Search

The purpose of the present note is to point out mathematical mistakes in some published papers on boundary value problems to prevent usage of mistaken results and methods by mathematicians. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we develop and analyze a mathematical model for spreading malaria, including treatment with transmission‐blocking drugs (TBDs). The paper's main aim is to demonstrate the impact the chosen model for demographic growth has on the disease's transmission and the effect of its treatment with TBDs. We calculate the model's control reproduction number and equilibria and perform a global stability analysis of the disease‐free equilibrium point. The mathematical analysis reveals that, depending on the model's demography, the model can exhibit forward, backward, and even some unconventional types of bifurcation, where disease elimination can occur for both small and large values of the reproduction number. We also conduct a numerical analysis to explore the short‐time behavior of the model. A key finding is that for one type of demographic growth, the population experienced a significantly higher disease burden than the others, and when exposed to high levels of treatment with TBDs, only this population succeeded in effectively eliminating the disease within a reasonable timeframe. [ABSTRACT FROM AUTHOR]

This paper is concerned with the well‐posedness of a diffusion–reaction system for a susceptible‐exposed‐infected‐recovered (SEIR) mathematical model. This model is written in terms of four nonlinear partial differential equations with nonlinear diffusions, depending on the total amount of the SEIR populations. The model aims at describing the spatio‐temporal spread of the COVID‐19 pandemic and is a variation of the one recently introduced, discussed, and tested in a paper by Viguerie et al (2020). Here, we deal with the mathematical analysis of the resulting Cauchy–Neumann problem: The existence of solutions is proved in a rather general setting, and a suitable time discretization procedure is employed. It is worth mentioning that the uniform boundedness of the discrete solution is shown by carefully exploiting the structure of the system. Uniform estimates and passage to the limit with respect to the time step allow to complete the existence proof. Then, two uniqueness theorems are offered, one in the case of a constant diffusion coefficient and the other for more regular data, in combination with a regularity result for the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an innovative hybrid discrete‐continuum model that provides a comprehensive framework for understanding the intricate interactions between tumor cells, immune cells, and the effects of immunotherapy. This study distinguishes itself by addressing the limitations of traditional diffusion models, which often fail to capture the irregular and nonlocal movements of cells within the complex tumor microenvironment. To overcome these challenges, we employ fractional time derivatives and the fractional Laplacian operator, offering a more accurate representation of anomalous diffusion processes that are critical in cancer dynamics. Our research begins with a rigorous mathematical analysis, where we establish the global existence of a unique mild solution, laying a solid theoretical foundation for the model. A key innovation of our study is the introduction of the “invasion threshold,” a critical parameter inspired by the next‐generation operator used in epidemiological modeling. This threshold provides a powerful tool for determining the existence and stability of equilibrium points, offering deep insights into the conditions that either promote or inhibit tumor growth under immunotherapeutic interventions. By integrating a hybrid discrete‐continuum approach, we capture the individual behavior of each cell, allowing for a more detailed exploration of the dynamic interplay within the tumor microenvironment. This model not only advances our understanding of tumor‐immune interactions but also holds potential for informing more effective therapeutic strategies, making a significant contribution to the field of mathematical oncology. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some explicit and recursive formulas for the n$$ n $$th q$$ q $$‐derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between commutative algebras and the properties of q$$ q $$‐differential operator. One of the formulas involves the determinant of the functions and their q$$ q $$‐derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, we used formulas to obtain some interesting q$$ q $$‐identities. [ABSTRACT FROM AUTHOR]

