We consider the integration of the special second-order initial value problem of the form . A recently introduced family of 7 stages, eighth-order methods, sharing constant coefficients, is used as base. This family is properly modified to derive phase fitted and zero dissipative methods (ie, trigonometric fitted) that are best suited for integrating oscillatory problems. Numerical tests over a set of problems shows enhanced performance when the purely linear part of the problems is rather large in comparison with the rest of nonlinear parts. An appendix implementing a MATLAB listing with the coefficients of the new method is also given.

]]>The aim of this paper is to establish a global existence result for a nonlinear reaction diffusion system with fractional Laplacians of different orders and a balance law. Our method of proof is based on a duality argument and a recent maximal regularity result due to Zhang.

]]>The qualifier “ephemeral” was proposed for continuous models of bodies, such as gases, for which the generally tacit axiom of permanence of material elements fails to apply. Consequently, to their scrutiny, a Eulerian (local) approach is mandatory, such as one adopted, eg, in molecular dynamics. Within the scheme of ephemeral continua, we discuss here 3 essential subclasses of bodies: (1) those undergoing energy-preserving processes (in this sense hyperelastic), (2) hypoelastic bodies inspired by a type proposed by Truesdell, and (3) a number of minor ones. We re-examine the essential issues of the general format focusing on the proposal of appropriate concepts of strains and strainings.

]]>In this paper, we construct a space of boundary values for minimal symmetric 1D Hamiltonian operator with defect index (1,1) (in limit-point case at *a*(*b*) and limit-circle case at *b*(*a*)) acting in the Hilbert space
In terms of boundary conditions at *a* and *b*, all maximal dissipative, accumulative, and self-adjoint extensions of the symmetric operator are given. Two classes of dissipative operators are studied. They are called “dissipative at *a*” and “dissipative at *b*.” For 2 cases, a self-adjoint dilation of dissipative operator and its incoming and outgoing spectral representations are constructed. These constructions allow us to establish the scattering matrix of dilation and a functional model of the dissipative operator. Further, we define the characteristic function of the dissipative operators in terms of the Weyl-Titchmarsh function of the corresponding self-adjoint operator. Finally, we prove theorems on completeness of the system of root vectors of the dissipative operators.

This paper studies the effects of the time change on the frequencies of specific terms connected to the document field in a given period. These specific terms are the field association (FA) terms. The paper also suggests a new method for automatic evaluation of the stabilization classes of FA terms to improve the precision of decision tree. The stabilization classes point out the popularity of list of FA terms depending on time change. Moreover, the suggested method manipulates the problem of the scattering of data numbers among classes to improve the performance of decision tree precision. The presented method is evaluated through conducting experiments by simulating the result of 1245 files, which are equivalent to 4.15 MB. The F-measure for increment, fairly steady, and decrement classes achieves %90.4, %99.3, and %38.6, sequentially.

]]>This paper is devoted to discuss a multidimensional backward heat conduction problem for time-fractional diffusion equation with inhomogeneous source. This problem is ill-posed. We use quasi-reversibility regularization method to solve this inverse problem. Moreover, the convergence estimates between regularization solution and the exact solution are obtained under the a priori and the a posteriori choice rules. Finally, the numerical examples for one-dimensional and two-dimensional cases are presented to show that our method is feasible and effective.

]]>In this paper, the problem of a nonlinear beam equation involving an integral term of the deformation energy, which is unknown before the solution, under different boundary conditions with simply supported, 2-end fixed, and cantilevered is investigated. We transform the governing equation into an integral equation and then solve it by using the sinusoidal functions, which are chosen both as the test functions and the bases of numerical solution. Because of the orthogonality of the sinusoidal functions, we can find the expansion coefficients of the numerical solution that are given in closed form by using the Drazin inversion formula. Furthermore, we introduce the concept of fourth-order and fifth-order boundary functions in the solution bases, which can greatly raise the accuracy over 4 orders than that using the partial boundary functions. The iterative algorithms converge very fast to find the highly accurate numerical solutions of the nonlinear beam equation, which are confirmed by 6 numerical examples.

]]>In this paper, we give a simple proof for the convergence of the deterministic particle swarm optimization algorithm under the weak chaotic assumption and remark that the weak chaotic assumption does not relax the stagnation assumption in essence. Under the spectral radius assumption, we propose a convergence criterion for the deterministic particle swarm optimization algorithm in terms of the personal best and neighborhood best position of the particle that incorporates the stagnation assumption or the weak chaotic assumption as a special case.

]]>In this paper, we study the zero-flux chemotaxis-system

where Ω is a bounded and smooth domain of
, *n*≥1, and where
, *k*,*μ*>0 and *α*≤1. For any *v*≥0, the chemotactic sensitivity function is assumed to behave as the prototype *χ*(*v*)=*χ*_{0}/(1+*a**v*)^{2}, with *a*≥0 and *χ*_{0}>0. We prove that for any nonnegative and sufficiently regular initial data *u*(*x*,0), the corresponding initial-boundary value problem admits a unique global bounded classical solution if *α*<1; indeed, for *α*=1, the same conclusion is obtained provided *μ* is large enough. Finally, we illustrate the range of dynamics present within the chemotaxis system in 1, 2, and 3 dimensions by means of numerical simulations.

A partial inverse problem for an integro-differential Sturm-Liouville operator on a star-shaped graph is studied. We suppose that the convolution kernels are known on all the edges of the graph except one and recover the kernel on the remaining edge from a part of the spectrum. We prove the uniqueness theorem for this problem and develop a constructive algorithm for its solution, based on the reduction of the inverse problem on the graph to the inverse problem on the interval by using the Riesz basis property of the special system of functions.

]]>This paper deals with the behavior of positive solutions to a nonautonomous reaction-diffusion system with homogeneous Neumann boundary conditions, which describes a two-species predator-prey system in which there is an infectious disease in prey. The sufficient condition on the permanence of the prey and the predator is established by combining the comparison principle with the results related to the corresponding ODE system. Some sufficient conditions for the spreading and vanishing of the disease are obtained. The global attractivity is also discussed by constructing a Lyapunov functional. Our results show that the disease is spreading if the transmission rate is suitably large, while if the transmission rate is small, the disease must be vanishing.

]]>In this paper, we show that the set of quaternionic polynomials is dense in the Bergman spaces of slice regular functions in the unit ball, both of the first and of the second kind. Several proofs are presented, including constructive methods based on the Taylor expansion and on the convolution polynomials. In the last case, quantitative estimates in terms of higher-order moduli of smoothness and of best approximation quantity are obtained.

]]>We are concerned with a family of dissipative active scalar equation on
. By using similar methods from the previous paper of Y. Giga et al. (see Introduction below), we construct a unique real, spatially almost periodic mild solution *θ* of satisfying . In this paper, we consider some countable sum-closed frequency sets (see Remark ). We show that the property of the solution is rather different from Chae et al and obtain that
with some initial data *θ*_{0} for all *t*≥0,
and 0≤*α*≤*ω*, where *ω* is a fixed constant. Furthermore, arranging the elements of a countable sum-closed frequency set *F*_{δ} as in Remark , we have for any 0≤*α*≤*ω* that
belongs to
, where *F*_{δ} is defined in or .

In this paper, an efficient numerical technique is applied to provide the approximate solution of nonlinear stochastic Itô-Volterra integral equations driven by fractional Brownian motion with Hurst parameter . The proposed method is based on the operational matrices of modification of hat functions (MHFs) and the collocation method. In this approach, by approximating functions that appear in the integral equation by MHFs and using Newton's-Cotes points, nonlinear integral equation is transformed to nonlinear system of algebraic equations. This nonlinear system is solved by using Newton's numerical method, and the approximate solution of integral equation is achieved. Some theorems related to error estimate and convergence analysis of the suggested scheme are also established. Finally, 2 illustrative examples are included to confirm applicability, efficiency, and accuracy of the proposed method. It should be noted that this scheme can be used to solve other appropriate problems, but some modifications are required.

