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A nonlinear cross-diffusion–fluid system with chemicals terms describing the dynamics of predator–prey living in a Newtonian fluid is proposed in this paper. The existence of weak solution for the proposed macro-scale system is proved on the basis of the Schauder fixed-point theory, a priori estimates, and compactness arguments. The proposed system is derived from the underlining description delivered by a kinetic-fluid theory model by multiscale approach. Finally, we discuss the computational results for the proposed macro-scale system in two dimensional space. [ABSTRACT FROM AUTHOR]

A novel messages spreading model is suggested in this paper. The model is a natural generalization of the SIS (susceptible-infective-susceptible) model, in which two relevant messages with same probability of acceptance may spread among nodes. One of the messages has a higher priority to be adopted than the other only in the sense that both messages act on the same node simultaneously. Node in the model is termed as supporter when it adopts either of messages. The transition probability allows that two kinds of supports may transform into each other with a certain rate, and it varies inversely with the associated levels which are discretely distributed in the symmetrical interval around original point. Results of numerical simulations show that individuals tend to believe the messages with a better consistency. If messages are conflicting with each other, the one with higher priority would be spread more and another would be ignored. Otherwise, the number of both supports remains at a uniformly higher level. Besides, in a network with lower connected degree, over a half of the individuals would keep neutral, and the message with lower priority becomes harder to diffuse than the prerogative one. This paper explores the propagation of multi-messages by considering their correlation degree, contributing to the understanding and predicting of the potential propagation trends. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution of fractional initial value problems (FIVP) in sense of Caputo’s definition for dynamical systems. Unlike for integer-order derivatives that have a single definition, there is more than one definition of non integer-order derivatives and the solution of an FIVP is definition-dependent. In this paper, the chief differences of the main definitions of fractional derivatives are revisited and a numerical algorithm to solve an FIVP for Caputo derivative is proposed. The main advantages of the algorithm are twofold: it can be initialized with integer-order derivatives, and it is faster than the corresponding standard algorithm. The performance of the proposed algorithm is illustrated with examples which suggest that it requires about half the computation time to achieve the same accuracy than the standard algorithm. [ABSTRACT FROM AUTHOR]

Highlight: [•] New results amend some oscillation criteria for dynamic equations on isolated time scales. [•] One can amend related oscillation results for differential equations using the method provided in this paper. [•] The results obtained in this paper complement oscillation theory of dynamic equations. [Copyright &y& Elsevier]

Highlights: [•] The paper presents new results on global stability of second-order interval CGNNs. [•] It is the first paper to study global stability of general second-order interval CGNNs. [•] The new results can also be used to verify the stability of first-order interval CGNNs. [ABSTRACT FROM AUTHOR]

In this paper, we propose an artificial neural network approach to identify the parameter in a non-linear subdiffusion model from additional data. Instead of determining the parameter in the time fractional diffusion model by its form itself, we approximate it in the form of an artificial neural network. The key point of this approach relies on the approximation capability of neural networks. We formulate this inverse problem as an optimal control one, and we demonstrate the existence of the solution for the control problem and provide a mathematical analysis and the derivation of optimal conditions. Moreover, various numerical tests of the regular and singular examples have shown that the artificial neural network method (ANN) is effective. This is reinforced by its numerical comparison with the gradient descent, the alternating direction multiplier method (ADMM), physics-informed neural network (PINN) and DeepONet method. • A new formulation of the inverse problem for a nonlinear subdiffusion model based on an artificial neural network. • Theoretical analysis of the inverse problem by optimal control formulation. • Proposition of an approach for solving the inverse problem by the gradient descent algorithm. • A qualitative and quantitative comparative study with existing methods. [ABSTRACT FROM AUTHOR]