Zika is a flavivirus that is transmitted to humans either through the bites of infected Aedes mosquitoes or through sexual transmission. Zika has been associated with congenital anomalies, like microcephalus. We developed and analyzed the fractional‐order Zika virus model in this paper, considering the vector transmission route with human influence. The model consists of four compartments: susceptible individuals are x1(t), infected individuals are x2(t), x3(t) shows susceptible mosquitos, and x4(t) shows the infected mosquitos. The fractional parameter is used to develop the system of complex nonlinear differential equations by using Caputo and Atangana–Baleanu derivative. The stability analysis as well as qualitative analysis of the fractional‐order model has been made and verify the non‐negative unique solution. Finally, numerical simulations of the model with Caputo and Atangana Baleanu are discussed to present the graphical results for different fractional‐order values as well as for the classical model. A comparison has been made to check the accuracy and effectiveness of the developed technique for our obtained results. This investigative research leads to the latest information sector included in the evolution of the Zika virus with the application of fractional analysis in population dynamics. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of weak solutions of the system that describes the two‐phase two‐component fluid flow in porous media. Both two‐phase and possible one‐phase flow regions are taken into account. Our research is based on a global pressure, an artificial variable that allows us to partially decouple the original equations. As a second primary unknown for the system, we choose the gas pseudo‐pressure, a persistent variable which coincides with the gas pressure in the two‐phase regions while it does not have physical meaning in one‐phase flow regions, when only the liquid phase is present. This allows us to introduce an another persistent variable that is an artificial variable in one‐phase flow regions and a physical variable in two‐phase flow regions—the capillary pseudopressure. We rewrite the system's equations in a fully equivalent form in terms of the global pressure and the gas‐pseudo pressure. In order to prove the existence of weak solutions of obtained system, we also use the capillary pseudo‐pressure. By using it, we can decouple obtained equations on the discrete level. This allows us to derive the existence result for weak solutions in more tractable way. [ABSTRACT FROM AUTHOR]

Cancer caused by malignant tumor is a very common and serious disease which can lead to people towards death. However, sometimes, cancer may be controlled if the cancer cells can be infected and split by particular virus with weak pathogenicity. To give a better understanding on tumor dynamics, in this paper, we deal with a reaction–diffusion tumor model, which comes from the process of tumor therapy with oncolytic virus. Our main objective is to analyze the biological compatibility of uninfected and infected tumor cells. In addition, we are concerned with the long‐time behavior, stability, and a priori estimates for the model. The compatibility conditions show that under certain range of parameters control, the infected tumor cells can coexist with normal tumor cells or be eliminated by oncolytic virus as time goes on; in the elimination case, it implies that the organism's complete recovery is highly possible as a result of the attack of oncolytic virus. An indirect idea in calculating fixed point index is proposed; correspondingly, a new condition on coexistence is presented. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the intrinsic impact of the predator‐taxis coefficient on the formation of spatial patterns in a predator–prey system with prey social behavior subject to Neumann boundary conditions. By treating the predator‐taxis coefficient as a potential critical parameter for Turing bifurcation, we observe that the Turing pattern is fully captured by three distinct critical thresholds. Meanwhile, utilizing weakly nonlinear analysis and the amplitude equation, we establish the direction of Turing bifurcation. Our mathematical analysis reveals that the inclusion of the predator‐taxis coefficient in the predator–prey system may lead to the emergence of either subcritical or supercritical Turing bifurcation. Our numerical experiments confirm the theoretical findings and exhibit various spatial patterns with different values of predator‐taxis coefficients. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is mathematical analysis on time‐periodic flows of electrons and holes in semiconductors. The flows appear in a situation that alternating‐current voltages are applied to devices. In this paper, we study the drift‐diffusion model for semiconductors in a three‐dimensional bounded domain and investigate the existence and stability of time‐periodic solutions. We first derive uniform‐in‐time estimates of time‐global solutions and then prove by the relative entropy method that the difference of any two solutions decays exponentially fast as time tends to infinity. These facts enable us to show the unique existence and global stability of time‐periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, by using subsuper solutions method, we study the existence of weak positive solution for a class of Kirrchoff elliptic systems in bounded domains with multiple parameters. [ABSTRACT FROM AUTHOR]