]]>The article presents a new general solution to a loaded differential equation and describes its properties. Solving a linear boundary value problem for loaded differential equation is reduced to the solving a system of linear algebraic equations with respect to the arbitrary vectors of general solution introduced. The system's coefficients and right sides are computed by solving the Cauchy problems for ordinary differential equations. Algorithms of constructing a new general solution and solving a linear boundary value problem for loaded differential equation are offered. Linear boundary value problem for the Fredholm integro-differential equation is approximated by the linear boundary value problem for loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are given. The paper proposes numerical and approximate methods of solving a linear boundary value problem for the Fredholm integro-differential equation and examines their convergence, stability, and accuracy.

]]>A Hamilton-Poisson system is an approach for the motion of a spacecraft around an asteroid or for the motion of an underwater vehicle. We construct a coordinate chart on the symplectic leaf which contains a specific generic equilibrium point and we establish stability conditions for this equilibrium point.

]]>In this paper, the Cauchy problem for the 3D diffusion approximation model in radiation hydrodynamics is considered. By using the embedding theorem and interpolation technique, we establish the global well-posedness of strong solutions in *H*^{2}.

In this paper, first of all, we consider a generalized game in choice form with 2 constraints and its corresponding equilibrium in choice. We assert new conditions under which the equilibrium in choice exists. As a consequence, we establish the existence of the equilibrium for generalized abstract economies. Then, we apply the obtained theorems to prove the existence of solutions for systems of quasi-equilibrium problems. We do this by considering new hypotheses for the properties of the involved correspondences. This approach leads us to results which differ a lot from the ones existing in literature.

]]>Foreground segmentation is a critical early step in most human-computer interaction applications notably in action and gesture recognition domain. In this paper, an approach to model background which based on luminance-invariant color with an adaptive Gaussian mixture is proposed to discriminate foreground object from their background in complex scene. Firstly, the background model is learned based on the spectral properties of shadows and scene activity. Secondly, the shadow with the hypotheses on color invariance is adaptively set up and updated. Finally, the log-likelihood measurement is to conduct the adaptation. Our experiments are performed on a wide range of practical applications of gesture and action recognition videos. Additionally, the proposed approach is efficient and more robust than premature state-of-the-art with no sacrificing real-time performance.

]]>This paper is devoted to the study of the differential systems in arbitrary Banach spaces that are obtained by mixing nonlinear evolutionary equations and generalized quasi-hemivariational inequalities (EEQHVI). We start by showing that the solution set of the quasi-hemivariational inequality associated to problem EEQHVI is nonempty, closed, and convex. Furthermore, we establish upper semicontinuity and measurability properties for this solution set. Then, based on them, we prove the existence of solutions for problem EEQHVI and the compactness of the set of corresponding trajectories of EEQHVI. These statements extend previous results in several directions, for instance, by dropping the boundedness requirement for the set of constraints and substantially relaxing monotonicity hypotheses.

]]>This paper studies the output tracking problem of Boolean control networks (BCNs) with impulsive effects via the algebraic state-space representation approach. The dynamics of BCNs with impulsive effects is converted to an algebraic form. Based on the algebraic form, some necessary and sufficient conditions are presented for the feedback output tracking control of BCNs with impulsive effects. These conditions contain constant reference signal case and time-varying reference signal case. The study of an illustrative example shows that the obtained new results are effective.

]]>In this article, we are interested by a system of heat equations with initial condition and zero Dirichlet boundary conditions. We prove a finite-time blow-up result for a large class of solutions with positive initial energy.

]]>The inverse scattering transform for the derivative nonlinear Schrödinger-type equation is studied via the Riemann-Hilbert approach. In the direct scattering process, the spectral analysis of the Lax pair is performed, from which a Riemann-Hilbert problem is established for the derivative nonlinear Schrödinger-type equation. In the inverse scattering process, *N*-soliton solutions of the derivative nonlinear Schrödinger-type equation are obtained by solving Riemann-Hilbert problems corresponding to the reflectionless cases. Moreover, the dynamics of the exact solutions are discussed.

We consider the Cauchy problem on a nonlinear conversation law with large initial data. By Green's function methods, energy methods, Fourier analysis, and frequency decomposition, we obtain the global existence and the optimal time-decay estimate of solutions.

]]>In this paper, we are concerned with a rumor propagation model with L vy noise. We first prove that there exists a positive global solution. Then, the asymptotic behaviors around the rumor-free equilibrium and rumor-epidemic equilibrium are obtained. Lastly, simulations verify our results.

]]>In this paper, we consider a Kudryashov-Sinelshchikov equation that describes pressure waves in a mixture of a liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer between liquid and gas bubbles. We show that this equation is rich in conservation laws. These conservation laws have been found by using the direct method of the multipliers. We apply the Lie group method to derive the symmetries of this equation. Then, by using the optimal system of 1-dimensional subalgebras we reduce the equation to ordinary differential equations. Finally, some exact wave solutions are obtained by applying the simplest equation method.

]]>The initial boundary value problem for a class of scalar nonautonomous conservation laws in 1 space dimension is proved to be well posed and stable with respect to variations in the flux. Targeting applications to traffic, the regularity assumptions on the flow are extended to a merely dependence on time. These results ensure, for instance, the well-posedness of a class of vehicular traffic models with time-dependent speed limits. A traffic management problem is then shown to admit an optimal solution.

]]>In this study, modelling and identification of prestress state in functionally graded plate within the framework of the Timoshenko theory are discussed. With the help of variational principles, statements of boundary problems for stationary vibration of inhomogeneous prestressed plates have been derived taking into account various factors of prestress state. The comparative analysis of classical and nonclassical models has been conducted. The effect of the prestress state factors on the solution characteristics has been estimated. New approaches to solving the inverse problems on a reconstruction of inhomogeneous prestress functions in a functionally graded plate have been proposed on the basis of derivation of reciprocity relations and iterative regularization. The results of numerical reconstruction experiments are presented; practical recommendations on a selection of frequency range for the purpose of getting the highest reconstruction accuracy are given.

]]>This paper is devoted to the study of the blow-up phenomena of following nonlinear reaction diffusion equations with Robin boundary conditions:

Here,
is a bounded convex domain with smooth boundary. With the aid of a differential inequality technique and maximum principles, we establish a blow-up or non–blow-up criterion under some appropriate assumptions on the functions *f*,*g*,*ρ*,*k*, and *u*_{0}. Moreover, we dedicate an upper bound and a lower bound for the blow-up time when blowup occurs.

We prove the controllability of the constant target to heat equations with the homogenous Neumann boundary condition via multiplicative controls. Our results show that the temperature of the surrounding medium plays a crucial role in the controllability of the heat transfer system.

]]>Population genetics is a scientific discipline that has extensively benefitted from mathematical modelling; since the Hardy-Weinberg law (1908) to date, many mathematical models have been designed to describe the genotype frequencies evolution in a population. Existing models differ in adopted hypothesis on evolutionary forces (such as, for example, mutation, selection, and migration) acting in the population. Mathematical analysis of population genetics models help to understand if the genetic population admits an equilibrium, ie, genotype frequencies that will not change over time. Nevertheless, the existence of an equilibrium is only an aspect of a more complex issue concerning the conditions that would allow or prevent populations to reach the equilibrium. This latter matter, much more complex, has been only partially investigated in population genetics studies. We here propose a new mathematical model to analyse the genotype frequencies distribution in a population over time and under two major evolutionary forces, namely, mutation and selection; the model allows for both infinite and finite populations. In this paper, we present our model and we analyse its convergence properties to the equilibrium genotype frequency; we also derive conditions allowing convergence. Moreover, we show that our model is a generalisation of the Hardy-Weinberg law and of subsequent models that allow for selection or mutation. Some examples of applications are reported at the end of the paper, and the code that simulates our model is available online at https://www.ding.unisannio.it/persone/docenti/del-vecchio for free use and testing.

