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Introduction: The moment quantities associated with the nonlinear Schrödinger equation offer important insights into the evolution dynamics of such dispersive wave partial differential equation (PDE) models. The effective dynamics of the moment quantities are amenable to both analytical and numerical treatments. Methods: In this paper, we present a data-driven approach associated with the "Sparse Identification of Nonlinear Dynamics" (SINDy) to capture the evolution behaviors of such moment quantities numerically. Results and Discussion: Our method is applied first to some well-known closed systems of ordinary differential equations (ODEs) which describe the evolution dynamics of relevant moment quantities. Our examples are, progressively, of increasing complexity and our findings explore different choices within the SINDy library. We also consider the potential discovery of coordinate transformations that lead to moment system closure. Finally, we extend considerations to settings where a closed analytical form of the moment dynamics is not available. [ABSTRACT FROM AUTHOR]

IntroductionThe moment quantities associated with the nonlinear Schrödinger equation offer important insights into the evolution dynamics of such dispersive wave partial differential equation (PDE) models. The effective dynamics of the moment quantities are amenable to both analytical and numerical treatments.MethodsIn this paper, we present a data-driven approach associated with the “Sparse Identification of Nonlinear Dynamics” (SINDy) to capture the evolution behaviors of such moment quantities numerically.Results and DiscussionOur method is applied first to some well-known closed systems of ordinary differential equations (ODEs) which describe the evolution dynamics of relevant moment quantities. Our examples are, progressively, of increasing complexity and our findings explore different choices within the SINDy library. We also consider the potential discovery of coordinate transformations that lead to moment system closure. Finally, we extend considerations to settings where a closed analytical form of the moment dynamics is not available.

Traveling Langmuir waves are numerically simulated in a warm plasma using a kinetic description of the process. Waves of this type can be excited by a powerful short laser pulse (the so-called wake wave). It is shown that the propagation velocities of perturbations at the moments of the distribution function of the second and third orders significantly exceed the propagation velocity of the charge density wave. The comparison of kinetic and hydrodynamic models on the example of the considered problem is the important aspect of this study. It is shown that the hydrodynamic model without taking into account the heat flux does not give a good approximation to the solution of the kinetic Vlasov equation. As a result, the closure problem for moment equations remains open. [ABSTRACT FROM AUTHOR]

Understanding the interplay between ecological processes and the evolutionary dynamics of quantitative traits in natural systems remains a major challenge. Two main theoretical frameworks are used to address this question, adaptive dynamics and quantitative genetics, both of which have strengths and limitations and are often used by distinct research communities to address different questions. In order to make progress, new theoretical developments are needed that integrate these approaches and strengthen the link to empirical data. Here, we discuss a novel theoretical framework that bridges the gap between quantitative genetics and adaptive dynamics approaches. 'Oligomorphic dynamics' can be used to analyse eco‐evolutionary dynamics across different time scales and extends quantitative genetics theory to account for multimodal trait distributions, the dynamical nature of genetic variance, the potential for disruptive selection due to ecological feedbacks, and the non‐normal or skewed trait distributions encountered in nature. Oligomorphic dynamics explicitly takes into account the effect of environmental feedback, such as frequency‐ and density‐dependent selection, on the dynamics of multi‐modal trait distributions and we argue it has the potential to facilitate a much tighter integration between eco‐evolutionary theory and empirical data. Understanding the interplay between ecological processes and the evolutionary dynamics of quantitative traits in natural systems remains a major challenge. Here, we discuss the 'oligomorphic dynamics' framework that bridges the gap between quantitative genetics and adaptive dynamics approaches, and can be used to analyse eco‐evolutionary dynamics across different time scales and to extend quantitative genetics theory to account for multimodal trait distributions, the dynamical nature of genetic variance, the potential for disruptive selection due to ecological feedbacks, and the non‐normal or skewed trait distributions encountered in nature. [ABSTRACT FROM AUTHOR]