• The paper proposes an overview of the modeling approaches for the mathematical description and analysis of processes that combine continuous and discontinuous behavior, namely impulsive differential equations, hybrid dynamical systems, and differential equations involving Dirac delta functions. • Insights are provided on the stability and attractivity analysis of hybrid behaviors, and essential differences are highlighted to the respective stability concepts for smooth dynamical systems. • Specific phenomena are discussed which are peculiar for hybrid behaviors, like beating or Zeno phenomenon, modeling of multiple impulses at a single time instance, death and splitting of solutions, etc. • With this, the paper attempts at bringing attention of the interested researchers to the methods available in other research communities and fostering the exchange of ideas and analysis techniques. We propose an overview of the modeling approaches for the mathematical description and analysis of processes that combine continuous and discontinuous behavior, namely impulsive differential equations, hybrid dynamical systems, and differential equations involving Dirac delta functions. These classes of systems are chosen due to their dominant prevalence in physics, mathematics, and control engineering research communities. A comparison of these frameworks is provided and their applicability depending on the character of the hybrid behavior is discussed. In particular, we show that special care should be taken when equations with Dirac delta function are interpreted as impulsive differential equations. We also provide insights on the stability and attractivity analysis of hybrid behaviors, highlight their essential differences to the respective stability concepts for smooth dynamical systems, and discuss specific phenomena which are peculiar for hybrid behaviors, like beating or Zeno phenomenon, modeling of multiple impulses at a single time instance, death and splitting of solutions, etc. With this, the paper attempts at bringing attention of the interested researchers to the methods available in other research communities and fostering the exchange of ideas and analysis techniques. [ABSTRACT FROM AUTHOR]

Abstract: A new model called Naming Game with Multiple Hearers (NGMH) is proposed in this paper. A naming game over a population of individuals aims to reach consensus on the name of an object through pair-wise local interactions among all the individuals. The proposed NGMH model describes the learning process of a new word, in a population with one speaker and multiple hearers, at each interaction towards convergence. The characteristics of NGMH are examined on three types of network topologies, namely ER random-graph network, WS small-world network, and BA scale-free network. Comparative analysis on the convergence time is performed, revealing that the topology with a larger average (node) degree can reach consensus faster than the others over the same population. It is found that, for a homogeneous network, the average degree is the limiting value of the number of hearers, which reduces the individual ability of learning new words, consequently decreasing the convergence time; for a scale-free network, this limiting value is the deviation of the average degree. It is also found that a network with a larger clustering coefficient takes longer time to converge; especially a small-word network with smallest rewiring possibility takes longest time to reach convergence. As more new nodes are being added to scale-free networks with different degree distributions, their convergence time appears to be robust against the network-size variation. Most new findings reported in this paper are different from that of the single-speaker/single-hearer naming games documented in the literature. [Copyright &y& Elsevier]

Abstract: In this letter, we show that the main results of the reply paper [1] are wrong. We demonstrate that the authors of Delavari et al. (2012) [1] have carried out an essential flaw in the proof approach of the system stability. Therefore, we prove that the defects of the paper [2] which have been described in the comment paper [3] are still continued. In this regard, we conclude that the results of our comment paper [3] are correct. [Copyright &y& Elsevier]

Abstract: In this paper, the problem of delay-dependent passivity analysis is studied for the singular Markov jump systems with time-varying/time-invariant delays. By use of the delay partitioning method, two delay-dependent passivity conditions are derived for the considered systems via two novel Lyapunov functionals including mode-dependent double integral terms. Some stochastic stability criteria are also given. All the results reported in this paper not only depend upon the time-delays, but also depend upon their partitioning, which aims at reducing the conservatism. Four numerical examples are proposed to show that the methods proposed here are less conservative than the existing ones thanks to the usage of the delay partitioning method and novel Lyapunov functionals. [Copyright &y& Elsevier]

Abstract: In this paper the original method of transformation of one random function to another one is suggested. The problem of transformation of one random function to another one is based on the NAFASS approach suggested previously by one of the authors (RRN) in paper [1]. The problem can be formulated as follows: is it possible to transform one random function to another one (the functional forms of the both functions are not known) during the fixed segment of time t 1? The solution of this problem shown in this paper gives a chance to manage with random functions that describe many complex systems, where the adequate model pretending on their functional or analytical description is not known. This transformation based on the successful solution of the Prony’s problem gives unique chances to manage with some chemical processes, technological processes and understand better the general behavior of the different complex systems which cannot be managed by the human being. Besides this solution another solution of this problem related to control of detrended random sequences is considered also. [Copyright &y& Elsevier]