]]>In this paper, we are concerned with optimal decay rates for higher-order spatial derivatives of classical solution to the compressible Navier-Stokes-Maxwell equations in three-dimensional whole space. If the initial perturbation is small in -norm, we apply the Fourier splitting method to establish optimal decay rates for the second-order spatial derivatives of a solution. As a by-product, the rate of classical solution converging to the constant equilibrium state in -norm is .

]]>Under the displacement and stress satisfying Riemann boundary value condition, the decoupled quasistatic linear thermoelasticity system is discussed on bounded simply connected domain. The quasistatic equilibrium equation is solved by using Riemann boundary value problem theory. Also decoupled temperature equation is studied by applying the contractive mapping principle. Finally, existence and analyticity of the solution are proved.

]]>This paper is devoted to the analysis of a linearized theta-Galerkin finite element method for the time-dependent coupled systems resulting from microsensor thermistor problems. Hereby, we focus on time discretization based on *θ*-time stepping scheme with
including the standard Crank-Nicolson (
) and the shifted Crank-Nicolson (
, where *δ* is the time-step) schemes. The semidiscrete formulation in space is presented and optimal error bounds in *L*^{2}-norm and the energy norm are established. For the fully discrete system, the optimal error estimates are derived for the standard Crank-Nicolson, the shifted Crank-Nicolson, and the general case where
with *k*=0,1 . Finally, numerical simulations that validate the theoretical findings are exhibited.

To protect fishery populations on the verge of extinction and sustain the biodiversity of the marine ecosystem, marine protected areas (MPA) are established to provide a refuge for fishery resource. However, the influence of current harvesting policies on the MPA is still unclear, and precise information of the biological parameters has yet to be conducted. In this paper, we consider a bioeconomic Gompertz population model with interval-value biological parameters in a 2-patch environment: a free fishing zone (open-access) and a protected zone (MPA) where fishing is strictly prohibited. First, the existence of the equilibrium is proved, and by virtue of Bendixson-dulac Theorem, the global stability of the nontrivial steady state is obtained. Then, the optimal harvesting policy is established by using Pontryagin's maximum principle. Finally, the results are illustrated with the help of some numerical examples. Our results show that the current harvesting policy is advantageous to the protection efficiency of an MPA on local fish populations.

]]>We study a class of microscopic models corresponding to the standard macroscopic logistic equation. The models at microscale refer to a number of interacting individuals and are in terms of linear evolution equations related to Markov jump processes. The asymptotic large time behavior for the microscopic models is obtained. Moreover, it is shown that any, even nonfactorized, initial probability density tends in the evolution to a factorized equilibrium probability density.

]]>This work addresses the study of the *L*^{p}-boundedness and compactness of abstract linear and nonlinear fractional integro-differential equations. The analysis is performed for the whole range of values of *p*, ie,
. In addition, theoretical results are complemented with illustrating particular cases of systems modeled by fractional evolution equations as heat conduction problems and problems arising in the theory of viscoelastic materials.

With the rapid growth of the amount of information stored on networks such as the internet, it is more difficult for information seekers to retrieve relevant information. This paper illustrates the design and improvement of a near neighborhood approach of information retrieval system to facilitate domain specific search. In exacting, a novel model depending on the notion of neighborhood system designed to rank documents according the searchers specific granularity requirements. The initial experiments confirm that our approach outperforms a classical vector-based information retrieval system. Our research work opens the door to the design and development of the next generation of internet search engines to alleviate the problem of information overload using more topological concepts.

]]>Fragmentation-coagulation processes, in which aggregates can break up or get together, often occur together with decay processes in which the components can be removed from the aggregates by a chemical reaction, evaporation, dissolution, or death. In this paper, we consider the discrete decay-fragmentation equation and prove the existence and uniqueness of physically meaningful solutions to this equation using the theory of semigroups of operators. In particular, we find conditions under which the solution semigroup is analytic, compact, and has the asynchronous exponential growth property. The theoretical analysis is illustrated by a number of numerical simulations.

]]>In this paper, we investigate the well-posedness and stability of mild solutions for a class of neutral impulsive stochastic integro-differential equations in a real separable Hilbert space. By the inequality technique combined with theory of resolvent operator, some sufficient conditions are established for the concerned problems. The obtained conclusions are completely new, which generalize and improve some existing results. An example is given to illustrate the effectiveness of our results.

]]>Lie group classification for 2 Burgers-type systems is obtained. Systems contain 2 arbitrary elements that depend on the 2 dependent variables. Equivalence transformations for the systems are derived. Examples of nonclassical reductions are given. A Hopf-Cole–type mapping that linearizes a nonlinear system is presented.

]]>The purpose of this paper is the presentation of a new extragradient algorithm in 2-uniformly convex real Banach spaces. We prove that the sequences generated by this algorithm converge strongly to a point in the solution set of split feasibility problem, which is also a common element of the solution set of a generalized equilibrium problem and fixed points of of two relatively nonexpansive mappings. We give a numerical example to investigate the behavior of the sequences generated by our algorithm.

]]>Bidirectional associative memory models are 2-layer heteroassociative networks. In this work, we prove the existence and the global exponential stability of the unique weighted pseudo–almost periodic solution of bidirectional associative memory neural networks with mixed time-varying delays and leakage time-varying delays on time-space scales. Some sufficient conditions are given for the existence, the convergence, and the global exponential stability of the weighted pseudo–almost periodic solution by using fixed-point theorem and differential inequality techniques. The results of this paper complement the previous outcomes. An example is given to show the effectiveness of the derived results via computer simulations.

]]>The generalized finite differences method allows the use of irregular clouds of nodes. The optimal values of the key parameters of the method vary depending on how the nodes in the cloud are distributed, and this can be complicated especially in 3D. Therefore, we establish 2 criteria to allow the automation of the selection process of the key parameters. These criteria depend on 2 discrete functions, one of them penalizes distances and the other one penalizes imbalances. In addition, we show how to generate irregular clouds of nodes more efficient than finer regular clouds of nodes. We propose an improved and more versatile h-adaptive method that allows adding, moving, and removing nodes. To decide which nodes to act on, we use an indicator of the error a posteriori. This h-adaptive method gives results more accurate than those ones presented for the generalized finite differences method so far and, in addition, with fewer nodes. In addition, this method can be used in time-dependant problems to increase the temporal step or to avoid instabilities. As an example, we apply it in problems of seismic waves propagation.

]]>Lie group classification for a diffusion-type system that has applications in plasma physics is derived. The classification depends on the values of 5 parameters that appear in the system. Similarity reductions are presented. Certain initial value problems are reduced to problems with the governing equations being ordinary differential equations. Examples of potential symmetries are also presented.

]]>In this paper we study the asymptotic behaviour of the solutions in linear models of population dynamics by means of the basic reproduction number *R*_{0}. Our aim is to give a practical approach to the computation of the basic reproduction number in continuous-time population models structured by age and/or space. The procedure is different depending on whether the density of newborns per time unit and the density of population belong to the same functional space or not. Three infinite-dimensional examples are illustrated: a transport model for a cell population, a model of spatial diffusion of individuals in a habitat, and a model of migration of individuals between age-structured local populations. For each model, we have highlighted the possible advantages of computing *R*_{0} instead of the Malthusian parameter.

We describe a mixture thin film as a membrane endowed with multiple out-of-tangent-plane vectors at each point, with vector sequence defined to within a permutation to account for the mixing of the mixture components. Such a description is motivated by a proposal for an atomistic-to-continuum derivation of a representation of multiatomic layer thin films, a view not requiring the introduction of phenomenological parameters. Differences between that proposal and the model discussed here are the definition of the values of the layer-descriptor map to within permutations and the explicit introduction of a bending-like term in the energy. The out-of-tangent-plane vectors satisfy a condition forbidding them to fall within the tangent plane after deformation. We consider a generic weakly surface-polyconvex membrane energy and a quadratic bending term involving the out-of-tangent-plane multiple vectors, a term which is also quasiconvex. Under appropriate energy growth assumptions and Dirichlet-type boundary conditions, we prove existence of ground states, i.e., equilibrium configurations described by the solutions to balance equations. An obvious corollary is the existence of equilibrium configurations of single out-of-tangent-plane vector Cosserat surfaces, a natural scheme for plates of simple materials.

