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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]

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]

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]

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]

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]

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 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 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]

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]

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]

• 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, pinning combining with impulsive control scheme is adopted to investigate the synchronization of complex-variable dynamical network. Based on the Lyapunov function method and mathematical analysis technique, sufficient conditions for achieving synchronization is first analytically derived. This result extends the condition derived for real-variable dynamical network to complex-variable network. Further, adaptive strategy is adopted to relax the restrictions on the impulsive intervals and reduce the control cost. Noticeably, the proposed adaptive pinning impulsive control scheme is universal for different dynamical networks to some extent. The impulsive instants are chosen by solving a series of maximum problems subject to the derived conditions. Several numerical simulations are performed to illustrate the effectiveness and correctness of the derived theoretical results. [ABSTRACT FROM AUTHOR]

• We give a (wider) vision of the state of the art with new references commented in the introduction. • We obtain a reduced model for elliptic membrane shells in normal damped response contact in thermoelasticity. • We show existence and uniqueness of both the three-dimensional and the two-dimensional limit problems. • We use the asymptotic expansion method to obtain the model. • We justify the validity of the model by providing a strong convergence result. The purpose of this paper is twofold. We first provide the mathematical analysis of a dynamic contact problem in thermoelasticity, when the contact is governed by a normal damped response function and the constitutive thermoelastic law is given by the Duhamel-Neumann relation. Under suitable hypotheses on data and using a Faedo-Galerkin strategy, we show the existence and uniqueness of solution for this problem. Then, we study the particular case when the deformable body is, in fact, a shell and use asymptotic analysis to study the convergence to a 2D limit problem when the thickness tends to zero. [ABSTRACT FROM AUTHOR]

• Stochastic noises are introduced into the model of Alzheimer's disease. • Analytic conditions for stochastic P-bifurcation are obtained. • A formula is presented for the mean switching time from a mild impairment state to a pathological state. • A disease index is proposed for the early warning of disease. • The results provide insightful suggestions to design he strategies to slow the progression of disease. Alzheimer's disease is a worldwide disease of dementia and is characterized by beta-amyloid plaques. Increasing evidences show that there is a positive feedback loop between the level of beta-amyloid and the level of calcium. In this paper, stochastic noises are incorporated into a minimal model of Alzheimer's disease which focuses upon the evolution of beta-amyloid and calcium. Mathematical analysis indicates that solutions of the model without stochastic noises converge either to a unique equilibrium or to bistable equilibria. Analytical conditions for the stochastic P-bifurcation are derived by means of technique of slow-fast dynamical systems. A formula is presented to approximate the mean switching time from a normal state to a pathological state. A disease index is also proposed to predict the risk to transit from a normal state to a disease state. Further numerical simulations reveal how the parameters influence the evolutionary outcomes of beta-amyloid and calcium. These results give new insights on the strategies to slow the development of Alzheimer's disease. [ABSTRACT FROM AUTHOR]

In this note, some comments on the paper [1] are made. [ABSTRACT FROM AUTHOR]