We construct bases of polynomials for the spaces of square-integrable harmonic functions that are orthogonal to the monogenic and antimonogenic -valued functions defined in a prolate or oblate spheroid.

]]>By comparing the class ratio deviation and restoring error of first-order accumulation with that of fractional-order accumulation, a gray model for monotonically increasing sequences can obtain optimal simulation accuracy via selecting a proper cumulative order. In this study, a gray model for increasing sequences with nonhomogeneous index trends based on fractional-order accumulation is proposed. To reduce the modeling error caused by the background value and to improve the prediction accuracy of the model, an optimized model using the 3/8 Simpson formula is constructed. Finally, the 2 proposed models are used to predict the total energy consumption in China and the monthly sales of new products in an enterprise. Compared with the GM(1,1) model based on fractional-order accumulation, the proposed model exhibits better simulation and prediction accuracy.

]]>This paper investigates the problem of exponential *H*_{∞} synchronization of discrete-time chaotic neural networks with time delays and stochastic perturbations. First, by using the Lyapunov-Krasovskii (Lyapunov) functional and output feedback controller, we establish the *H*_{∞} performance of exponential synchronization in the mean square of master-slave systems, which is analyzed using a matrix inequality approach. Second, the parameters of a desired output feedback controller can be achieved by solving a linear matrix inequality. Finally, 2 simulated examples are presented to show the effectiveness of the theoretical results.

The purpose of this paper is to establish unique solvability for a certain generalized boundary-value problem for a loaded third-order integro-differential equation with variable coefficients. Moreover, the method of integral equations is applied to obtain an equation related to the Riemann-Liouville operators.

]]>In this work we study the classical Lie symmetries and the consevations laws of a generalized Dullin-Gottwald-Holm equation with arbitrary coefficients.

]]>This paper is devoted to investigate synchronization and antisynchronization of *N*-coupled general fractional-order complex chaotic systems described by a unified mathematical expression with ring connection. By means of the direct design method, the appropriate controllers are designed to transform the fractional-order error dynamical system into a nonlinear system with antisymmetric structure. Thus, by using the recently established result for the Caputo fractional derivative of a quadratic function and a fractional-order extension of the Lyapunov direct method, several stability criteria are derived to ensure the occurrence of synchronization and antisynchronization among *N*-coupled fractional-order complex chaotic systems. Moreover, numerical simulations are performed to illustrate the effectiveness of the proposed design.

The purpose of this paper is to describe the oscillatory properties of second-order Euler-type half-linear differential equations with perturbations in both terms. All but one perturbations in each term are considered to be given by finite sums of periodic continuous functions, while coefficients in the last perturbations are considered to be general continuous functions. Since the periodic behavior of the coefficients enables us to solve the oscillation and non-oscillation of the considered equations, including the so-called critical case, we determine the oscillatory properties of the equations with the last general perturbations. As the main result, we prove that the studied equations are conditionally oscillatory in the considered very general setting. The novelty of our results is illustrated by many examples, and we give concrete new corollaries as well. Note that the obtained results are new even in the case of linear equations.

]]>In this paper, we give new oscillation criteria for forced sublinear impulsive differential equations of the form

where *γ*∈(0,1), under the assumption that associated homogenous linear equation

is nonoscillatory.

]]>We study the periodicity of multipatch dispersal predator-prey system with Holling type-II functional response in this paper. By providing a new method, we overcome the difficulty to get the priori bounds estimation of unknown solutions of operator equation *L**u*=*λ**N**u*. Graph theory with coincidence degree theory is used, and a sufficient criterion for the periodicity of the system is obtained. The criterion presented in this paper is closely related with topological structure of dispersal network and can be verified easily. Finally, a numerical example is also provided to verify the effectiveness of theoretical results.

The Heisenberg ferromagnetic spin chain equation is investigated. By applying the improved F-expansion method (Exp-function method) and the Jacobi elliptic method, respectively, a series of exact solutions is constructed. The parametric conditions of the existence for the solutions are presented. These solutions comprise periodic wave solutions, doubly periodic wave solutions, and dark and bright soliton solutions, which are expressed in several different function forms, namely, Jacobi elliptic function, trigonometric function, hyperbolic function, and exponential function. The results illustrate that the Exp-function method is a powerful symbolic algorithm to look for new solutions for the nonlinear evolution systems.

]]>This paper considers the 2-species chemotaxis-Stokes system with competitive kinetics

under homogeneous Neumann boundary conditions in a 3-dimensional bounded domain
with smooth boundary. Both chemotaxis-fluid systems and 2-species chemotaxis systems with competitive terms were studied by many mathematicians. However, there have not been rich results on coupled 2-species–fluid systems. Recently, global existence and asymptotic stability in the above problem with (*u*·∇)*u* in the fluid equation were established in the 2-dimensional case. The purpose of this paper is to give results for global existence, boundedness, and stabilization of solutions to the above system in the 3-dimensional case when
is sufficiently small.