]]>We propose a simple delay mathematical model for the dynamics of AIDS-related cancers with treatment of HIV and chemotherapy. The main goals are to study the effects of the delay and of treatment (HAART and chemotherapy) in cancer cells growth. The model was simulated for several biologically reasonable values of the delay, of HAART efficacies and of chemotherapeutic drugs decay rates. The results of the simulations reveal an epidemiologically well-defined model. Important inferences are drawn for designing future treatment protocols.

]]>Quaternions, introduced by Hamilton in 1843 as a generalization of complex numbers, have found, in more recent years, a wealth of applications in a number of different areas that motivated the design of efficient methods for numerically approximating the zeros of quaternionic polynomials. In fact, one can find in the literature recent contributions to this subject based on the use of complex techniques, but numerical methods relying on quaternion arithmetic remain scarce. In this paper, we propose a Weierstrass-like method for finding simultaneously *all* the zeros of unilateral quaternionic polynomials. The convergence analysis and several numerical examples illustrating the performance of the method are also presented.

In this paper, we study the following Schrödinger-Kirchhoff–type equation with critical or supercritical growth

where *a*>0, *b*>0, *λ*>0, and *p*≥6. Under some suitable conditions, we prove that the equation has a nontrivial solution for small *λ*>0 by variational method. Moreover, we regard *b* as a parameter and obtain a convergence property of the nontrivial solution as *b*↘0. Our main contribution is related to the fact that we are able to deal with the case *p*>6.

In this paper, we consider the following nonlinear Choquard equation driven by fractional Laplacian

where *V*(*x*) is a nonnegative continuous potential function, 0<*s*<1, *N*≥2, (*N*−4*s*)^{+}<*α*<*N*, and *λ* is a positive parameter. By variational methods, we prove the existence of least energy solution which localizes near the bottom of potential well *i**n**t*(*V*^{−1}(0)) as *λ* large enough.

A semilinear parabolic problem is considered in a thin 3-D star-shaped junction that consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter

The purpose is to study the asymptotic behavior of the solution *u*_{ε} as *ε*0, ie, when the star-shaped junction is transformed in a graph. In addition, the passage to the limit is accompanied by special intensity factors
and
in nonlinear perturbed Robin boundary conditions.

We establish qualitatively different cases in the asymptotic behavior of the solution depending on the value of the parameters {*α*_{i}}and {*β*_{i}}. Using the multiscale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter *ε*0. Namely, in each case, we derive the limit problem (*ε*=0)on the graph with the corresponding Kirchhoff transmission conditions (untypical in some cases) at the vertex, define other terms of the asymptotic approximation and prove appropriate asymptotic estimates that justify these coupling conditions at the vertex, and show the impact of the local geometric heterogeneity of the node and physical processes in the node on some properties of the solution.

This paper deals with finite-time stabilization results of delayed Cohen-Grossberg BAM neural networks under suitable control schemes. We propose a state-feedback controller together with an adaptive-feedback controller to stabilize the system of delayed Cohen-Grossberg BAM neural networks. Stabilization conditions are derived by using Lyapunov function and some algebraic conditions. We also estimate the upper bound of settling time functional for the stabilization, which depends on the controller schemes and system parameters. Two illustrative examples and numerical simulations are given to validate the success of the derived theoretical results.

]]>A new defect-correction method based on the pressure projection for the stationary Navier-Stokes equations is proposed in this paper. A local stabilized technique based on the pressure projection is used in both defect step and correction step. The stability and convergence of this new method is analyzed detailedly. Finally, numerical examples confirm our theory analysis and validate high efficiency and good stability of this new method.

]]>In this paper, we consider stabilization of a 1-dimensional wave equation with variable coefficient where non-collocated boundary observation suffers from an arbitrary time delay. Since input and output are non-collocated with each other, it is more complex to design the observer system. After showing well-posedness of the open-loop system, the observer and predictor systems are constructed to give the estimated state feedback controller. Different from the partial differential equation with constant coefficients, the variable coefficient causes mathematical difficulties of the stabilization problem. By the approach of Riesz basis property, it is shown that the closed-loop system is stable exponentially. Numerical simulations demonstrate the effect of the stable controller. This paper is devoted to the wave equation with variable coefficients generalized of that with constant coefficients for delayed observation and non-collocated control.

]]>This paper is concerned with laminated beams modeled from the well-established Timoshenko system with time delays and boundary feedbacks. By using semigroup method, we prove the global well-posedness of solutions. Assuming the weights of the delay are small, we establish the exponential decay of energy to the system by using an appropriate Lyapunov functional.

]]>In this article, a finite element scheme based on the Newton method is proposed to approximate the solution of a nonlocal coupled system of parabolic problem. The Crank-Nicolson method is used for time discretization. Well-posedness of the problem is discussed at continuous and discrete levels. We derive a priori error estimates for both semidiscrete and fully discrete formulations. Results based on usual finite element method are provided to confirm the theoretical estimates.

]]>This work presents a new model of the fractional Black-Scholes equation by using the right fractional derivatives to model the terminal value problem. Through nondimensionalization and variable replacements, we convert the terminal value problem into an initial value problem for a fractional convection diffusion equation. Then the problem is solved by using the Fourier-Laplace transform. The fundamental solutions of the derived initial value problem are given and simulated and display a slow anomalous diffusion in the fractional case.

]]>In this paper, the existence of antiperiodic solutions for fourth-order impulsive differential equation is obtained by variational approaches and results on the auxiliary system. It is interesting that there is no growth restraint on nonlinear terms and impulsive terms. Besides, any minimizing sequence is bounded in a closed convex set of a space composed of Lipschitzian functions with the appearance of antiperiodic boundary value conditions.

]]>In this paper, we study relative controllability of fractional differential equations with pure delay. Delayed Gram-type matrix criterion and rank criterion for relative controllability are established with the help of the explicit solution formula. An example is given to illustrate our theoretical results.

]]>No abstract is available for this article.

]]>In this paper, we obtain the existence of at least two nontrivial solutions for a Robin-type differential inclusion problem involving *p*(*x*)-Laplacian type operator and nonsmooth potentials. Our approach is variational, and it is based on the nonsmooth critical point theory for locally Lipschitz functions. Copyright © 2013 John Wiley & Sons, Ltd.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

This paper is concerned with the following nonlinear fractional Schrödinger equation

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

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

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

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

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

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

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

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

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

A two-dimensional sparse-data tomographic problem is studied. The target is assumed to be a homogeneous object bounded by a smooth curve. A nonuniform rational basis splines (NURBS) curve is used as a computational representation of the boundary. This approach conveniently provides the result in a format readily compatible with computer-aided design software. However, the linear tomography task becomes a nonlinear inverse problem because of the NURBS-based parameterization. Therefore, Bayesian inversion with Markov chain Monte Carlo sampling is used for calculating an estimate of the NURBS control points. The reconstruction method is tested with both simulated data and measured X-ray projection data. The proposed method recovers the shape and the attenuation coefficient significantly better than the baseline algorithm (optimally thresholded total variation regularization), but at the cost of heavier computation.

]]>In this paper, the 2D Navier-Stokes-Voight equations with 3 delays in
is considered. By using the Faedo-Galerkin method, Lions-Aubin lemma, and Arzelà-Ascoli theorem, we establish the global well-posedness of solutions and the existence of pullback attractors in *H*^{1}.