Numerically solving the Boltzmann equation is computationally expensive in part due to the number of variables the distribution function depends upon. Another contributor to the complexity of the Boltzmann Equation is the quadratic collision operator describing changes in the distribution function due to colliding particle pairs. Solving it as efficiently as possible has been a topic of recent research, e.g. Cai and Torrilhon (Phys Fluids 31(12):126105, 2019. ), Wang and Cai (J Comput Phys 397:108815, 2019. ), Cai et al. (Comput Fluids 200:104456, 2020. ). In this paper we exploit results from representation theory to find a very efficient algorithm both in terms of memory and computational time for the evaluation of the quadratic collision operator. With this novel approach we are also able to provide a meaningful interpretation of its structure. [ABSTRACT FROM AUTHOR]

In free surface flows, shallow water models simplify the flow conditions by assuming a constant velocity profile over the water depth. Recently developed Shallow Water Moment Equations allow for variations of the velocity profile at the expense of a more complex PDE system. The resulting equations can become stiff depending on the friction parameters, which leads to severe time step constraints of standard numerical schemes. In this paper, we apply Projective Integration schemes to stiff Shallow Water Moment Equations to overcome the time step constraints in the stiff regime and accelerate the numerical computations while still achieving high accuracy. In different dam break and smooth wave test cases, we obtain a speedup of up to 55 with respect to standard schemes. [ABSTRACT FROM AUTHOR]

Stiff hyperbolic balance laws exhibit large spectral gaps, especially if the relaxation term significantly varies in space. Using examples from rarefied gases and the general form of the underlying balance law model, we perform a detailed spectral analysis of the semi-discrete model that reveals the spectral gaps. Based on that, we show the inefficiency of standard time integration schemes expressed by a severe restriction of the CFL number. We then develop the first spatially adaptive projective integration schemes to overcome the prohibitive time step constraints of standard time integration schemes. The new schemes use different time integration methods in different parts of the computational domain, determined by the spatially varying value of the relaxation time. We use our analytical results to derive accurate stability bounds for the involved parameters and show that the severe time step constraint can be overcome. The new adaptive schemes show good accuracy in a numerical test case and can obtain a large speedup with respect to standard schemes. [ABSTRACT FROM AUTHOR]

We introduce Sim.DiffProc, an R package for symbolic and numerical computations on scalar and multivariate systems of stochastic differential equations (SDEs). It provides users with a wide range of tools to simulate, estimate, analyze, and visualize the dynamics of these systems in both forms, Itô and Stratonovich. One of Sim.DiffProc key features is to implement the Monte Carlo method for the iterative evaluation and approximation of an interesting quantity at a fixed time on SDEs with parallel computing, on multiple processors on a single machine or a cluster of computers, which is an important tool to improve capacity and speed-up calculations. We also provide an easy-to-use interface for symbolic calculation and numerical approximation of the first and central second-order moments of SDEs (i.e., mean, variance and covariance), by solving a system of ordinary differential equations, which yields insights into the dynamics of stochastic systems. The final result object of Monte Carlo and moment equations can be derived and presented in terms of LATEX math expressions and visualized in terms of LATEX tables. Furthermore, we illustrate various features of the package by proposing a general bivariate nonlinear dynamic system of Haken-Zwanzig, driven by additive, linear and nonlinear multiplicative noises. In addition, we consider the particular case of a scalar SDE driven by three independent Wiener processes. The Monte Carlo simulation thereof is obtained through a transformation to a system of three equations. We also study some important applications of SDEs in different fields.

The Fletcher-Ponnambalam (FP) reservoir optimization method has been successfully applied to several case studies in the literature. As an explicit stochastic optimization problem, it requires the assumption of a probability distribution for the reservoir inflows in order to solve several integrals regarding the statistical moment equations for storage as well as probabilities of containment (storage within minimum and maximum levels), deficit (below minimum), and spill (above maximum). This technical note proposes an alternative way to approximate the solution of such integrals using the sample of historical inflow records. This does not involve the consideration of any distribution and leads to far shorter expressions as well as to less coding effort. The new formulation is applied to solve a water reservoir operation optimization problem for a major dam in Brazil. The results compare well with those from the current FP formulation and with those obtained by a Monte Carlo simulation of the monthly reservoir operation. [ABSTRACT FROM AUTHOR]