Abstract: The aim of this letter is to confirm the achievements in and answer all the mentioned comments in . Meanwhile some other drawbacks in the comment paper are also presented in this paper. [Copyright &y& Elsevier]

Abstract: This paper is motivated from some recent papers treating the problem of the existence of a solution for impulsive differential equations with fractional derivative. We firstly show that the formula of solutions in cited papers are incorrect. Secondly, we reconsider a class of impulsive fractional differential equations and introduce a correct formula of solutions for a impulsive Cauchy problem with Caputo fractional derivative. Further, some sufficient conditions for existence of the solutions are established by applying fixed point methods. Some examples are given to illustrate the results. [Copyright &y& Elsevier]

Abstract: We slightly modify definition in paper [Xu Y, Zhou W, Fang J, Sun W. Topology identification and adaptive synchronization of uncertain complex networks with adaptive double scaling functions. Commun Nonlinear Sci Numer Simulat 2011;16:3337–3343], so remark stated problem is resolved in paper . Finally, a more general theorem is given. [Copyright &y& Elsevier]

Abstract: In this paper, a new epidemic SIS model with nonlinear infectivity, as well as birth and death of nodes and edges, is investigated on heterogeneous networks. The global behavior of the model is studied mathematically. When the basic reproductive number is less than or equal to unity, it is verified that the disease dies out; otherwise, the model solutions lead to positive steady states. This paper provides a concise mathematical analysis to verify the global dynamics of the model. [Copyright &y& Elsevier]

Abstract: This paper deal with the study of the existence of solutions of nonlinear fractional integrodifferential equations with impulsive conditions in Banach spaces, by means of fixed point principle. The purpose of the paper is to generalize the results of and correct an error in the example included in . [Copyright &y& Elsevier]

Abstract: In this paper, we consider an almost periodic Schoener’s competition model with delays and impulsive effects. Sufficient conditions which guarantee the permanence of the model and the existence of a unique uniformly asymptotically stable positive almost periodic solution are obtained. The result of this paper is completely new. An suitable example is employed to illustrate the feasibility of the main results. [Copyright &y& Elsevier]

Highlights • Mathematical model for low grade gliomas that fits well to medical data. • Net growth coefficient is split in proliferation and natural death rates. • Global stability of tumour free equilibrium under suitable assumptions is proved. • Sensitivity analysis reveals the impact of model parameters on the solution. Abstract Gliomas are the most frequent type of primary brain tumour. Low-grade gliomas (LGGs) in particular are infiltrative and incurable with a slow evolution that eventually causes death. In this paper, we propose a mathematical model for the growth of LGGs and its response to chemotherapy. We validate our model with medical data and show that the proposed model describes real patients' data quite well. A mathematical analysis of the model shows the existence of a unique non-negative solution. We further investigate the stability of steady-state solutions. In particular, we demonstrate the global stability of a tumour-free equilibrium in the case of sufficiently strong constant and asymptotically periodic treatment. A sensitivity analysis of the model indicates that the proliferation rate has the biggest impact on solutions of the model. We also numerically investigate the stability of the fitting procedure. [ABSTRACT FROM AUTHOR]

Highlights • A new extended Painlevé hierarchy. • Bäcklund and auto-Bäcklund transformations, nesting, and other properties. • Importance of underlying structure of equations. Abstract In a recent paper we introduced an extended second Painlevé hierarchy and studied its properties. The approach developed in order to derive these results is widely applicable. Here we use it to obtain a second example of an extended Painlevé hierarchy. We also give results on Bäcklund transformations, auto-Bäcklund transformations and other properties of this and related hierarchies of ordinary differential equations, as well as on the nesting of equations whereby we obtain relations between systems of different orders but of the same form. [ABSTRACT FROM AUTHOR]