The study of discrete model of movement on a single contour, equivalent to *C**A* 184 in terms of Wolfram cellular automata classification, was conducted in the late 1990s and early 2000s. In similar formulations of problems for contour networks, conflicts of movement take place in nodes. These problems have been poorly studied so far. However, they are very interesting for scientific and research applications relating to the flow theory (traffic, development of new materials, energy metabolism in the body). In this paper, results are obtained for the simplest contour network. This network is a pair of contours with common node. The supporter of the system consists of 2 contours with a common point (node). Particles move on their contour in accordance with rule 184 or 240 of Wolfram cellular automata. We consider the system with a common cell (simple node) or common point between 2 cells on each contour (alternating node). We have developed approaches to the study of two-contour system such that these approaches can be used to the study of contour networks with more complex architecture. We have obtained the following criterion of that the system comes to the state of free movement (self-organization). The criterion of self-organization is inequality *ρ*_{1}+*ρ*_{2}≤1/2 in the case of rule CA 184 and simple node and inequality *ρ*_{1}+*ρ*_{2}≤1 in the case of rule CA 240 and alternating node. In the general case, presence of self-organization depends on the initial state of the system. Velocities of particles have been found for different rules of movement. Prospects of future research and possible application are outlined.

Dengue is a vector-borne viral disease increasing dramatically over the past years due to improvement in human mobility. In this work, a multipatch model for dengue transmission dynamics is studied, and by that, the control efforts to minimize the disease spread by host and vector control are investigated. For this model, the basic reproduction number is derived, giving a choice for parameters in the endemic case. The multipatch system models the host movement within the patches, which coupled via a residence-time budgeting matrix *P*. Numerical results confirm that the control mechanism embedded in incidence rates of the disease transmission effectively reduces the spread of the disease.

In this paper, the existence and multiplicity of nontrivial solutions are obtained for nonlinear fractional differential systems with *p*-Laplacian by combining the properties of fractional calculus with critical point theory. Firstly, we present a result that a class of *p*-Laplacian fractional differential systems exists infinitely many solutions under the famous Ambrosetti-Rabinowitz condition. Then, a criterion is given to guarantee that the fractional systems exist at least 1 nontrivial solution without satisfying Ambrosetti-Rabinowitz condition. Our results generalize some existing results in the literature.

In this paper, we consider a generalized Fisher equation with exponential diffusion from the point of view of the theory of symmetry reductions in partial differential equations. The generalized Fisher-type equation arises in the theory of population dynamics. These types of equations have appeared in many fields of study such as in the reaction-diffusion equations, in heat transfer problems, in biology, and in chemical kinetics. By using the symmetry classification, simplified by equivalence transformations, for a special family of Fisher equations, all the reductions are derived from the optimal system of subalgebras and symmetry reductions are used to obtain exact solutions.

]]>This paper describes the solution of group classification problem for heat and mass transfer equations with respect to 3 transport coefficients. Two coefficients depend on temperature and concentration, and the thermal diffusivity coefficient is the function of only one of these state parameters. The forms of the arbitrary elements providing the additional transformations are found. Examples of exact solutions of the governing equations are constructed.

]]>In this paper, we consider an initial boundary value problem for the 3-dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and resistivity coefficients over a bounded smooth domain. Global in time unique strong solution is proved to exist when the *L*^{2} norms of initial vorticity and current density are both suitably small with arbitrary large initial density, and the vacuum of initial density is also allowed.

Finally, we revisit the Navier-Stokes model without electromagnetic effect. We find that this initial boundary problem also admits a unique global strong solution under other conditions. In particular, we prove small kinetic-energy strong solution exists globally in time, which extends the recent result of Huang and Wang.

In this article, we consider the asymptotic behavior of the classical solution to the 3-dimensional Vlasov-Poisson plasma interacting repulsively with *N* point charges. The large time behavior in terms of diameters of its velocity-spatial supports is improved to *O*(*t*^{2/3+ϵ}) for any *ϵ*>0.

We present new exact solutions and reduced differential systems of the Navier-Stokes equations of incompressible viscous fluid flow. We apply the method of semi-invariant manifolds, introduced earlier as a modification of the Lie invariance method. We show that many known solutions of the Navier-Stokes equations are, in fact, semi-invariant and that the reduced differential systems we derive using semi-invariant manifolds generalize previously obtained results that used ad hoc methods. Many of our semi-invariant solutions solve decoupled systems in triangular form that are effectively linear. We also obtain several new reductions of Navier-Stokes to a single nonlinear partial differential equation. In some cases, we can solve reduced systems and generate new analytic solutions of the Navier-Stokes equations or find their approximations, and physical interpretation.

]]>This paper is concerned with the global dynamics of a Holling-Tanner predator-prey model with periodic coefficients. We establish sufficient conditions for the existence of a positive solution and its global asymptotic stability. The stability conditions are first given in average form and afterward as pointwise estimates. In the autonomous case, the previous criteria lead to a known result.

]]>The study of collision-induced breakage phenomenon in the particulate process has much current interest. This is an important process arising in many engineering disciplines. In this work, the existence of continuous solution of the pure collisional breakage model is developed beneath some restrictions on the breakage kernels. Furthermore, the mass conservation and uniqueness of solution are investigated in the absence of “shattering transition.” The underlying theory is based on the compactness result of Arzelà-Ascoli and Banach contraction mapping principle.