Nonequilibrium heat and mass transfer processes through liquid-vapor interfaces are studied through solutions of the Enskog-Vlasov (EV) equation and the corresponding system of EV26 moment equations. These models fully resolve liquid and vapor bulk regions and the diffuse interface connecting both. With that, evaporation and heat transfer processes can be studied without the need of modeled interface relations. Comparison of numerical results shows qualitative agreement of moment simulations with DSMC solutions of the EV equation, but quantitative differences. Interface resistivities for jump interface conditions are determined from the simulations, which show marked differences to those found from classical kinetic theory, where dimensionless resistivities are constants. In contrast, the EV models give temperature dependent resistivities, some negative off-diagonal resistivities, and indicate non-linear behavior where resistivities depend on mass and heat fluxes through the interface. In summary, the results point to the urgent need for systematic evaluation of resistivities over a wide range of conditions between weak and strong nonequilibrium, close to and far from the critical point. • Enskog-Vlasov model to describe nonequilibrium processes in liquid-vapor systems. • 26 moment equations for liquid and vapor bulk phases and the diffuse interface, beyond Navier-Stokes-Fourier equations. • Numerical solution for EV kinetic equation and its 26 moment system. • Interface resistivities obtained from simulations. • Results indicate rich behavior of resistivities including temperature dependence and nonlinearities. [ABSTRACT FROM AUTHOR]

We study the Darcy boundary value problem with lognormal permeability field. We adopt a perturbation approach, expanding the solution in Taylor series around the nominal value of the coefficient, and approximating the expected value of the stochastic solution of the PDE by the expected value of its Taylor polynomial. The recursive deterministic equation satisfied by the expected value of the Taylor polynomial (first moment equation) is formally derived. Well-posedness and regularity results for the recursion are proved to hold in Sobolev space-valued Hölder spaces with mixed regularity. The recursive first moment equation is then discretized by means of a sparse approximation technique, and the convergence rates are derived. [ABSTRACT FROM AUTHOR]

Compartmentalization of biochemical processes underlies all biological systems, from the organelle to the tissue scale. Theoretical models to study the interplay between noisy reaction dynamics and compartmentalization are sparse, and typically very challenging to analyze computationally. Recent studies have made progress toward addressing this problem in the context of specific biological systems, but a general and sufficiently effective approach remains lacking. In this work, we propose a mathematical framework based on counting processes that allows us to study dynamic compartment populations with arbitrary interactions and internal biochemistry. We derive an efficient description of the dynamics in terms of differential equations which capture the statistics of the population. We demonstrate the relevance of our approach by analyzing models inspired by different biological processes, including subcellular compartmentalization and tissue homeostasis. [ABSTRACT FROM AUTHOR]

In this work, we make a further step in bringing together different approaches to non-equilibrium thermodynamics. The structure of the moment hierarchy derived from the Boltzmann equation is at the heart of rational extended thermodynamics (RET, developed by Ingo Müller and Tommaso Ruggeri). Whereas the full moment hierarchy has the structure expressed in the general equation for the nonequilibrium reversible--irreversible coupling (GENERIC), the Poisson bracket structure of reversible dynamics postulated in that approach is a major obstacle for truncating moment hierarchies, which seems to work only in exceptional cases (most importantly, for the five moments associated with conservation laws). The practical importance of truncated moment hierarchies in rarefied gas dynamics and microfluidics motivates us to develop a new strategy for establishing the full GENERIC structure of truncated moment equations, based on non-entropy-producing irreversible processes associated with Casimir symmetry. Detailed results are given for the special case of 10 moments. [ABSTRACT FROM AUTHOR]