This paper explores the dynamics analysis of a human immunodeficiency virus (HIV) model with saturated cytotoxic T lymphocyte (CTL) immune response and Beddington–DeAngelis infection rate. There are two time delays in the model to describe the time needed for infection of cell and CTL immune response generation, respectively. We obtain two thresholds and three possible equilibria from the model. By analyzing the corresponding characteristic equations, we study the stabilities of equilibrium and the effect of delays on CTL immune response. The results indicate that when immune delay is present, the steady state of equilibrium is disrupted and leads to a Hopf bifurcation. Finally, we use sensitivity analyses to show the effect of parameters on thresholds and numerical simulations to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Leslie–Gower predator–prey model in which the predator can only be captured when its population size exceeds a critical value; the mean growth rate of the prey in the absence of the predator is subject to a semi‐saturation rate that affects its birth curve, and the interaction between the two species is defined by a Holling II predation functional with alternative food for the predator. Since the proposed model is equivalent to a Filippov system, its mathematical analysis leads to a local study of the equilibria in each vector field corresponding to the proposed model, in addition to the study of the stability of its pseudo‐equilibria located on the curve separating the two vector fields. In particular, the model could have between one and three pseudo‐equilibria and at least one limit cycle surrounding one or two inner equilibria, locally unstable points. [ABSTRACT FROM AUTHOR]

This special issue of the Journal Mathematical Methods in the Applied Sciences on the topic of Modelling, Analysis, and Applications is devoted to the development of mathematical modelling, analysis, and applications, from theoretical and numerical perspectives, involving different applied sciences and engineering. It collects invited contributions, from selected quality full papers submitted to the International Conference on Mathematical Analysis and Applications in Science and Engineering (ICMA2SC2022). The conference took place in Porto, Portugal in 27–29 June 2022 (https://www.isep.ipp.pt/Page/ViewPage/ICMASC). [ABSTRACT FROM AUTHOR]

In this paper, we study an age‐structured reaction‐diffusion‐advection population model. First, we use a non‐densely defined operator to the linear age‐structured reaction‐diffusion‐advection population model in a patchy environment. By spectral analysis, we obtain the asynchronous exponential growth of the population model. Then we consider nonlinear death rate and birth rate, which all depend on the function related to the generalized total population, and we prove the existence of a steady state of the system. Finally, we study the age‐structured reaction‐diffusion‐advection population model in non‐autonomous situations. We give the comparison principle and prove the eventual compactness of semiflow by using integrated semigroup. We also prove the existence of compact attractors under the periodic situation. [ABSTRACT FROM AUTHOR]

This paper examines the effects of treatment on tumor‐immune dynamics based on a conceptual two‐dimensional model. In absence of treatment, this model exhibits regimes of "active" and "dormant" tumor. Increasing intensity of chemotherapy entails a significant complication of tumor‐immune interaction with parameter zones of monostability, bistability, and tristability where "active," "dormant," and "zero" tumor states can coexist. A detailed description of bifurcations, attractors, and their basins is given. Based on this mathematical analysis, we discuss efficiency of various programs of treatment combining chemotherapy and radiotherapy. [ABSTRACT FROM AUTHOR]

This paper deals with a certain condenser capacity in an anisotropic environment. More precisely, we are going to investigate a free boundary problem for a class of anisotropic equations on a ring domain [ABSTRACT FROM AUTHOR]

This paper is concerned with mathematical analysis for the viscoelastic von Karman equation with memory and time‐varying delay. Based on the existence and uniqueness of the strong solution established by Galerkin method, the finite dimensional global attractor for the system without time‐varying delay has been shown by using the quasi‐stability theory. The difficulties for our desired well‐posedness are the limiting of fourth‐order derivative terms and the von Karman bracket term, which need some delicate estimates to achieve. The key point for the dynamics is to obtain the quasi‐stability for gradient system, which needs the higher regularity of trajectory especially the von Karman bracket. [ABSTRACT FROM AUTHOR]