]]>In this paper, a collocation spectral numerical algorithm is presented for solving nonlinear systems of fractional partial differential equations subject to different types of conditions. A proposed error analysis investigates the convergence of the mentioned algorithm. Some numerical examples confirm the efficiency and accuracy of the method.

]]>This paper is devotedto prove the existence of one or multiple solutions for a family of nonlinear fourth-order boundary value problems.

We use several fixed point theorems, previously developed by the author and her coauthor, to prove the existence of solutions of some simply supported beam problems. To finish the work, a particular case is studied, and the existence of multiple solutions is proved for 2 different particular nonlinear functions.

An exact knowledge of the mechanical and optical properties of crystals allows not only for theoretical advances, but it is also a useful tool to asses crystal quality in the technological processes of growth and production of advanced crystals like, eg, scintillators. In this paper, we study the elasto-optic behavior of tetragonal crystals to evaluate the photoelastic constants, associated to various states of stress, in terms of the components of the piezo-optic tensor. Moreover, we arrive at a generalization, for tetragonal crystals, of the Brewster law for optically isotropic materials.

]]>The initial-boundary value problem for an equation of ion sound waves in plasma is considered. A theorem on nonextendable solution is proved. The blow-up phenomena are studied. The sufficient blow-up conditions and the blow-up time are analysed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow-up more accurately.

]]>We consider a chemotaxis consumption system with singular sensitivity
, *v*_{t}=*ε*Δ*v*−*u**v* in a bounded domain
with *χ*,*α*,*ε*>0. The global existence of classical solutions is obtained with *n*=1. Moreover, for any global classical solution (*u*,*v*) to the case of *n*,*α*≥1, it is shown that *v* converges to 0 in the *L*^{∞}-norm as *t**∞* with the decay rate established whenever *ε*∈(*ε*_{0},1) with
.

During the 2013-2015 Ebola virus disease outbreak, admission into a health facility depended on the availability of hospital beds and health personnel. The limited number of such important logistics contributed to the escalation of the epidemic. We use a compartmental model to study the dynamics of Ebola virus disease when there is a limited number of beds for patients. We use a non-linear hospitalisation rate and formulate the rate at which the time-dependent number of available beds evolves. The model shows a backward bifurcation. Simulation results show that bed supply in Ebola treatment units contribute to the reduction of the number of individuals infected by Ebola virus. The model fitting results suggest that a timely supply of sufficient beds to Ebola treatment units limits the spread of the disease. Despite the fact that bed supplies to Ebola treatment units are not in themselves a control measure, they contribute to the reduction of the disease spread, by keeping the infectious in one place, during their infectious period.These results have important implications to the management and control of the disease.

]]>In this paper, we propose an improved human T-cell leukemia virus type 1 infection model with mitotic division of actively infected cells and delayed cytotoxic T lymphocyte immune response. By constructing suitable Lyapunov functional and using LaSalle invariance principle, we investigate the global stability of the infection-free equilibrium of the system. Our results show that the time delay can change stability behavior of the infection equilibrium and lead to the existence of Hopf bifurcations. Finally, numerical simulations are conducted to illustrate the applications of the main results.

]]>Adaptive Fourier decomposition (AFD, precisely 1-D AFD or Core-AFD) was originated for the goal of positive frequency representations of signals. It achieved the goal and at the same time offered fast decompositions of signals. There then arose several types of AFDs. The AFD merged with the greedy algorithm idea, and in particular, motivated the so-called pre-orthogonal greedy algorithm (pre-OGA) that was proven to be the most efficient greedy algorithm. The cost of the advantages of the AFD-type decompositions is, however, the high computational complexity due to the involvement of maximal selections of the dictionary parameters. The present paper constructs one novel method to perform the 1-D AFD algorithm. We make use of the FFT algorithm to reduce the algorithm complexity, from the original
to
, where *N* denotes the number of the discretization points on the unit circle and *M* denotes the number of points in [0,1). This greatly enhances the applicability of AFD. Experiments are performed to show the high efficiency of the proposed algorithm.

The purpose of the present paper is to define the GBS (Generalized Boolean Sum) operators associated with the two-dimensional Bernstein-Durrmeyer operators introduced by Zhou 1992 and study its approximation properties. Furthermore, we show the convergence and comparison of convergence with the GBS of the Bernstein-Kantorovich operators proposed by Deshwal et al 2017 by numerical examples and illustrations.

]]>In this paper, we study the existence and concentration behavior of positive solutions for the following Kirchhoff type equation:

where *ɛ* is a positive parameter, *a* and *b* are positive constants, and 3<*p*<5. Let
denotes the ground energy function associated with
,
, where
is regard as a parameter. Suppose that the potential *V*(*x*) decays to zero at infinity like |*x*|^{−α} with 0<*α*≤2, we prove the existence of positive solutions *u*_{ɛ} belonging to
for vanishing or unbounded *K*(*x*) when *ɛ* > 0 small. Furthermore, we show that the solution *u*_{ɛ} concentrates at the minimum points of
as *ɛ*0^{+}.