Microsilica particles arise as a byproduct of silicon furnace operation, created inside high temperature ames due to the combustion reaction of silicon monoxide with oxygen. These nanoparticles, which grow as silicon dioxide vapor condenses on the surface of existing particles, are used in a variety of composite materials. The size and quality of the particles affect the performance of the material used for such applications, and hence control of these quantities is of importance to manufacturers. Motivated by this, we derive a mathematical model that connects local thermal and chemical concentrations conditions to the formation and growth of microsilica particles. We consider two distinct reductions of our general model: the case of initially well-mixed or spatially homogeneous chemical species (modeling the region within the ame or reaction zone), and the case of initially spatially separated chemical species, in which diffusion will play a dominant role in providing material to a combustion front (modeling a larger cross section, which contains a reaction zone with limiting quantities of fuel which must diffuse into the reaction zone). In both cases, we provide asymptotic solutions for the temperature, chemical concentrations, and number density function of microsilica particles in the oxygen rich limit, and compare them to numerical simulations. Motivated by realistic furnace control mechanisms, we treat the relative quantity of oxygen to other fuel components and the saturation concentration of silicon dioxide as control parameters, discussing how each may be used to modify the properties (such as size and abundance) of microsilica particles formed. One physically interesting finding is the theoretical description of a bimodal distribution for microsilica particle size which was previously observed in experiments. [ABSTRACT FROM AUTHOR]

Consider independent observations (X i , R i) with random or fixed ranks R i , while conditional on R i , the random variable X i has the same distribution as the R i -th order statistic within a random sample of size k from an unknown distribution function F. Such observation schemes are well known from ranked set sampling and judgment post-stratification. Within a general, not necessarily balanced setting we derive and compare the asymptotic distributions of three different estimators of the distribution function F: a stratified estimator, a nonparametric maximum-likelihood estimator and a moment-based estimator. Our functional central limit theorems generalize and refine previous asymptotic analyses. In addition, we discuss briefly pointwise and simultaneous confidence intervals for the distribution function with guaranteed coverage probability for finite sample sizes. The methods are illustrated with a real data example, and the potential impact of imperfect rankings is investigated in a small simulation experiment. [ABSTRACT FROM AUTHOR]

Abstract Stochastic reaction network modelling is widely utilized to describe the probabilistic dynamics of biochemical systems in general, and gene interaction networks in particular. The statistical analysis of the response of these systems to perturbation inputs is typically dependent on specific perturbation models. Motivated by reporter gene systems, widely utilized in biology to monitor gene activity in individual cells, we address the analysis of reaction networks with state-affine rates in presence of an input process. We develop a generalization of the so-called moment equations that precisely accounts for the first- and second-order moments of arbitrary inputs without the need for a model of the input process, as well as spectral relationships between the network input and state. We then apply these results to develop a method for the reconstruction of the autocovariance function of gene activity from reporter gene population-snapshot data, a crucial step toward the investigation of gene regulation, and demonstrate its performance on a simulated case study. [ABSTRACT FROM AUTHOR]

The shock structures of a 13 moment generalized hydrodynamics system of rarefied gases are simulated. These are first order hyperbolic equations derived from the Boltzmann equation. The investigated moment system stands out due to having an entropy evolution. In addition, a particular interest arises from the fact that the equations not only contain nonconservative products, but also provide the key to solving this mathematical and numerical issue by means of a simple substitution utilizing the physical entropy evolution. The apparent success of this method warrants investigation and provides a new perspective and starting point for finding general approaches to nonconservative products and irreversible processes. Furthermore, the system shows physically accurate results for low Mach numbers and is able to reveal the nonequilibrium entropy profile across a shock wave. [ABSTRACT FROM AUTHOR]

The paper proposes an approximate closure procedure for hierarchies of macroscopic equations for rarefied gases, derived as moment equations from the Boltzmann equation in kinetic theory of gases. The procedure is based upon application of the maximum entropy principle. If the exact minimizer is exploited, moments of the distribution function may diverge, unless the restriction on the structure of the moments is introduced. In this paper, a perturbative approach is proposed by restricting the set of admissible functions in the variational problem. This leads to an approximate minimizer, but the procedure can be applied to an arbitrary choice of the moments.