We introduce, in this paper, a delayed hybrid epidemic model describing the evolution of heroin addiction in a given population. The main objective of our mathematical analysis will be to provide the global behavior of the solutions. To the best of our knowledge, there are no results about the global dynamics of such a model containing both age structure and distributed time delay. We will prove that our model has threshold dynamics in terms of the basic reproduction number R0$$ {R}_0 $$. It will be established that for R0≤1$$ {R}_0\le 1 $$, the addiction‐free equilibrium is globally stable and for R0>1$$ {R}_0>1 $$, the endemic equilibrium is globally stable. The theoretical results are checked numerically. Especially, we focus on the influence of the addiction rate on the population dynamic. [ABSTRACT FROM AUTHOR]

Threshold noise reduction methods of vibration signals have been widely researched and used. However, these methods are less efficient in such situation, including requiring a time-consuming and subjective to manual editing because different degree of noise signal requires selecting different characterization for filtering. In this paper, an efficient denoising method based on PDE for mechanical vibration signals time-frequency distribution is investigated, in which, a one-dimensional vibration signal is transformed into 2D time-frequency domain by using Gabor transform. This enables (i) simultaneously utilize both time and frequency characteristic for effectively multiple dimension signal denosing and (ii) isotropic and anisotropic characteristics to be imposed by employing PDE, which explicitly fit with the local structure of time-frequency signal. This paper analyzes the basic methods of isotropic and anisotropic diffusion filtering, investigates the anisotropic diffusion method based on local feature structure of 2D information, and conducts a set of comparative tests. Experiments show that this proposed method has a better performance of denoising than that of thresholding. At the same time, it is more handy than that of other methods, such as independent component analysis. Finally, problems and ways of improving the PDE-based filter method are analyzed in this paper. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, using the nonlinear capacity method, we derive sufficient conditions for the nonexistence of global weak solutions for some differential inequalities of Sobolev type posed in an exterior domain of R N, N ≥ 3. The obtained results are extensions of those established by Korpusov and Sveshnikov in the case of the entire space R N. [ABSTRACT FROM AUTHOR]

In this paper, we make a rigorous mathematical analysis of the radiative heating of a semitransparent body made of a glass by a black radiative source surrounding it. This requires the study of the coupling between quasi‐steady radiative‐transfer boundary value problems with nonhomogeneous reflectivity boundary conditions (one for each wavelength band in the semitransparent electromagnetic spectrum of the glass) and a nonlinear heat‐conduction evolution equation with a nonlinear Robin boundary condition, which takes into account those wavelengths for which the glass behaves like an opaque body. We prove existence and uniqueness of the solution and give also uniform bounds on the solution, ie, on the absolute temperature distribution inside the body and on the radiative intensities. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a rumor propagation model with L évy 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. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of linear second-order matrix descriptor differential equations of Apostol-Kolodner type with constant coefficients. In the new approach, we propose a different transformation from what Kalogeropoulos et al. (2009) have used in their recent paper. However, similarly with them, the Weierstrass canonical form has been considered, and the analytical formula for the solution of this general class is derived naturally for consistent initial conditions. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence, nonexistence, uniqueness, and non-uniqueness of solutions for a singular parabolic equation in one dimension case. For different cases of some constant m of the equation, we study the uniqueness of solutions for m > − 1, the nonexistence of solutions for m ≤ − p, and the existence and non-uniqueness of solutions for − p < m ≤ − 1. The novelty of this paper lies in the study of nonexistence. Moreover, the other results of this paper extend some recent works. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Impulsive reaction–diffusion system is a novel assemblage of tools for better understanding ecological dynamics. This paper is devoted to mathematical and biological analysis of an impulsive tritrophic periodic plankton–fish system with diffusion and Beddington–DeAngelis functional response in lakes. On the basis of comparison theory and method of upper and lower solution, we have rigorously analyzed ultimate boundedness and permanence of the reaction–diffusion system under impulsive effects. Significantly, the existence and globally asymptotic stability of a unique positive periodic solution are verified by Brouwer's fixed point theorem and auxiliary function method. Additionally, two particular scenarios are chosen to illustrate our results. It is found that impulsive effects can influence lake ecological dynamics and regulate lake biological structure. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence of positive solutions for a fourth-order impulsive boundary value problem with integral boundary conditions on time scales. Existence results of at least two and three positive solutions are established via the double fixed point theorem and six functionals fixed point theorem, respectively. Also, an example is given to illustrate the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Ren and Zeng (2013) introduced a new kind of q-Bernstein-Schurer operators and studied some approximation properties. Acu et al. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q-Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K-functional. Next, we introduce the bivariate case of q-Bernstein-Schurer-Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K-functional. Finally, we define the generalized Boolean sum operators of the q-Bernstein-Schurer-Kantorovich type and investigate the approximation of the Bögel continuous and Bögel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