In this paper, we consider the 2D Boussinesq system with variable kinematic viscosity in the velocity equation and with weak damping effect to instead of the regularity effect for the thermal conductivity. Even if without thermal diffusion in the temperature equation, we establish the global well-posedness for the 2D Boussinesq system with general initial data.

]]>In this paper, we consider a rotating Euler-Bernoulli beam. The beam is made of a viscoelastic material, and it is subject to undesirable vibrations. Under a suitable control torque applied at the motor, we prove the arbitrary stabilization of the system for a large class of relaxation functions by using the multiplier method and some ideas introduced by Tatar (J. Math. Phys. 52:013502, 2011).

]]>In this paper, we consider the uniqueness problems of finite-order meromorphic solutions to Painlevé equation. Our result says that such solutions *w* are uniquely determined by their poles and the zeros of *w*−*e*_{j} (counting multiplicities) for 2 finite complex numbers *e*_{1}≠*e*_{2}. As applications, we derive 2 uniqueness theorems about the Weierstrass *℘* function and Jacobi elliptic function *s**n*, respectively.

In this paper, we consider the Bresse-Cattaneo system with a frictional damping term and prove some optimal decay results for the *L*^{2}-norm of the solution and its higher order derivatives. In fact, we show that there is a completely new stability number *δ* that controls the decay rate of the solution. To prove our results, we use the energy method in the Fourier space to build some very delicate Lyapunov functionals that give the desired results. We also prove the optimality of the results by using the eigenvalues expansion method. In addition, we show that for the absence of the frictional damping term, the solution of our problem does not decay at all. This result improves some early results

In this paper, we introduce (*p*,*q*)-Sturm-Liouville problems and prove that their solutions are orthogonal with respect to a (*p*,*q*)-integral space. We then present some illustrative examples for this kind of problems and obtain the (*p*,*q*)-hypergeometric representation of the polynomial solutions together with their 3-term recurrence relations. We also compute the norm square value of the polynomial solutions and obtain their limiting cases in the sequel.

In this paper, the existence and multiplicity of positive solutions are obtained for a class of Kirchhoff type problems with two singular terms and sign-changing potential by the Nehari method.

]]>This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem for the nonlinear diffusion equation in an *unbounded* domain
(
), written as

which represents the porous media, the fast diffusion equations, etc, where *β* is a single-valued maximal monotone function on
, and *T*>0. In Kurima and Yokota (J Differential Equations 2017; 263:2024-2050 and Adv Math Sci Appl 2017; 26:221-242) existence and uniqueness of solutions for were directly proved under a growth condition for *β* even though the Stefan problem was excluded from examples of . This paper completely removes the growth condition for *β* by confirming Cauchy's criterion for solutions of the following approximate problem _{ε} with approximate parameter *ε*>0:

The purpose of the study is to analyze the time-fractional reaction-diffusion equation with nonlocal boundary condition. The proposed model is used to predict the invasion of tumor and its growth. Further, we establish the existence and uniqueness of a weak solution of the proposed model using the Faedo-Galerkin method and compactness arguments.

]]>We investigate the nature of Allee thresholds and basins of attraction in a predation model with double Allee effect in the prey and a competition behaviour in the predator. From a mathematical perspective, this implies to find and characterise the corresponding basin boundaries in phase space. This is typically a major challenge since the objects that act as boundaries between 2 different basins are invariant manifolds of the system, which may also undergo topological changes at bifurcations. For this goal, we make an extensive use of analytical tools from dynamical systems theory and numerical bifurcation analysis and determine the full bifurcation diagram. Local bifurcations include saddle-node, transcritical and Hopf bifurcations, while global phenomena include homoclinic bifurcations, heteroclinic connections, and heteroclinic cycles. We identify the Allee threshold to be either a limit cycle, a homoclinic orbit, or the stable manifold of an equilibrium. This strategy based on bifurcation and invariant manifold analysis allows us to identify the mathematical mechanisms that produce rearrangements of separatrices in phase space. In this way, we give a full geometrical explanation of how the Allee threshold and basins of attraction undergo critical transitions. This approach is complemented with a study of the dynamics near infinity. In this way, we determine the conditions such that the basins of attraction are bounded or unbounded sets in phase space. All in all, these results allow us to show a complete description of phase portraits, extinction thresholds, and basins of attraction of our model under variation of parameters.

]]>It is easy to write down entire solutions of the Helmholtz equation: Examples are plane waves and Herglotz wavefunctions. We are interested in the far-field behaviour of these solutions motivated by the following question: When is it legitimate to split the far field of such an entire solution into the sum of an incoming spherical wave and an outgoing spherical wave? We review the relevant literature (there are disjoint physical and mathematical threads), and then we answer the question for Herglotz wavefunctions, using a combination of the 2-dimensional method of stationary phase and some explicit examples.