Highlights • We proposed a class of improved stochastic theta methods, driven by the error corrected and exponential error corrected ODEs solvers, to solve a general class of linear and nonlinear stochastic differential equations. • Using the Itô–Taylor expansion under the Lipschitz conditions and linear growth bounds, we analyzed the mean–square convergence of the proposed scheme in the strong sense. • We investigated the numerical stability properties of a linear test equation with real parameters, based on the Descarte's rule of signs, and sufficient conditions for the mean–square stability of solutions are provided. • Numerical examples are reported to confirm the theoretical results, and to illustrate the efficiency of the proposed methods for solving one and two dimensionals stochastic differential equations. Abstract In this paper, we present a family of stochastic theta methods modified by ODEs solvers for stochastic differential equations. This class of methods constructed by adding error correction and exponential error correction terms to the traditional stochastic theta methods. Using the Itô–Taylor expansion, analyzed mean-square convergence under the Lipschitz conditions and linear growth bounds. Also, we concern mean-square stability analysis of our proposed methods. Numerical examples are presented to demonstrate the efficiency of these methods for the pathwise approximation solution of some stochastic differential equations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study and classify the conservation laws of the combined nonlinear KdV, Camassa–Holm, Hunter–Saxton and the inviscid Burgers equation which arises in, inter alia, shallow water equations. It is shown that these can be obtained by variational methods but the main focus of the paper is the construction of the conservation laws as a consequence of the interplay between symmetry generators and ‘multipliers’, particularly, the higher-order ones. [Copyright &y& Elsevier]

Abstract: This paper discusses topology identification and adaptive synchronization of uncertain complex networks with adaptive the scaling functions. In comparison with those of the existing scaling function synchronization, the scaling function can be identified by adaptive laws in this paper. Moreover, the uncertain network topological structure are identified simultaneously in the process of synchronization. Illustrative examples are presented to demonstrate the application of the theoretical results. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the adaptive synchronization problem for a class of stochastic delayed neural networks. Based on the LaSalle invariant principle of stochastic differential delay equations and the stochastic analysis theory as well as the adaptive feedback control technique, a linear matrix inequality approach is developed to derive some novel sufficient conditions achieving complete synchronization of unidirectionally coupled stochastic delayed neural networks. In particular, the synchronization criterion considered in this paper is the globally almost surely asymptotic stability of the error dynamical system, which has seldom been applied to investigate the synchronization problem. Moreover, the delays proposed in this paper are time-varying delays and distributed delays, which have rarely been used to study the synchronization problem for coupled stochastic delayed neural networks. Therefore, the results obtained in this paper are more general and useful than those given in the previous literature. Finally, two numerical examples and their simulations are provided to demonstrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are concerned with the stochastic differential delay equations with Poisson jump (SDDEsPJ). As stochastic differential equations, most SDDEsPJ cannot be solved explicitly. Therefore, numerical solutions have become an important issue in the study of SDDEsPJ. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEsPJ when the drift and diffusion coefficients are Taylor approximations. [Copyright &y& Elsevier]

Abstract: In this paper, the homotopy analysis method (HAM) is applied to numerically approximate the eigenvalues of the second and fourth-order Sturm–Liouville problems. These eigenvalues are calculated by starting the HAM algorithm with one initial guess. In this paper, it can be observed that the auxiliary parameter ℏ, which controls the convergence of the HAM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more important qualitative difference in analysis between HAM and other methods. [ABSTRACT FROM AUTHOR]

Abstract: The above mentioned paper contains some fundamental mistakes and misinterpretations along with a false conclusion. Applying the optimal homotopy asymptotic method (OHAM) in an incorrect manner, Niu and Wang have drawn the false conclusion that this approach is not efficient in practice because it is time-consuming for high-order of approximation. We emphasized the presence of some evident mistakes and misinterpretations in their paper and we proved that OHAM is very efficient in practice since we solved all three examples analyzed by Niu and Wang using only the first-order of approximation, which yields accurate results. We demonstrate that OHAM does not need high-orders of approximation as Niu and Wang suggests and we show that the main strength of OHAM is its rapid convergence, contradicting Niu and Wang’s assumption. [Copyright &y& Elsevier]