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

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

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

]]>In this paper, we study the following Kirchhoff-type equation with critical or supercritical growth

where *a*>0, *b*>0, *λ*>0, *p*≥6 and *f* is a continuous superlinear but subcritical nonlinearity. When *V* and *f* are asymptotically periodic in *x*, we prove that the equation has a ground state solution for small *λ*>0 by Nehari method. Moreover, we regard *b* as a parameter and obtain a convergence property of the ground state solution as *b*↘0. Our main contribution is related to the fact that we are able to deal with the case *p*>6.

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

This paper mainly focus on the exponential stabilization problem of coupled systems on networks with mixed time-varying delays. Periodically intermittent control is used to control the coupled systems on networks with mixed time-varying delays. Moreover, based on the graph theory and Lyapunov method, two different kinds of stabilization criteria are derived, which are in the form of Lyapunov-type theorem and coefficients-type criterion, respectively. These laws reveal that the stability has a close relationship with the topology structure of the networks. In addition, as a subsequent result, a decision theorem is also presented. It is straightforward to show the stability of original system can be determined by that of modified system with added absolute value into the coupling weighted-value matrix. Finally, the feasibility and validity of the obtained results are demonstrated by several numerical simulation figures.

]]>In this paper, we introduce fractional order into an ecoepidemiological model, where predator consumes disproportionately large number of infected preys following type 2 response function. We prove different mathematical results like existence, uniqueness, nonnegativity, and boundedness of the solutions of fractional order system. We also prove the local and global stability of different equilibrium points of the system. The results are illustrated with several examples.

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

]]>In this paper, we investigate the existence and multiplicity of harmonic and subharmonic solutions for second-order quasilinear equation

where
, *g* satisfies the superlinear condition at infinity. We prove that the given equation possesses harmonic and subharmonic solutions by using the phase-plane analysis methods and a generalized version of the Poincaré-Birkhoff twist theorem.

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

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

]]>In this paper, we provide a detailed convergence analysis for fully discrete second-order (in both time and space) numerical schemes for nonlocal Allen-Cahn and nonlocal Cahn-Hilliard equations. The unconditional unique solvability and energy stability ensures *ℓ*^{4} stability. The convergence analysis for the nonlocal Allen-Cahn equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nonlocal Cahn-Hilliard equation, because of the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken, and an *H*^{−1} inner-product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a priori
bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang (eg, 1964), instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an *O*(*s*^{3}+*h*^{4}) convergence in
norm, in which *s* and *h* denote the time step and spatial mesh sizes, respectively. This in turn leads to the necessary bound under a standard constraint *s*≤*C**h*. Here, we also prove convergence of the scheme in the maximum norm under the same constraint.

In this paper, we study the existence of a positive ground state solution to the following coupled system of nonlinear Schrödinger equations:

where the nonlinearities *f*_{1}(*x*,*s*) and *f*_{2}(*x*,*s*) are superlinear at infinity and have exponential critical growth of the Trudinger-Moser type. The potentials *V*_{1}(*x*) and *V*_{2}(*x*) are nonnegative and satisfy a condition involving the coupling term *λ*(*x*), namely, *λ*(*x*)^{2}<*δ*^{2}*V*_{1}(*x*)*V*_{2}(*x*) for some 0<*δ*<1. For this purpose, we use the minimization technique over the Nehari manifold and strong maximum principle to get a positive ground state solution. Moreover, by using a bootstrap argument and *L*^{q}-estimates, we get regularity and asymptotic behavior.

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

]]>In this paper, the problem of stability in terms of two measures is considered for a class of stochastic partial differential delay equations with switching. Sufficient conditions for stability in terms of two measures are obtained based on the technique of constructing a proper approximating strong solution system and conducting a limiting type of argument to pass on stability of strong solutions to mild ones. In particular, the stochastic stability under the fixed-index sequence monotonicity condition and under the average dwell-time switching are considered.

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

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

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

]]>We investigate the asymptotic periodicity, *L*^{p}-boundedness, classical (resp., strong) solutions, and the topological structure of solutions set of strongly damped semilinear wave equations. The theoretical results are well complemented with a set of very illustrating applications.

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

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

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

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

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

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

]]>A well-known result on pathwise uniqueness of the solution of stochastic differential equations in is the Yamada-Watanabe theorem. We have extended this result by replacing the Lipschitz assumption on the drift coefficient by much weaker assumption of semi-monotonicity.

We consider 2 transmission problems. The first problem has 2 damping mechanisms acting in the same part of the body, one of frictional type and other of Kelvin-Voigt type. In this case, we show that, even though it has too much dissipation, the semigroup is not exponentially stable. The second problem also has those damping terms but they act in complementary parts of the body. For this case, we show that the semigroup is exponentially stable and it is not analytic.

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

In this study, first, a formula for regularized sums of eigenvalues of a Sturm-Liouville problem with retarded argument at 2 points of discontinuity which contains a spectral parameter in the boundary conditions is obtained. After that, oscillation properties of the related problem is investigated. Finally, under the condition that a subset of nodal points is dense in definition set, the potential function is determined.

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

]]>In this paper, we prove the large-time behavior, as time tends to infinity, of solutions in and for a system modeling the nematic liquid crystal flow, which consists of a subsystem of the compressible Navier-Stokes equations coupling with a subsystem including a heat flow equation for harmonic maps.

]]>The issue of justifying the eddy current approximation of Maxwell's equations is reconsidered in the time-dependent setting. Convergence of the solution operators is shown in the sense of strong operator limits.

]]>In this paper, our main purpose is to establish the existence results of positive solutions for a *p*−*q*-Laplacian system involving concave-convex nonlinearities:

where Ω is a bounded domain in *R*^{N}, *λ*,*θ*>0 and 1<*r*<*q*<*p*<*N*. We assume 1<*α*,*β* and
is the critical Sobolev exponent and △_{s}·=div(|∇·|^{s−2}∇·) is the s-Laplacian operator. The main results are obtained by variational methods.

We study the shape derivative of the strongly singular volume integral operator that describes time-harmonic electromagnetic scattering from homogeneous medium. We show the existence and a representation of the derivative, and we deduce a characterization of the shape derivative of the solution to the diffraction problem as a solution to a volume integral equation of the second kind.

]]>In this paper, we consider a class of asymptotically linear second-order Hamiltonian system with resonance at infinity. We will use Morse theory combined with the technique of penalized functionals to obtain the existence of rotating periodic solutions.

]]>We introduce the notions of the pseudospherical normal Darboux images for the curve on a lightlike surface in Minkowski 3-space and study these Darboux images by using technics of the singularity theory. Furthermore, we give a relation between these Darboux images and Darboux frame from the viewpoint of Legendrian dualities.

]]>In this paper, we consider the multiplicity of solutions of the *p*-Laplacian problems involving supercritical Sobolev growth and exponential growth in
via Ricceri principle. By means of the truncation combining with Moser iteration, we can extend the result about the subcritical growth to the supercritical and exponential growth.

In this paper, we consider a nonautonomous impulsive plankton model with mutual help of preys. Sufficient conditions ensuring permanence and global attractivity of the model are established by the relation between solutions of impulsive system and corresponding nonimpulsive system. Also, we propose the conditions for which the species of system are driven to extinction. Numerical simulations are given to verify the main results.

]]>This paper deals with an adaptation of the Poincaré-Lindstedt method for the determination of periodic orbits in three-dimensional nonlinear differential systems. We describe here a general symbolic algorithm to implement the method and apply it to compute periodic solutions in a three-dimensional Lotka-Volterra system modeling a chain food interaction. The sufficient conditions to make secular terms disappear from the approximate series solution are given in the paper.

]]>In this paper, 2 extragradient methods for solving differential variational inequality (DVI) problems are presented, and the convergence conditions are derived. It is shown that the presented extragradient methods have weaker convergence conditions in comparison with the basic fixed-point algorithm for solving DVIs. Then the linear complementarity systems, as an important and practical special case of DVIs, are considered, and the convergence conditions of the presented extragradient methods are adapted for them. In addition, an upper bound for the Lipschitz constant of linear complementarity systems is introduced. This upper bound can be used for adjusting the parameters of the extragradient methods, to accelerate the convergence speed. Finally, 4 illustrative examples are considered to support the theoretical results.