Calibrating parameters is a crucial problem within quantitative modeling approaches to reaction networks. Existing methods for stochastic models rely either on statistical sampling or can only be applied to small systems. Here, we present an inference procedure for stochastic models in equilibrium that is based on a moment matching scheme with optimal weighting and that can be used with high-throughput data like the one collected by flow cytometry. Our method does not require an approximation of the underlying equilibrium probability distribution and, if reaction rate constants have to be learned, the optimal values can be computed by solving a linear system of equations. We discuss important practical issues such as the selection of the moments and evaluate the effectiveness of the proposed approach on three case studies. [ABSTRACT FROM AUTHOR]

We present a new method for calculating the time-transient behavior of stochastic reaction networks. We first derive the set of equations for the moments of the master probability distribution. We then linearize these equations calculating the Jacobian matrix around the stationary probability distribution. In order to demonstrate the method, we present examples of stochastic reaction networks and compute their dynamic behavior. We find that the calculations are accurate and significantly more efficient than stochastic simulation algorithms based on Gillespie’s algorithms. [ABSTRACT FROM AUTHOR]

The paper investigates the navigation problem of following a moving target, using a mathematical model described by a system of differential equations with random parameters. The differential equations, which employ controls for following the target, are solved by a new approach using moment equations. Simulations are presented to test effectiveness of the approach. [ABSTRACT FROM AUTHOR]

In many cases, it is difficult to find a solution to a system of difference equations with random structure in a closed form. Thus, a random process, which is the solution to such a system, can be described in another way, for example, by its moments. In this paper, we consider systems of linear difference equations whose coefficients depend on a random Markov or semi-Markov chain with jumps. The moment equations are derived for such a system when the random structure is determined by a Markov chain with jumps. As an example, three processes: Threats to security in cyberspace, radiocarbon dating, and stability of the foreign currency exchange market are modelled by systems of difference equations with random parameters that depend on a semi-Markov or Markov process. The moment equations are used to obtain the conditions under which the processes are stable.

In this paper, an attempt has been made to understand the activity of mean membrane voltage and subsidiary system variables with moment equations (i.e., mean, variance and covariance’s) under noisy environment. We consider a biophysically plausible modified Hindmarsh–Rose (H–R) neural system injected by an applied current exhibiting spiking–bursting phenomenon. The effects of predominant parameters on the dynamical behavior of a modified H–R system are investigated. Numerically, it exhibits period-doubling, period halving bifurcation and chaos phenomena. Further, a nonlinear system has been analyzed for the first and second order moments with additive stochastic perturbations. It has been solved using fourth order Runge–Kutta method and noisy systems by Euler’s scheme. It has been demonstrated that the firing properties of neurons to evoke an action potential in a certain parameter space of the large exact systems can be estimated using an approximated model. Strong stimulation can cause a change in increase or decrease of the firing patterns. Corresponding to a fixed set of parameter values, the firing behavior and dynamical differences of the collective variables of a large, exact and approximated systems are investigated. [ABSTRACT FROM AUTHOR]

This paper presents a robust implementation of the maximum-entropy closure in the context of rarefied gas dynamics. Moment systems supplied with the maximum-entropy closure have attractive mathematical properties: They are hyperbolic in the interior of the domain of definition of the dual minimization problem and endowed with an entropy law. In contrast to Grad’s classical closure theory, the maximum-entropy closure allows for applications to strongly non-equilibrium gas flows. The 35-moment system studied in this paper includes as basis functions all monomials up to order four, so that evolution equations for important non-equilibrium quantities, such as the stress tensor and heat flux vector, are contained in the system. To remove the singularity in the maximum-entropy closure, we consider a bounded underlying velocity domain and approximate moments of the reconstructed maximum-entropy distribution with a fixed, block-wise Gauss–Legendre quadrature rule. The convex dual minimization problem is solved with a Newton type algorithm. We show that the Hessian matrix used in the Newton iteration can become ill-conditioned even for equilibrium states if monomial basis functions are used. To improve the robustness of the Newton iteration, we consider partially and fully adaptive basis algorithms and demonstrate that the 35-moment system allows for accurate and robust simulations of non-equilibrium rarefied gas flows in the transition regime by applying the model to one-dimensional gas processes, including a continuous shock-structure problem. [ABSTRACT FROM AUTHOR]

