of the paper is that these conditions are independent of the initial values unlike in the deterministic model. Thus, we can interpret our results as some kind of stabilization produced by both types of noise. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we focus on a diffuse interface model named by Hele–Shaw–Cahn–Hilliard system, which describes a two-phase Hele–Shaw flow with matched densities and arbitrary viscosity contrast in a bounded domain. The diffuse interface thickness is measured by *ϵ*, and the mobility coefficient (the diffusional Peclet number) is *ϵ*^{α}. We will prove rigorously that the global weak solutions of the Hele–Shaw–Cahn–Hilliard system converge to a varifold solution of the sharp interface model as *ϵ*0 in the case of 0≤*α* < 1. Copyright © 2016 John Wiley & Sons, Ltd.