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

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

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

In this paper, Sturmian comparison theory is developed for the pair of second-order differential equations; first of which is the nonlinear differential equations of the form [ABSTRACT FROM AUTHOR]

In this paper, we study the following second-order dynamical system: [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method called the fractional natural decomposition method (FNDM). We give the proof of new theorems of the FNDM, and we extend the natural transform method to fractional derivatives. We apply the FNDM to construct analytical and approximate solutions of the nonlinear time-fractional Harry Dym equation and the nonlinear time-fractional Fisher's equation. The fractional derivatives are described in the Caputo sense. The effectiveness of the FNDM is numerically confirmed. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

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

This paper is concerned with a generalized Arzela-Ascoli's lemma, which has been extensively applied in almost periodic problems by the continuation theorem of degree theory. We give a counter example to show that this lemma is incorrect, and there is a gap in the proof of some existing literature, where the addressed generalized Arzela-Ascoli's lemma was used. Moreover, we make some final comments and introduce an open problem. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, three conservative finite volume element schemes are proposed and compared for the modif ied Korteweg-de Vries equation, especially with regard to their accuracy and conservative properties. The schemes are constructed basing on the discrete variational derivative method and the finite volume element method to inherit the properties of the original equation. The theoretical analysis show that three schemes are conservative under suitable boundary conditions as well as unconditionally linear stability. Numerical experiments are given to confirm the theoretical results and the capacity of the proposed methods for capturing the solitary wave phenomena. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, a nonperiodic discrete nonlinear equation with Jacobi operators is considered. By using the critical point theory, we establish some new sufficient conditions on the existence and multiplicity of homoclinic solutions. Recent results are generalized and significantly improved. Furthermore, our results greatly improve some existing ones even for some special cases. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we study the global existence and asymptotic behavior of the Boussinesq-Burgers system subject to the Dirichlet boundary conditions. Based on the L p( p > 2) estimates of the solution, which are different from the standard L2-based energy methods, we show that the classical solutions exist globally and converge to their boundary data at an exponential decay rate as time goes to infinity for large initial data. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, the issue of stability of multi-group coupled systems on networks with multi-diffusion (MCSNMs) is mainly analyzed. Utilizing graph theory, a novel and practical method of constructing a proper Lyapunov function for the MCSNMs is presented. Furthermore, based on the graph-theoretic approach and the proposed Lyapunov function, sufficient criteria, in the term of Lyapunov function and coefficients of the system, respectively, are derived to ensure the stability of the MCSNMs. Apart from accessibility to checking, the proposed results can generalize the corresponding results published in a previous time. Finally, the effectiveness and feasibility of the obtained results are demonstrated by a numerical example. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The fractional derivative with nonsingular kernel has been widely used to many physical fields which was shown to offer a new insight into the mathematical modeling of natural phenomena. In this paper, we first study the well‐posedness and regularity of the multidimensional variable‐order time‐fractional reaction–diffusion equation with Caputo–Fabrizio fractional derivative and then present and analyze a Galerkin finite element approximation to the proposed model based on the proved smoothing properties of the solutions. To improve the computational efficiency of the variable‐order Caputo–Fabrizio derivative, we approximate the kernel of the fractional derivative by the K‐term truncation of its Taylor expansion at a fixed fractional order to develop a fast evaluation scheme which significantly reduces the memory requirement from O(N) to O(K) and the computational complexity from O(N2) to O(K2N) where N refers to the number of time steps. We accordingly develop a fast Galerkin finite element method to the proposed model. Numerical experiments are carried out to substantiate the theoretical findings and to show the performance of the fast method. [ABSTRACT FROM AUTHOR]