]]>Population aging is rapidly increasing in developing countries; thus, covering medical needs for breast cancer diagnosis and treatment is a priority in Latin America. We describe an approach for integrating differential expression analysis, biological pathway enrichment, in silico transcription-binding sites and network topology, to find key genes that may be used as biomarkers or therapeutic targets. This approach is based on publicly available data from microarrays of the MCF-7 breast cancer cell line treated with estrogen. We found significant estrogen-responsive genes, which were used as nodes to construct networks based on protein-protein interactions reported in the literature. Then, we conducted a topology analysis of the networks, revealing the most-connected nodes, ie, those responsible for maintaining the network structure corresponding to genes with well-acknowledged functions in G1/S cell cycle transition, such as cyclin-dependent kinase 2 (CDK2), which has been proposed as a therapeutic target in classical biochemical studies. In addition, analyses of biological pathway enrichment and in silico transcription-binding sites support the biological meaning and importance of these key genes and help to decide the best target genes. Therefore, we postulate that the integrative bioinformatics approach shown here, unlike the classical bioinformatics approach that only includes differentially expressed genes and enriched biological pathways, can be applied as an approach for finding novel biomarkers and/or therapeutic target genes for nonresponsive treatments.

]]>In this paper, the mathematical modeling and trajectory planning of a 3D rotating manipulator composed of a rotating-prismatic joint and multiple rigid links is considered. Possible trajectories of the end effector of the manipulator—following a sequence of 3D target points under the action of 2 external driving torques and an axial force—are modeled using zenithal gnomic projections and polar piecewise interpolants expressed as polynomial Hermite-type functions. Because of the geometry of the manipulator, the time-dependent generalized coordinates are associated with the spherical coordinates named the radial distance related to the manipulator length, and the polar and azimuthal angles describing the left and right and, respectively, up and down motion of the manipulator. The polar trajectories (left and right motion) of the end effector are generated using a inverse geometric transformation applied to the polar piecewise interpolants that approximate the gnomic projective trajectory of the 3D via-points. The gnomic via-points—located on a projective plane situated on the northern hemisphere—are seen from the manipulator base location, which represents the center of rotation of the extensible manipulator. The related azimuthal trajectory (up and down motion) is generated by polar piecewise interpolants on the azimuthal angles. Smoothness of the polygonal trajectory is obtained through the use of piecewise interpolants with continuous derivatives, while the kinematics and dynamics implementation of the model is well suited to computer implementation (easy calculation of kinematics variables) and simulation. To verify the approach and validate the model, a numerical example—implemented in Matlab—is presented, and the results are discussed.

]]>In this work, the modified Green function technique for the exterior Dirichlet problem in linear thermoelasticity is presented. Expressing the solution of the problem as a double-layer potential of an unknown density, we form the associated boundary integral equation that describes the problem. Exploiting that the discrete spectrum of the irregular values of the associated integral equation is identified with the spectrum of eigenvalues of the corresponding interior homogeneous Neumann problem for the transverse part of the elastic displacement field, we introduce a modification of the fundamental solution of the elastic field. We establish the sufficient conditions that the coefficients of the modification must satisfy to overcome the problem of nonuniqueness for the thermoelastic problem.

]]>The study of contractions of Lie algebras is profusely extended in the last decades. In this paper we study the graded contractions of some lower-dimensional filiform Lie algebras which have not been studied earlier. Particularly, we deal with graded contractions of model filiform Lie algebras of dimension less than or equal to 6 and with the ones of a nonmodel 6-dimensional filiform Lie algebra.

]]>In this paper, we will establish the bounded solutions, periodic solutions, quasiperiodic solutions, almost periodic solutions, and almost automorphic solutions for linearly coupled complex cubic-quintic Ginzburg-Landau equations, under suitable conditions. The main difficulty is the nonlinear terms in the equations that are not Lipschitz-continuity, traditional methods cannot deal with the difficulty in our problem. We overcome this difficulty by the Galerkin approach, energy estimate method, and refined inequality technique.

]]>The presence of a steady-state distribution is an important issue in the modelization of cell populations. In this paper, we analyse, from a numerical point of view, the approach to the stable size distribution for a size-structured balance model with an asymmetric division rate. To this end, we introduce a second-order numerical method on the basis of the integration along the characteristic curves over the natural grid. We validate the interest of the scheme by means of a detailed analysis of convergence.

]]>Let *L* be the *n*-th order linear differential operator *L**y*=*ϕ*_{0}*y*^{(n)}+*ϕ*_{1}*y*^{(n−1)}+⋯+*ϕ*_{n}*y* with variable coefficients. A representation is given for *n* linearly independent solutions of *L**y*=*λ**r**y* as power series in *λ*, generalizing the SPPS (spectral parameter power series) solution that has been previously developed for *n*=2. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a solution system for *λ*=0. It is shown how to obtain such an initializing system working upwards from equations of lower order. The values of the successive derivatives of the power series solutions at the basepoint of integration are given, which provides a technique for numerical solution of *n*-th order initial value problems and spectral problems.