Abstract: In this paper, we study the characteristic exponent for a second order linear differential equation if . As an application, we calculate the characteristic exponents of Eq. when . The obtained results can be seen as a good generalization of Shi and coworkers . We end this paper with some examples which show the simplicity and the high accuracy of our result. [Copyright &y& Elsevier]

Abstract: Based on a new approach presented in detail in this paper one can find new evidences of existence of the fractional kinetics not only in the frequency range. One can find rather general principles of detection of different collective motions in temperature region. These principles can be expressed in terms of an algorithm (defined in the paper as an approach). This approach includes some steps that help to separate a couple of the neighboring collective motions (expressed in the frequency range as a linear combination of two power-law exponents) from each other and establish the temperature evolution of the extreme point that follows to the generalized Vogel–Fulcher–Tamman (VFT)-equation. This experimentally confirmed fact gives new evidences for supporting of the theory of dielectric relaxation based on the fractional kinetics on the frequency/temperature domain. As an example for verification of this new approach the ibuprofen complex permittivity data measured in the wide frequency/temperature range were chosen. The reason of such selection was the following. It helps to compare the conventional study of this complex substance recently published in and use possibilities of the developed approach that can add some new features to the picture obtained in the frame of the conventional treatment. We suppose that possibilities presented by new approach will be extremely useful for detection of different collective motions in other substances studied by the method of broadband dielectric spectroscopy (BDS). [Copyright &y& Elsevier]

Abstract: The -expansion method can be used for constructing exact travelling wave solutions of real nonlinear evolution equations. In this paper, we improve the -expansion method and explore new application of this method to the complex KdV equation. New types of exact travelling wave solutions of the complex KdV equation are found. Some exact solutions of the complex KdV equation obtained before are special cases of our results in this paper. [Copyright &y& Elsevier]

Abstract: The sign of unknown input coefficients is assumed to be known in most papers about the input uncertainties. In this paper, a Nussbaum gain method is adopted to cope with the situation that both the sign and the value of input are unknown. And the unknown parameters can be estimated under the situation of unknown sign of control. The synchronization is achieved for a class of hyperchaotic systems with unknown parameters and input uncertainties by adopting of the Nussbaum gain method and the global terminal adaptive method. And the conclusions are made as follows: First, the proposed method is effective in the situation that the sign of input is unknown. Second, the estimation of unknown parameters can be achieved only when the number of unknown parameters satisfied some condition and no uncertainty exist in the input of systems. Third, the unknown parameters cannot be estimated correctly with common adaptive method when there are input uncertainties in the system. But the Nussbaum gain method can get good result in the estimation of unknown parameters. At last, numerical simulations are done to show the effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: This paper obtains the adiabatic variation of the soliton velocity, in presence of perturbation terms, of the phi-four model and the nonlinear Klein–Gordon equations. There are three types of models of the nonlinear Klein–Gordon equation, with power law nonlinearity, that are studied in this paper. The soliton perturbation theory is utilized to carry out this investigation. [Copyright &y& Elsevier]

Abstract: The many types of public sheet-handling machines must handle sheets (e.g. tickets, sheet of paper) under various conditions at high speed. It is therefore necessary to understand the sheet’s behavior in order to avoid problems associated with handling it. This paper develops a nonlinear simulation that determines the behavior of a sheet handled in a machine. The simulation is based on a discretized model of the sheet using lumped masses and springs. The simulation results are validated with experimental data from a sheet-handling machine fed with various types of sheets. The experimental results agree well with simulation, showing that this method is useful and practical to predict the behavior of flexible sheets handling in a feeding machine. [Copyright &y& Elsevier]

Abstract: This paper studies the adiabatic dynamics of topological solitons in presence of perturbation terms. The solitons due to sine-Gordon equation, double sine-Gordon equation, sine–cosine Gordon equation and double sine–cosine Gordon equations are studied, in this paper. The adiabatic variation of soliton velocity is obtained in this paper by soliton perturbation theory. [Copyright &y& Elsevier]