]]>We define an abstract setting to treat essential spectra of unbounded coupled operator matrix. We prove a well-posedness result and develop a spectral theory which also allows us to prove an amelioration to many earlier works. We point out that a concrete example from integro-differential equation fit into this abstract framework involving a general class of regular operator in *L*_{1} spaces.

In this paper, we give some new properties of the presented asynchronous algorithms of theta scheme combined with finite elements methods (App. Math. Comp., 217 (2011), 6443-6450) for an evolutionary implicit 2-sided obstacle problem to prove the existence and uniqueness of the discrete solution. Furthermore, an error estimate on the uniform norm is given.

]]>Aveiro method is a sparse representation method in reproducing kernel Hilbert spaces, which gives orthogonal projections in linear combinations of reproducing kernels over uniqueness sets. It, however, suffers from determination of uniqueness sets in the underlying reproducing kernel Hilbert space. In fact, in general spaces, uniqueness sets are not easy to be identified, let alone the convergence speed aspect with Aveiro method. To avoid those difficulties, we propose an new Aveiro method based on a dictionary and the matching pursuit idea. What we do, in fact, are more: The new Aveiro method will be in relation to the recently proposed, the so-called pre-orthogonal greedy algorithm involving completion of a given dictionary. The new method is called Aveiro method under complete dictionary. The complete dictionary consists of all directional derivatives of the underlying reproducing kernels. We show that, under the boundary vanishing condition bring available for the classical Hardy and Paley-Wiener spaces, the complete dictionary enables an efficient expansion of any given element in the Hilbert space. The proposed method reveals new and advanced aspects in both the Aveiro method and the greedy algorithm.

]]>In this paper, we study the following fractional Schrödinger equations:

- (1)

where (−△)^{α} is the fractional Laplacian operator with
, 0≤*s*≤2*α*, *λ*>0, *κ* and *β* are real parameter.
is the critical Sobolev exponent. We prove a fractional Sobolev-Hardy inequality and use it together with concentration compact theory to get a ground state solution. Moreover, concentration behaviors of nontrivial solutions are obtained when the coefficient of the potential function tends to infinity.

Let Γ be a simple closed curve that bounds the finite domain *D*, *z*=*z*(*ζ*)=*z*(*r**e*^{iϑ}) be the conformal mapping of the circle {*ζ*:|*ζ*|<1} onto the domain *D*. Furthermore, let the functions *A*(*z*), *B*(*z*) be given on *D* and *U*^{s,2}(*A*;*B*;*D*) be the set of regular solutions of the equation
.

We call the Smirnov class *E*^{p(t)}(*A*;*B*;*D*) the set of those generalized functions *W*in *D* for which

where *p*(*t*) is a positive measurable function on Γ.

We consider the Riemann-Hilbert problem: Define a function *W*(*z*) from the class *E*^{p(t)}(*A*;*B*;*D*) for which the equality,

is fulfilled almost everywhere on Γ.

It is assumed that Γ is a piecewise-smooth curve without external peaks;
, *p* is Log Hölder continuous and

the function
belongs to the class *A*(*p*(*t*);Γ), which is the generalization of the well-known Simonenko class *A*(*p*;Γ), where *p* is constant.

The solvability conditions are established, and solutions are constructed.

In this paper, we introduce a *q*-analog of 1-dimensional Dirac equation. We investigate the existence and uniqueness of the solution of this equation. Later, we discuss some spectral properties of the problem, such as formally self-adjointness, the case that the eigenvalues are real, orthogonality of eigenfunctions, Green function, existence of a countable sequence of eigenvalues, and eigenfunctions forming an orthonormal basis of
. Finally, we give some examples.

In this paper, a leader-following consensus of discrete-time multi-agent systems with nonlinear intrinsic dynamics is investigated. We propose and prove conditions ensuring a leader-following consensus. Numerical examples are given to illustrate our results.

]]>Shannon and Zipf-Mandelbrot entropies have many applications in many applied sciences, for example, in information theory, biology and economics, etc. In this paper, we consider two refinements of the well-know Jensen inequality and obtain different bounds for Shannon and Zipf-Mandelbrot entropies. First of all, we use some convex functions and manipulate the weights and domain of the functions and deduce results for Shannon entropy. We also discuss their particular cases. By using Zipf-Mandelbrot laws for different parameters in Shannon entropies results, we obtain bounds for Zipf-Mandelbrot entropy. The idea used in this paper for obtaining the results may stimulate further research in this area, particularly for Zipf-Mandelbrot entropy.

]]>In this short paper, the initial value problem for the Navier-Stokes equations with the Coriolis force is investigated. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large-scale phenomena. We prove that existence of uniform global large solutions to the Navier-Stokes equations with the Coriolis force for a class of special initial data. The results obtained in this paper are different from the previous 2 types of results.

]]>In this paper, we introduce and investigate the functions of (*μ*,*ν*)-pseudo *S*-asymptotically *ω*-periodic of class *r*(class infinity). We systematically explore the properties of these functions in Banach space including composition theorems. As applications, we establish some sufficient criteria for (*μ*,*ν*)-pseudo *S*-asymptotic *ω*-periodicity of (nonautonomous) semilinear integro-differential equations with finite or infinite delay. Finally, some interesting examples are presented to illustrate the main findings.

The model of pollution for a system of 3 lakes interconnected by channels is extended using Caputo-Hadamard fractional derivatives of different orders *α*_{i}∈(0,1), *i*=1,2,3. A numerical approach based on ln-shifted Legendre polynomials is proposed to solve the considered fractional model. No discretization is needed in our approach. Some numerical experiments are provided to illustrate the presented method.

This paper deals with the attraction-repulsion chemotaxis system with nonlinear diffusion *u*_{t}=∇·(*D*(*u*)∇*u*)−∇·(*u**χ*(*v*)∇*v*)+∇·(*u*^{γ}*ξ*(*w*)∇*w*), *τ*_{1}*v*_{t}=Δ*v*−*α*_{1}*v*+*β*_{1}*u*, *τ*_{2}*w*_{t}=Δ*w*−*α*_{2}*w*+*β*_{2}*u*, subject to the homogenous Neumann boundary conditions, in a smooth bounded domain
, where the coefficients *α*_{i}, *β*_{i}, and *τ*_{i}∈{0,1}(*i*=1,2) are positive. The function *D* fulfills *D*(*u*)⩾*C*_{D}*u*^{m−1} for all *u*>0 with certain *C*_{D}>0 and *m*>1. For the parabolic-elliptic-elliptic case in the sense that *τ*_{1}=*τ*_{2}=0 and *γ*=1, we obtain that for any
and all sufficiently smooth initial data *u*_{0}, the model possesses at least one global weak solution under suitable conditions on the functions *χ* and *ξ*. Under the assumption
, it is also proved that for the parabolic-parabolic-elliptic case in the sense that *τ*_{1}=1, *τ*_{2}=0, and *γ*⩾2, the system possesses at least one global weak solution under different assumptions on the functions *χ* and *ξ*.

This paper reports a modified homotopy perturbation algorithm, called the domain decomposition homotopy perturbation method (DDHPM), for solving two-point singular boundary value problems arising in science and engineering. The essence of the approach is to split the domain of the problem into a number of nonoverlapping subdomains. In each subdomain, a method based on a combination of HPM and integral equation formalism is implemented. The boundary condition at the right endpoint of each inner subdomain is established before deriving an iterative scheme for the components of the solution series. The accuracy and efficiency of the DDHPM are demonstrated by 4 examples (2 nonlinear and 2 linear). In comparison with the traditional HPM, the proposed domain decomposition HPM is highly accurate.