Previous works on the convergence of numerical methods for the Boussinesq problem were conducted, while the optimal *L*^{2}-norm error estimates for the velocity and temperature are still lacked. In this paper, the backward Euler scheme is used to discrete the time terms, standard Galerkin finite element method is adopted to approximate the variables. The MINI element is used to approximate the velocity and pressure, the temperature field is simulated by the linear polynomial. Under some restriction on the time step, we firstly present the optimal *L*^{2} error estimates of approximate solutions. Secondly, two-level method based on Stokes iteration for the Boussinesq problem is developed and the corresponding convergence results are presented. By this method, the original problem is decoupled into two small linear subproblems. Compared with the standard Galerkin method, the two-level method not only keeps good accuracy but also saves a lot of computational cost. Finally, some numerical examples are provided to support the established theoretical analysis.

This paper is devoted to the Cauchy problem for the nonlinear Schrodinger equation with time-dependent fractional damping term. We prove the local existence result, and we study the global existence and blow-up solutions.

]]>An inverse problem of determining a time-dependent source term from the total energy measurement of the system (the over-specified condition) for a space-time fractional diffusion equation is considered. The space-time fractional diffusion equation is obtained from classical diffusion equation by replacing time derivative with fractional-order time derivative and Sturm-Liouville operator by fractional-order Sturm-Liouville operator. The existence and uniqueness results are proved by using eigenfunction expansion method. Several special cases are discussed, and particular examples are provided.

]]>We consider a laminar boundary-layer flow of a viscous and incompressible fluid past a moving wedge in which the wedge is moving either in the direction of the mainstream flow or opposite to it. The mainstream flows outside the boundary layer are approximated by a power of the distance from the leading boundary layer. The variable pressure gradient is imposed on the boundary layer so that the system admits similarity solutions. The model is described using 3-dimensional boundary-layer equations that contains 2 physical parameters: pressure gradient (*β*) and shear-to-strain-rate ratio parameter (*α*). Two methods are used: a linear asymptotic analysis in the neighborhood of the edge of the boundary layer and the Keller-box numerical method for the full nonlinear system. The results show that the flow field is divided into near-field region (mainly dominated by viscous forces) and far-field region (mainstream flows); the velocity profiles form through an interaction between 2 regions. Also, all simulations show that the subsequent dynamics involving overshoot and undershoot of the solutions for varying parameter characterizing 3-dimensional flows. The pressure gradient (favorable) has a tendency of decreasing the boundary-layer thickness in which the velocity profiles are benign. The wall shear stresses increase unboundedly for increasing *α* when the wedge is moving in the *x*-direction, while the case is different when it is moving in the *y*-direction. Further, both analysis show that 3-dimensional boundary-layer solutions exist in the range −1<*α*<*∞*. These are some interesting results linked to an important class of boundary-layer flows.

This paper concerns itself with the development of an a priori error analysis of an eddy current problem when applying the well-known hybridizable discontinuous Galerkin (HDG) method. Up to the authors' knowledge, this kind of theoretical result has not been proved for this kind of problems. We consider nontrivial domains and heterogeneous media which contain conductor and insulating materials. When dealing with these domains, it is necessary to impose the divergence-free condition explicitly in the insulator, what is done by means of a suitable Lagrange multiplier in that material. In the end, we deduce an equivalent HDG formulation that includes as unknowns the tangential and normal trace of a vector field. This represents a reduction in the degrees of freedom when compares with the standard DG methods. For this scheme, we conduct a consistency and local conservative analysis as well as its unique solvability. After that, we introduce suitable projection operators that help us to deduce the expected a priori error estimate, which provides estimated rates of convergence when additional regularity on the exact solution is assumed.

]]>In this work we consider the higher order fractional differential equation with derivative defined in the sense of Katugampola. We present some equivalent integral form of the considered boundary value problem and using properties of an appropriate Green function and prove fractional counterpart of the Lyapunov inequality.

]]>In the present paper, exact solutions of fractional nonlinear Schrödinger equations have been derived by using two methods: Lie group analysis and invariant subspace method via Riemann-Liouvill derivative. In the sense of Lie point symmetry analysis method, all of the symmetries of the Schrödinger equations are obtained, and these operators are applied to find corresponding solutions. In one case, we show that Schrödinger equation can be reduced to an equation that is related to the Erdelyi-Kober functional derivative. The invariant subspace method for constructing exact solutions is presented for considered equations.

]]>The purpose of this work is to investigate the problem of solutions to the time-fractional Navier-Stokes equations with Caputo derivative operators. We obtain the existence and uniqueness of the solutions to each approximate equation, as well as the convergence of the approximate solutions. Furthermore, we present some convergence results for the Faedo-Galerkin approximations of the given problems.

]]>This paper focuses on the construction of periodic solutions of nonlinear beam equations on the *d*-dimensional tori. For a large set of frequencies, we demonstrate that an equivalent form of the nonlinear equations can be obtained by a para-differential conjugation. Given the nonresonant conditions on each finite dimensional subspaces, it is shown that the periodic solutions can be constructed for the block diagonal equation by a classical iteration scheme.