Abstract: In this paper, the definition of finite-time control is presented. The system under consideration is subject to time-varying norm-bounded exogenous disturbance. The main aim of this paper is focused on the design a state feedback controller which ensures that the closed-loop system is finite-time bounded (FTB) and reduces the effect of the disturbance input on the controlled output to a prescribed level. A sufficient condition is presented for the solvability of this problem, which can be reduced to a feasibility problem involving linear matrix inequalities (LMIs). A detailed solving method is proposed for the restricted linear matrix inequalities. Finally, examples are given to show the validity of the methodology. [Copyright &y& Elsevier]

Abstract: In this paper, the exact solution of average path length in Barabási–Albert model is given. The average path length is an important property of networks and attracts much attention in many areas. The Barabási–Albert model, also called scale free model, is a popular model used in modeling real systems. Hence it is valuable for us to examine the average path length of scale free model. There are two answers, regarding the exact solution for the average path length of scale free networks, already provided by Newman and Bollobas respectively. As Newman proposed, the average path length grows as log(n) with the network size n. However, Bollobas suggested that while it was true when m =1, the answer changed to log(n)/log(log(n)) when m >1. In this paper, as we propose, the exact solution of average path length of BA model should approach log(n)/log(log(n)) regardless the value of m. Finally, the simulation is presented to show the validity of our result. [Copyright &y& Elsevier]

The unbalanced magnetic pull results in the nonlinearity of the system for the permanent magnet synchronous motor rotor having mixed eccentricity. The mass unbalance and static eccentricity form multi-frequency excitation, which results in the coupling effect of forward and backward whirling motions on the steady-state of super harmonic resonance. The analysis is difficult due to the phases of complex amplitudes of two motions which is no longer time invariant compared with the main resonance. To solve this problem, an analysis method of complex amplitudes is presented in this paper. The rotor is considered a Jeffcott rotor, and multi-scale method is applied to this system. A mathematical analysis method of the complex amplitude phases is employed for their time variant characteristics. The results show that both phases are linear time variant and their variation rates are both equal in magnitude to the difference between twice the excitation frequency and natural frequency of the generating system. The characteristics lead to the frequency shift of the two motions by the difference value from the natural frequency of the generating system. The numerical analysis reveals that the phases are independent of the initial condition and hence the complex amplitudes. The analysis solves two problems: the analytical solution of the rotor system carrying phases information can be expressed, which is in good agreement with the numerical result, and the solution construction is analyzed easily, which indicates the contribution of every component of the analytical solution. The phase characteristics analysis is the prerequisite for the frequency characteristics which can be easily carried out, and the coupling mechanism of the two motions is explained. • A mathematical analysis method is presented to solve the problem of the time variant complex amplitude characteristics. • The mode coupling mechanisms of the forward and backward whirling motions, mass unbalance and static eccentricity are explained. • The frequency shift is discovered which is caused by linear time variant phase of the complex amplitude. [ABSTRACT FROM AUTHOR]

• Numerical analysis of the condition equations with the exact factorization of wavefunctions to molecular time-dependent Schrödinger equations. • Identification of mathematical issues. • Development of possible cures. In this paper, we are interested in the numerical analysis of the conditional and marginal equations within the Exact Factorization (EF) of wavefunctions to molecular time-dependent Schrödinger equations. After an analysis of toy-versions of the conditional equations, we provide a detailed mathematical analysis of elaborated numerical methods approximating this equation. The purpose of the paper is hence to provide some useful and precise informations on the EF from the computational point of view. Some illustrating numerical computations are provided along the paper. [ABSTRACT FROM AUTHOR]