]]>In this paper, we present a new algorithm for solving the split common null point and common fixed point problem, to find a point that belongs to the common element of common zero points of an infinite family of maximal monotone operators and common fixed points of an infinite family of demicontractive mappings such that its image under a linear transformation belongs to the common zero points of another infinite family of maximal monotone operators and its image under another linear transformation belongs to the common fixed point of another infinite family of demicontractive mappings in the image space. We establish strong convergence for the algorithm to find a unique solution of the variational inequality, which is the optimality condition for the minimization problem. As special cases, we shall use our results to study the split equilibrium problems and the split optimization problems.

]]>In this article, we consider the Cauchy problem to Keller-Segel equations coupled to the incompressible Navier-Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, let *u*_{F}:=*e*^{tΔ}*u*_{0}; we prove that there exist 2 positive constants *σ*_{0} and *C*_{0} such that if the gravitational potential
and the initial data (*u*_{0},*n*_{0},*c*_{0}) satisfy

for some *p*,*q* with
and
, then the global solutions can be established in critical Besov spaces.

The swarm behaviour can be controlled by different localizations of attractants (food pieces) and repellents (dangerous places), which, respectively, attract and repel the swarm propagation. If we assume that at each time step, the swarm can find out not more than *p*−1 attractants (
), then the swarm behaviour can be coded by *p*-adic integers, ie, by the numbers of the ring **Z**_{p}. Each swarm propagation has the following 2 stages: (1) the discover of localizations of neighbour attractants and repellents and (2) the logistical optimization of the road system connecting all the reachable attractants and avoiding all the neighbour repellents. In the meanwhile, at the discovering stage, the swarm builds some direct roads and, at the logistical stage, the transporting network of the swarm gets loops (circles) and it permanently changes. So, at the first stage, the behaviour can be expressed by some linear *p*-adic valued strings. At the second stage, it is expressed by non-linear modifications of *p*-adic valued strings. The second stage cannot be described by conventional algebraic tools; therefore, I have introduced the so-called non-linear group theory for describing both stages in the swarm propagation.

We study the well-posedness and dynamic behavior for the KdV-Burgers equation with a force
on **R**. We establish *L*^{p}−*L*^{q} estimates of the evolution
, as an application we obtain the local well-posedness. Then the global well-posedness follows from a uniform estimate for solutions as *t*goes to infinity. Next, we prove the asymptotical regularity of solutions in space
and
by the smoothing effect of
. The regularity and the asymptotical compactness in *L*^{2} yields the asymptotical compactness in
by an interpolation arguement. Finally, we conclude the existence of an
globalattractor.

In this study, we consider the stability of tumor model by using the standard differential geometric method that is known as Kosambi-Cartan-Chern (KCC) theory or Jacobi stability analysis. In the KCC theory, we describe the time evolution of tumor model in geometric terms. We obtain nonlinear connection, Berwald connection and KCC invariants. The second KCC invariant gives the Jacobi stability properties of tumor model. We found that the equilibrium points are Jacobi unstable for positive coordinates. We also discussed the time evolution of components of deviation tensor and the behavior of deviation vector near the equilibrium points.

]]>In this paper we consider a periodic 2-dimensional quasi-geostrophic equations with subcritical dissipation. We show the global existence and uniqueness of the solution for small initial data in the Lei-Lin-Gevrey spaces . Moreover, we establish an exponential type explosion in finite time of this solution.

]]>In the present article, we study the temperature effects on two-phase immiscible incompressible flow through a porous medium. The mathematical model is given by a coupled system of 2-phase flow equations and an energy balance equation. The model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy-Muskat and the capillary pressure laws. The problem is written in terms of the phase formulation; ie, the saturation of one phase, the pressure of the second phase, and the temperature are primary unknowns. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result to pass to the limit in nonlinear terms.

]]>We give 2 widest Mehler's formulas for the univariate complex Hermite polynomials
, by performing double summations involving the products
and
. They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level *m*. The second Mehler's formula generalizes the one appearing as a particular case of the so-called Kibble-Slepian formula. The proofs we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.

We consider the stabilization of the electromagneto-elastic system with Wentzell conditions in a bounded domain of . Using the multiplier method we prove an exponential stability result under some geometric condition. Previous results of this type have recently been obtained for the coupled Maxwell/wave system with Wentzell conditions by H. Kasri and A. Heminna (Evol Equ and Control Theo 5: 235-250, 2016)

]]>We investigate the uniform regularity and vanishing viscosity limit for the incompressible chemotaxis-Navier-Stokes system with Navier boundary condition for velocity field and Neumann boundary condition for cell density and chemical concentration in a 3D bounded domain. It is shown that there exists a unique strong solution of the incompressible chemotaxis-Navier-Stokes system in a finite time interval, which is independent of the viscosity coefficient. Moreover, this solution is uniformly bounded in a conormal Sobolev space, which allows us to take the vanishing viscosity limit to obtain the incompressible chemotaxis-Euler system.

]]>We consider a Cahn-Hilliard–type equation with degenerate mobility and single-well potential of Lennard-Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn-Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension *d*=1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion.

In this paper, we develop a high-order finite difference scheme for the solution of a time fractional partial integro-differential equation with a weakly singular kernel. The fractional derivative is used in the Riemann-Liouville sense. We prove the unconditional stability and convergence of scheme using energy method and show that the convergence order is . We provide some numerical experiments to confirm the efficiency of suggested scheme. The results of numerical experiments are compared with analytical solutions to show the efficiency of proposed scheme. It is illustrated that the numerical results are in good agreement with theoretical ones.

]]>We use a particle method to study a Vlasov-type equation with local alignment, which was proposed by Sebastien Motsch and Eitan Tadmor [*J. Statist. Phys.*, 141(2011), pp. 923-947]. For *N*-particle system, we study the unconditional flocking behavior for a weighted Motsch-Tadmor model and a model with a “tail”. When *N* goes to infinity, global existence and stability (hence uniqueness) of measure valued solutions to the kinetic equation of this model are obtained. We also prove that measure valued solutions converge to a flock. The main tool we use in this paper is Monge-Kantorovich-Rubinstein distance.

In the spectral Petrov-Galerkin methods, the trial and test functions are required to satisfy particular boundary conditions. By a suitable linear combination of orthogonal polynomials, a basis, that is called the modal basis, is obtained. In this paper, we extend this idea to the nonorthogonal dual Bernstein polynomials. A compact general formula is derived for the modal basis functions based on dual Bernstein polynomials. Then, we present a Bernstein-spectral Petrov-Galerkin method for a class of time fractional partial differential equations with Caputo derivative. It is shown that the method leads to banded sparse linear systems for problems with constant coefficients. Some numerical examples are provided to show the efficiency and the spectral accuracy of the method.

]]>In this paper, we study persistent piecewise linear multidimensional random motions. Their velocities, switching at Poisson times, are uniformly distributed on a sphere. The changes of direction are accompanied with subsequent jumps of random length and of uniformly distributed orientation.

In this paper, we obtain some useful properties and formulae of distributions of these processes. In particular, we get these distributions in the cases of jumps with Gaussian and exponential distributions of jump magnitudes.

In this article, we consider and investigate the cases when the retailer's capitals are restricted and when the supplier offers another kind of 2-level trade credit. This means that the supplier offers 2-level trade credit for the retailer to settle the account and the retailer's capitals are restricted, so the retailer decides to pay off the unpaid balance as follows: Firstly, the retailer decides to pay off the unpaid balance at the end of the first credit period if the retailer can pay off all accounts and, in addition, the retailer can use the sales revenue to earn interest throughout the replenishment cycle time. Secondly, the retailer decides to pay off all accounts either after the end of the first credit period, but before the second credit period, or after the second credit period if the retailer cannot pay off the unpaid balance at the end of the first credit period. Additionally, the delay will incur interest charges on the unpaid and overdue balance due to the difference between the interest earned and the interest charged. Consequently, the main purpose of this article is to characterize the optimal solution processes and (in accordance with the functional behavior of the cost function) to search for the optimal replenishment cycle time. Finally, numerical examples are given to illustrate the theoretical results which are proven in this article by means of mathematical solution procedures.