In this paper, we study the properties of a large class of zeta functions that arises in geometric analysis and mathematical physics. They are attached to some elliptic operators. This method can be used to evaluate explicitly the special values of zeta functions of elliptic operators defined on some symmetric spaces. [ABSTRACT FROM AUTHOR]

In general, a proportional function is obviously not antiperiodic, yet a very interesting fact in this paper shows that it is possible there is an antiperiodic solution for some proportional delayed dynamical systems. We deal with the issue of antiperiodic solutions for RNNs (recurrent neural networks) incorporating multiproportional delays. Employing Lyapunov method, inequality techniques and concise mathematical analysis proof, sufficient criteria on the existence of antiperiodic solutions including its uniqueness and exponential stability are built up. The obtained results provide us some lights for designing a stable RNNs and complement some earlier publications. In addition, simulations show that the theoretical antiperiodic dynamics are in excellent agreement with the numerically observed behavior. [ABSTRACT FROM AUTHOR]

In this paper, we investigate with a time fractional‐order derivative in a three‐species predator‐prey model with the presence of prey social behavior. A new approximation for predator‐prey interaction in the presence of prey social behavior has been considered. For the model analysis, the study has been divided into two principal parts. First of all, we study the local stability of the equilibria and the existence of Hopf bifurcation. Then, for the numerical analysis, the Caputo fractional derivative operator is utilized to approximate the numerical solution of the model. An excellent agreement is seen between the numerical results and the theoretical predictions. [ABSTRACT FROM AUTHOR]

This paper is concerned with a reaction‐advection‐diffusion system, which is a continuous version of statistical model for criminal activity proposed by Short et al (Math. Models Methods Appl. Sci., 8 (2008), 1249‐1267). From very recently results, it is known that the classical solution of the corresponding initial‐boundary value problem exists globally. On the other hand, Rodriguez (Phys. D, 260 (2013), 191‐200) asserted the uniform‐in‐time boundedness of classical solution; however, there is a gap in the proof. In this work, we first solve this open problem by establishing some crucial a priori estimates. Then, invoking the uniform boundedness, we present that the classical solution converges exponentially to the steady state. Our results are consistent with the experiment observation and numerical simulation in the existing literatures. [ABSTRACT FROM AUTHOR]

This paper examines the initial-value problem for the nonhomogeneous incompressible nematic liquid crystals system with vacuum. This paper establishes two main results. The first result is involved with the global strong solutions to the 2D liquid crystals system in a bounded smooth domain. Our second result is concerned with the small data global existence result about the 3D system in the whole space. In addition, the local existence and a blow-up criterion of strong solutions are also mentioned. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we study the longtime dynamics of the non-autonomous Boussinesq-type equation with critical nonlinearity, and time-dependent external forcing, which is translation bounded but not translation compact. We prove the existence of a uniform attractor in [ABSTRACT FROM AUTHOR]

In this paper, we consider the coupled system of Kirchhoff-type equations: [ABSTRACT FROM AUTHOR]

In this paper, the stabilized mixed finite element methods are presented for the Navier‐Stokes equations with damping. The existence and uniqueness of the weak solutions are proven by use of the Brouwer fixed‐point theorem. Then, optimal error estimates for the H1‐norm and L2‐norm of the velocity and the L2‐norm of the pressure are derived. Moreover, on the basis of the optimal L2‐norm error estimate of the velocity, a stabilized two‐step method is proposed, which is more efficient than the usual stabilized methods. Finally, two numerical examples are implemented to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]