In this paper, we suggest an efficient method, based on the well-known multi-step differential transform method to considerably reduce the number of arithmetic operations of differential transform method. The proposed method is heavily depended on the solution of two nonlinear systems which are exactly solved and the closed-form expressions are derived, fortunately. The present method is suitable for solving the governing equations of oscillatory systems. This fact is thoroughly shown by several nonlinear numerical examples. Moreover, the number of arithmetic operations is calculated for all methods implemented in the article, i.e., the proposed method and two previous methods, and it is clearly illustrated that the present method is really efficient. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the global stability of classical solutions to an initial and boundary value problem (IBVP) of a nonlinear PDE system, converted from a model of reinforced random walks, subject to time-dependent boundary conditions. It is shown that under certain integrability conditions on the boundary data, classical solutions to the IBVP exist globally in time and the differences between the solutions and their corresponding ansatz, determined by the initial and boundary conditions, converge to zero, as time goes to infinity. Though the final states of the boundary functions are required to match, the boundary values do not necessarily equal to each other at any finite time. In addition, there is no smallness restriction on the magnitude of the initial perturbations. Furthermore, numerical simulations are performed to investigate the long-time dynamics of the IBVP when the final states of the boundary functions do not match or the boundary functions are periodic in time. • We studied the global stability of classical solutions to an initial and boundary value problem. • Under certain conditions on the boundary data, classical solutions exist globally in time. • The differences between the solutions and their corresponding ansatz converge to zero. • There is no smallness restriction on the magnitude of the initial perturbations. • Numerical simulations are consistent with mathematical analysis. [ABSTRACT FROM AUTHOR]

The aim of this paper is to design a rescaling algorithm for the numerical solution to the system of two porous medium equations defined on two different components of the real line, that are connected by the nonlinear contact condition. The algorithm is based on the self-similarity of solutions on different scales and it presents a space-time adaptable method producing more exact numerical solution in the area of the interface between the components, whereas the number of grid points stays fixed. [ABSTRACT FROM AUTHOR]

Our goal of this paper is of a purely theoretical question, however which would be fundamental in computational partial differential equations: Can a linear solution-structure for the equation of motion for an infinite nonlinear beam be directly manipulated for constructing its nonlinear solution? Here, the equation of motion is modeled as mathematically a fourth -order nonlinear partial differential equation. To answer the question, a pseudo -parameter is firstly introduced to modify the equation of motion. And then, an integral formalism for the modified equation is found here, being taken as a linear solution-structure. It enables us to formulate a nonlinear integral equation of second kind, equivalent to the original equation of motion. The fixed point approach, applied to the integral equation, results in proposing a new iterative solution procedure for constructing the nonlinear solution of the original beam equation of motion, which consists luckily of just the simple regular numerical integration for its iterative process; i.e., it appears to be fairly simple as well as straightforward to apply. A mathematical analysis is carried out on both natures of convergence and uniqueness of the iterative procedure by proving a contractive character of a nonlinear operator. It follows conclusively , therefore , that it would be one of the useful nonlinear strategies for integrating the equation of motion for a nonlinear infinite beam, whereby the preceding question may be answered. In addition, it may be worth noticing that the pseudo-parameter introduced here has double roles; firstly, it connects the original beam equation of motion with the integral equation, second, it is related with the convergence of the iterative method proposed here. [ABSTRACT FROM AUTHOR]

Abstract: By means of a so-called extended homoclinic test method, Guo et al. (2009) obtained exact periodic solitary-wave solutions of the modified Korteweg-de Vries (MKdV) equation. We point out that the bilinear form of MKdV equation given in above paper is not true, so that the obtained solutions in the above paper are not actual solutions of the equation. In this paper, we apply an extended ansatz to obtain some new exact periodic solitary-wave solutions to MKdV equation. [Copyright &y& Elsevier]