This paper deals with the energy decay estimates and infinite blow-up phenomena for a strongly damped semilinear wave equation with logarithmic nonlinear source term under null Dirichlet boundary condition. By constructing a new family of potential wells, together with logarithmic Sobolev inequality and perturbation energy technique, we establish sufficient conditions to guarantee the solution exists globally or occurs infinite blow-up and derive the polynomial or exponential energy decay estimates under some appropriate conditions.

]]>Topology is the most important branch of modern mathematics, which plays an important role in applications. In this paper, we use the concept of the topology, based on the concept of multiset to solve an important problems in life (DNA and RNA mutation) to detect diseases and help biologists in the treatment of diseases. Also, we introduce a new theory that explains if there is an existence of a mutation or not, and we have a set of metric functions through which we examine the congruence and similarity and dissimilarity between “types,” which may be a strings of bits, vectors, DNA or RNA sequences, …, etc. Finally, we will introduce a theory that which can be used to know the existence and place of mutation.

]]>A general nonlinear model of illiquid markets with feedback effects is considered. This equation with 2 free functional parameters contains as partial cases the classical Black-Scholes equation, Schönbucher-Wilmott equation, and Sircar-Papanicolaou equation of option pricing. We obtain here the complete group classification of the equation, and for every parameters specification we obtain the principal Lie algebra and its optimal system of 1-dimensional subalgerbras. For every such subalgebra we calculate the invariant submodel and invariant solution, when it is possible. Thus, the series of invariant submodels and invariant solutions are derived for the considered nonlinear model.

]]>No abstract is available for this article.

]]>We propose and investigate a delayed model that studies the relationship between HIV and the immune system during the natural course of infection and in the context of antiviral treatment regimes. Sufficient criteria for local asymptotic stability of the infected and viral free equilibria are given. An optimal control problem with time delays both in state variables (incubation delay) and control (pharmacological delay) is then formulated and analyzed, where the objective consists to find the optimal treatment strategy that maximizes the number of uninfected *C**D*4^{ + } T cells as well as cytotoxic T lymphocyte immune response cells, keeping the drug therapy as low as possible. Copyright © 2016 John Wiley & Sons, Ltd.

Graph theory is a fundamental tool in the study of economic issues, and input–output tables are one of the main examples. We use the interpretation of the labour market through networks to obtain a better understanding on its overall functioning. One benefit of the network perspective is that a large body of mathematics exists to help analyze many forms of networks models. If an economic system has obtained a suitable model, then it becomes possible to utilize relevant mathematical tools, such as graph theory, to better understand the way the labour market works. This interpretation allows us to employ the concepts of coverage, invariance, orbit and the structural functions supply–demand and competition and interpret them from the point of view of circular flow. In this paper, we aim to interpret the labour market through networks that are represented by graphs and where characteristic concepts of chaos theory such as cover, invariance and orbits interact with the concept circular flow. Finally, an example of this approach to labour markets is described, and some conclusions are drawn. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we propose an eco-epidemiological predator–prey model, modeling the spread of infectious keratoconjunctivitis among domestic and wild ungulates, during the summer season, when they intermigle in high mountain pastures. The disease can be treated in the domestic animals, but for the wild herbivores, it leads to blindness, with consequent death. The model shows that the disease can lead infected herbivores or their predators to extinction, even if it does not affect the latter. Boundedness of solutions and equilibria feasibility are obtained. Stability around the different equilibrium points is analyzed through eigenvalues and the Routh–Hurwitz criterion. Simulations are carried out to support the theoretical results. Sensitivity with respect of some parameters is investigated. The prey vaccination as control measure is introduced and simulated, although at present, the vaccine is not yet available, but just being developed. It would then possibly eradicate the infection in the domestic animals, which are considered a disease reservoir. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this study, the generation of smooth trajectories of the end effector of a rotating extensible manipulator arm is considered. Possible trajectories are modelled using Cartesian and polar piecewise cubic interpolants expressed as polynomial Hermite-type functions. The use of polar piecewise cubic interpolants devises continuous first-order and – in some cases – second-order derivatives and allows easy calculation of kinematics variables such as velocity and acceleration. Moreover, the manipulator equations of motion can be easily handled, and the constrained trajectory of the non-active end of the manipulator derived directly from the position of the end-effector. To verify the proposed approach, numerical simulations are conducted for two different configurations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper describes the procedure of extracting information about the dynamics of highway traffic speed. The wavelet shrinkage is used to diminish the effect of the noise. Afterwards, the dynamical properties of the system are estimated through the 0–1 test for chaos, Lyapunov exponents and the notion of Shannon entropy. The results indicate the strong chaotic dynamics in the traffic speed data. In addition to that, the predictability of the system is related to the values of the maximal Lyapunov exponent and Shannon entropy. The higher those values are, the worse the predictability of the system is. Furthermore, it is shown that Shannon entropy can be used to detect changes in dynamics on different time scales. 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 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.

]]>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.

]]>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.

]]>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 © 2017 John Wiley & Sons, Ltd.