]]>The goal in the paper is to advertise Dunkl extension of Szász beta-type operators. We initiate approximation features via acknowledged Korovkin and weighted Korovkin theorem and obtain the convergence rate from the point of modulus of continuity, second-order modulus of continuity, the Lipschitz class functions, Peetre's *K*-functional, and modulus of weighted continuity by Dunkl generalization of Szász beta-type operators.

In this paper, we study constraint minimizers of the following *L*^{2}−critical minimization problem:

where *E*(*u*) is the Schrödinger-Poisson-Slater functional

and *N* denotes the mass of the particles in the Schrödinger-Poisson-Slater system. We prove that *e*(*N*) admits minimizers for
and, however, no minimizers for *N*>*N*^{∗}, where *Q*(*x*) is the unique positive solution of
in
. Some results on the existence and nonexistence of minimizers for *e*(*N*^{∗}) are also established. Further, when *e*(*N*^{∗}) does not admit minimizers, the limit behavior of minimizers as *N*↗*N*^{∗} is also analyzed rigorously.

In living cells, we can observe a variety of complex network systems such as metabolic network. Studying their sensitivity is one of the main approaches for understanding the dynamics of these biological systems. The study of the sensitivity is done by increasing/decreasing, or knocking out separately, each enzyme mediating a reaction in the system and then observing the responses in the concentrations of chemicals or their fluxes. However, because of the complexity of the systems, it has been unclear how the network structures influence/determine the responses of the systems. In this study, we focus on monomolecular networks at steady state and establish a simple criterion for determining regions of influence when any one of the reaction rates is perturbed through sensitivity experiments of enzyme knock-out type. Specifically, we study the network response to perturbations of a reaction rate *j*^{∗} and describe which other reaction rates
respond by non-zero reaction flux, at steady state. Non-zero responses of
to *j*^{∗} are called flux-influence of *j*^{∗} on
. The main and most important aspect of this analysis lies in the reaction graph approach, in which the chemical reaction networks are modelled by a directed graph. Our whole analysis is function-free, ie, in particular, our approach allows a graph theoretical description of sensitivity of chemical reaction networks. We emphasize that the analysis does not require numerical input but is based on the graph structure only. Our specific goal here is to address a topological characterization of the flux-influence relation in the network. In fact we characterize and describe the whole set of reactions influenced by a perturbation of any specific reaction.

The determination of a space-dependent source term along with the solution for a 1-dimensional time fractional diffusion equation with nonlocal boundary conditions involving a parameter *β*>0 is considered. The fractional derivative is generalization of the Riemann-Liouville and Caputo fractional derivatives usually known as Hilfer fractional derivative. We proved existence and uniqueness results for the solution of the inverse problem while over-specified datum at 2 different time is given. The over-specified datum at 2 time allows us to avoid initial condition in terms of fractional integral associated with Hilfer fractional derivative.

In the present paper, we construct a new sequence of Bernstein-Kantorovich operators depending on a parameter *α*. The uniform convergence of the operators and rate of convergence in local and global sense in terms of first- and second-order modulus of continuity are studied. Some graphs and numerical results presenting the advantages of our construction are obtained. The last section is devoted to bivariate generalization of Bernstein-Kantorovich operators and their approximation behaviors.

This paper is concerned with the large time behavior of solutions to the initial value problem for the damped wave equations with nonlinear convection in one-dimensional whole space. In 2007, Ueda and Kawashima showed that the solution tends to a self similar solution of the Burgers equation. However, they did not mention that their decay estimate is optimal or not. Under this situation, the aim of this paper was to find out the sharp decay estimate by studying the second asymptotic profile of solutions. The explicit representation formula and the decay estimates of the solution for the linearized equation including the lower order term play crucial roles in our analysis.

]]>The purpose of this paper is to study the mixed Dirichlet-Neumann boundary value problem for the semilinear Darcy-Forchheimer-Brinkman system in *L*_{p}-based Besov spaces on a bounded Lipschitz domain in
, with *p* in a neighborhood of 2. This system is obtained by adding the semilinear term |**u**|**u** to the linear Brinkman equation. First, we provide some results about equivalence between the Gagliardo and nontangential traces, as well as between the weak canonical conormal derivatives and the nontangential conormal derivatives. Various mapping and invertibility properties of some integral operators of potential theory for the linear Brinkman system, and well-posedness results for the Dirichlet and Neumann problems in *L*_{p}-based Besov spaces on bounded Lipschitz domains in
(*n*≥3) are also presented. Then, using integral potential operators, we show the well-posedness in *L*_{2}-based Sobolev spaces for the mixed problem of Dirichlet-Neumann type for the linear Brinkman system on a bounded Lipschitz domain in
(*n*≥3). Further, by using some stability results of Fredholm and invertibility properties and exploring invertibility of the associated Neumann-to-Dirichlet operator, we extend the well-posedness property to some *L*_{p}-based Sobolev spaces. Next, we use the well-posedness result in the linear case combined with a fixed point theorem to show the existence and uniqueness for a mixed boundary value problem of Dirichlet and Neumann type for the semilinear Darcy-Forchheimer-Brinkman system in *L*_{p}-based Besov spaces, with *p*∈(2−*ε*,2+*ε*) and some parameter *ε*>0.

This paper deals with the oscillation of the fourth-order linear delay differential equation with a negative middle term under the assumption that all solutions of the auxiliary third-order differential equation are nonoscillatory. Examples are included to illustrate the importance of results obtained.

]]>The self-adaptive intelligence gray predictive model (SAIGM) has an alterable-flexible model structure, and it can build a dynamic structure to fit different external environments by adjusting the parameter values of SAIGM. However, the order number of the raw SAIGM model is not optimal, which is an integer. For this, a new SAIGM model with the fractional order accumulating operator (SAIGM_FO) was proposed in this paper. Specifically, the final restored expression of SAIGM_FO was deduced in detail, and the parameter estimation method of SAIGM_FO was studied. After that, the Particle Swarm Optimization algorithm was used to optimize the order number of SAIGM_FO, and some steps were provided. Finally, the SAIGM_FO model was applied to simulate China's electricity consumption from 2001 to 2008 and forecast it during 2009 to 2015, and the mean relative simulation and prediction percentage errors of the new model were only 0.860% and 2.661%, in comparison with the ones obtained from the raw SAIGM model, the GM(1, 1) model with the optimal fractional order accumulating operator and the GM(1, 1) model, which were (1.201%, 5.321%), (1.356%, 3.324%), and (2.013%, 23.944%), respectively. The findings showed both the simulation and the prediction performance of the proposed SAIGM_FO model were the best among the 4 models.

]]>We prove that the Volterra-Gause system of predator-prey type exhibits 2 kinds of zero-Hopf bifurcations for convenient values of their parameters. In the first, 1 periodic solution bifurcates from a zero-Hopf equilibrium, and in the second, 4 periodic solutions bifurcate from another zero-Hopf equilibrium. This study is done using the averaging theory of second order.

]]>In this paper, we consider the integration of the special second-order initial value problem. Hybrid Numerov methods are used, which are constructed in the sense of Runge-Kutta ones. Thus, the Taylor expansions at the internal points are matched properly in the final expression. A new family of such methods attaining eighth algebraic order is given at a cost of only 7 function evaluations per step. The new family provides us with an extra parameter, which is used to increase phase-lag order to 18. We proceed with numerical tests over a standard set of problems for these cases. Appendices implementing the symbolic construction of the methods and derivation of their coefficients are also given.

]]>The nonlinear versions of Sturm-Picone comparison theorem as well as Leighton's variational lemma and Leighton's theorem for regular and singular nonlinear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions are established. Although discontinuity of the solutions causes some difficulties, these new comparison theorems cover the old ones where impulse effects are dropped.

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