This paper deals with a reaction–diffusion–advection system that characterizes the interactions between predators and prey in open advective environments, subject to an unidirectional flow, such as streams or rivers. To explore the influence of advection rates on the dynamics of this system, we study its dynamical classification by taking advection rates of two species as the variable parameters. It turns out that there exist some critical curves in the parameter plane of advection rates, which classify the dynamics of this system into several scenarios: (i) coexistence, (ii) persistence of prey only, (iii) persistence of predators only, (iv) extinction of both species. Furthermore, some qualitative properties of these critical curves are given by rigorous mathematical analysis and numerical computations. Our analytical and numerical results show that predators can invade successfully when they take relatively small advection rates. Especially, specialist predators always coexist with the prey when they invade successfully because they must keep pace with the prey for food. Nevertheless, generalist predators can take over the habitats when the prey's advection rate is suitably large, which implies that the prey's advection can facilitate the invasion of generalist predators. In addition, we numerically observe that the boundary loss rate has a significant influence on the dynamical classification. All of these results provide us with some deep insights into the dynamics of this system. • Dynamical classification of a general predator–prey model in open advective environments is established in the parameter plane of advection rates. • There exist some critical curves, which classify the dynamics of this system into several scenarios. • Some qualitative properties of these critical curves are given by rigorous mathematical analysis and numerical computations. • The results show that predators are more likely to invade when their advection rates are relatively small, and the prey's advection can facilitate the invasion of generalist predators. • The boundary loss rate has a significant influence on the dynamical classification. [ABSTRACT FROM AUTHOR]

In this paper, a novel type of network called network sharing common nonlinearity comprising both autonomous and non-autonomous oscillators have been investigated. We propose that these networks are robust for operating at desired modes i.e., chaotic or periodic by altering the v–i characteristics of common nonlinear element alone. The dynamics of these networks were examined through numerical, analytical, experimental and Multisim simulations. [ABSTRACT FROM AUTHOR]

This paper studies the collective behavior in a network of nonlinear systems with antagonistic interactions and switching topologies. The concept of modulus synchronization is introduced to characterize the case that the moduli of corresponding components of the agent (node) states reach a synchronization. The network topologies are modeled by a set of directed signed graphs. When all directed signed graphs are structurally balanced and the nonlinear system satisfies a one-sided Lipschitz condition, by using matrix measure and contraction theory, we show that modulus synchronization can be evaluated by the time average of some matrix measures. These matrices are about the second smallest eigenvalue of undirected graphs corresponding to directed signed graphs. Finally, we present two numerical examples to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper extends a homogenous network model proposed by Haldar and Mishra (2014) into a heterogeneous one by taking into consideration the topology property of the Internet. The dynamics of this new model are investigated by studying the stability of its equilibria using mathematical methods. The qualitative analyses show that, because of the effect of the Internet topology, the results of the model exhibit several distinct features as compared to those of the original model. Some numerical experiments are also conducted to account for the potential scenarios of the model. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic autonomous mutualism model. The local and global existence and uniqueness of the positive solution are derived with any positive initial value. Then we obtain that the positive solution of the system is stochastically ultimate bounded. Furthermore, we establish sufficient conditions for stochastic permanence and extinction of the system. The sufficient criterion for the system to be persistent in the mean are established as well. [ABSTRACT FROM AUTHOR]

In this paper the generalized differential transform method is applied to obtain an approximate solution of linear and nonlinear differential equation of fractional order with boundary conditions. Several numerical examples are considered and comparisons with the existing solution techniques are reported. Results show that the method is effective, easier to implement and very accurate when applied for the solution of fractional boundary values problems. [ABSTRACT FROM AUTHOR]

In this paper, we present a new SIS model with delay on scale-free networks. The model is suitable to describe some epidemics which are not only transmitted by a vector but also spread between individuals by direct contacts. In view of the biological relevance and real spreading process, we introduce a delay to denote average incubation period of disease in a vector. By mathematical analysis, we obtain the epidemic threshold and prove the global stability of equilibria. The simulation shows the delay will effect the epidemic spreading. Finally, we investigate and compare two major immunization strategies, uniform immunization and targeted immunization. [ABSTRACT FROM AUTHOR]

In this paper we systematically investigate the regulation of self-sustained target waves in excitable small-world networks (ESWNs) based on the dominant phase-advanced driving (DPAD) method. Three aspects of regulation, oscillation period regulation, oscillation center regulation and oscillation suppression, have been studied in detail and the following results are obtained: (i) The oscillation period can be regulated by manipulating the length of oscillation source within certain limits. (ii) Arbitrary oscillation center regulation can be achieved successfully. (iii) Two necessary steps are revealed to suppress the oscillation effectively. [ABSTRACT FROM AUTHOR]