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

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

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

]]>This paper investigates the asymptotic behavior of a delayed Nicholson-type system involving patch structure and nonlinear density-dependent mortality terms. Based on the fluctuation lemma and some differential inequality techniques, delay-dependent criteria are deduced for the global attractivity of the addressed system, which indicate that small delays have no effect on the attractivity of the system. Meanwhile, an illustrative numerical example is given to illustrate the feasibility of the theoretical results.

]]>In this paper, we study a class of fractional nonlinear impulsive switched coupled evolution equations. Existence and uniqueness of solutions as well as Hyers-Ulam stability results are presented. An example is provided for the verification of our results.

]]>We propose and study a strongly coupled PDE-ODE-ODE system modeling cancer cell invasion through a tissue network under the go-or-grow hypothesis asserting that cancer cells can either move or proliferate. Hence, our setting features 2 interacting cell populations with their mutual transitions and involves tissue-dependent degenerate diffusion and haptotaxis for the moving subpopulation. The proliferating cells and the tissue evolution are characterized by way of ODEs for the respective densities. We prove the global existence of weak solutions and illustrate the model behaviour by numerical simulations in a 2-dimensional setting. The numerical results recover qualitatively the infiltrative patterns observed histologically and moreover allow to establish a qualitative relationship between the structure of the tissue and the expansion of the tumour, thereby paying heed to its heterogeneity.

]]>In this paper, we study the well posed-ness of Cauchy problem for a class of hyperbolic equation with characteristic degeneration on the initial hyperplane. By a delicate analysis of two integral operators in terms of Bessel functions, we give the uniform weighted estimates of solutions to the linear problem with a parameter *m*∈(0,1) and establish local and global existences of solution to the semilinear equation. Meanwhile, we derive the existence of solutions to semilinear generalized Euler-Poisson-Darboux equation with a negative parameter *α*∈(−1,0).

In this paper, we investigate the generalized Stieltjes-Wigert and *q*-Laguerre polynomials. We derive the second- and the third-order nonlinear difference equations for the subleading coefficients of these polynomials and use them to find a few terms of the formal expansions in powers of *q*^{n/2}. We also show how the recurrence coefficients in the three-term recurrence relation for these polynomials can be computed efficiently by using the nonlinear difference equations for the subleading coefficient. Moreover, we obtain systems of difference equations with one of the equations being *q*-discrete Painlevé III or V equations and analyze them by a singularity confinement. We also discuss certain generalized weights.

The detection of image edges is of great importance in image processing. One of the efficient implementations for this image recovery problem is based on the identification of sharp jumps of the gray function of the image. Mathematically, this problem can be modeled by the numerical differentiation of the gray function with 2 variables. For this ill-posed problem with nonsmooth solution, we investigate the regularization schemes with total variation and *L*^{1} penalty term, respectively. We prove that the regularizing parameter under the Tikhonov regularization framework can be uniquely chosen in terms of the Morozov's discrepancy principle and then establish the convergence rate of the regularizing solutions in terms of the Bregman distance. The discrete schemes are performed by the lagged diffusivity fixed point iteration, with numerical implementations showing the validity of the proposed scheme.

In this paper, we study the spatial behavior of solutions to the equations obtained by taking formal Taylor approximations to the heat conduction dual-phase-lag and 3-phase-lag theories, reflecting Saint-Venant's principle. In a recent paper, 2 families of cases for high-order partial differential equations were studied. Here, we investigate a third family of cases, which corresponds to the fact that a certain condition on the time derivative must be satisfied. We also study the spatial behavior of a thermoelastic problem. We obtain a Phragmén-Lindelöf alternative for the solutions in both cases. The main tool to handle these problems is the use of an exponentially weighted Poincaré inequality.

]]>Singularity analysis of a parallel manipulator (PM) is important in its trajectory planning and workspace design. This paper presents a new method based on geometric algebra (GA) for the singularity analysis of the 3/6-SPS Gough-Stewart PM, where S denotes a spherical joint and P a prismatic pair. The 6 line vectors associated with the SPS limb are expressed using GA. An analytic singular polynomial is then derived as the coefficient of the outer product of all 6 line vectors. This polynomial provides an overall description of the singularity of the 3/6-SPS Gough-Stewart PM. Position-singularity loci and orientation-singularity loci can be drawn based on this polynomial, the latter of which have seldom been addressed. It is also shown that the proposed GA-based method is geometrically intuitive and computationally efficient.

]]>In this paper, we obtain a Cauchy-type integral representation to solve certain Beltrami equations and apply it for research of the Riemann boundary value problem for that equations on nonrectifiable contours. While more or less studied for piecewise smooth contours, for the case of nonrectifiable curve, this is a pioneer result. There is only one relatively simple type of Beltrami equation that is solved under such condition (see below). Our next goal is to achieve solutions for more types of the Beltrami equations on nonrectifiable curves.

]]>In this article, we introduce the local fractional integral iterative method and the local fractional new iterative method for solving the local fractional differential equations. Also, we perform a comparison between the results obtained by these 2 local fractional methods with the results obtained by some other local fractional methods. The obtained results illustrate the significant features of the 2 methods that are both very effective and straightforward for solving the differential equations with local fractional derivative compared with the other local fractional methods.

]]>