The goal of the present paper is to construct a relativistic extended thermodynamics (ET) theory of rarefied polyatomic gases. This is achieved by adopting the closure procedure for the 14 generalized moments of a distribution function that, as in the classical case, depends on an additional continuous variable representing the energy of the internal modes of a molecule. This permits the theory to take into account the energy exchange between translational modes and internal modes of a molecule in binary collisions. The theory includes the ET theory of monatomic gases as a singular limit and converges to the corresponding ET theory of polyatomic gases in the classical limit. In contrast to a monatomic gas in which the order of magnitude of the dynamical pressure is O ( 1 / c 2 ) , the dynamical pressure in a polyatomic gas is not small due to the internal motion of molecules as is the case in the classical limit. Therefore the present theory might result particularly useful in cosmology to describe some aspects of the post-recombination era. [ABSTRACT FROM AUTHOR]

Ethylene was homopolymerized over Ziegler-Natta catalyst and the homopolymerization was modeled using moment equations. Mechanism was modeled according to five different reaction centers of catalyst. For each center, there are different reaction rate coefficients; therefore the final product of each center would be expected to be different. Modeling results showed good conformity to the experimental results. According to the results obtained, the molecular weight distribution of each active center follows a Schultz-Flory distribution. However, the molecular weight distribution ofpolymer produced is much broader than a Schultz-Flory distribution. Besides, the order of polymerization with regards to monomer concentration is different for each center and it is higher than unity. Moreover, the catalyst active centers deteriorate in the presence of hydrogen and consequently catalyst yield drops. Nevertheless, polymerization kinetics is not affected much by hydrogen. Hydrogen also reduces polymer molecular weight since it is a strong transfer agent in olefin polymerizations. Notwithstanding, it does not affect polydispersity index. Finally, by increasing the cocatalyst concentration the activity of active centers is not changed, while it lessens the molecular weight as a transfer agent.

In current work a comprehensive mechanism based on intermediate radical termination theory is assumed for RAFT polymerization of styrene over cumyl dithiobenzoate as RAFT agent. Rate constants for addition (ka) and fragmentation reactions (kf) are set to 6×106 and 5×104 respectively, which lead to an equilibrium constant value of K = ka/kf = 1.2 x 102. Moment equations method was used to model this mechanism and the results were compared to experimental data to verify modeling. The effects of changing RAFT agent concentration on conversion, molecular weight and polydispersity index of the final product were investigated through the modeling. According to the results, the likelihood of living polymerization increases with raising RAFT agent concentration which leads to linearity of conversion and molecular weight curves and therefore lowering the polydispersity index and narrowing the molecular weight distribution.

Mathematical models are of fundamental importance in the understanding of complex population dynamics. For instance, they can be used to predict the population evolution starting from different initial conditions or to test how a system responds to external perturbations. For this analysis to be meaningful in real applications, however, it is of paramount importance to choose an appropriate model structure and to infer the model parameters from measured data. While many parameter inference methods are available for models based on deterministic ordinary differential equations, the same does not hold for more detailed individual-based models. Here we consider, in particular, stochastic models in which the time evolution of the species abundances is described by a continuous-time Markov chain. These models are governed by a master equation that is typically difficult to solve. Consequently, traditional inference methods that rely on iterative evaluation of parameter likelihoods are computationally intractable. The aim of this paper is to present recent advances in parameter inference for continuous-time Markov chain models, based on a moment closure approximation of the parameter likelihood, and to investigate how these results can help in understanding, and ultimately controlling, complex systems in ecology. Specifically, we illustrate through an agricultural pest case study how parameters of a stochastic individual-based model can be identified from measured data and how the resulting model can be used to solve an optimal control problem in a stochastic setting. In particular, we show how the matter of determining the optimal combination of two different pest control methods can be formulated as a chance constrained optimization problem where the control action is modeled as a state reset, leading to a hybrid system formulation.

We study optimal control problems for linear systems with prescribed initial and terminal states. We analyze the exact penalization of the terminal constraints. We show that for systems that are exactly controllable, the norm-minimal exact control can be computed as the solution of an optimization problem without terminal constraint but with a nonsmooth penalization of the end conditions in the objective function, if the penalty parameter is sufficiently large. We describe the application of the method for hyperbolic and parabolic systems of partial differential equations, considering the wave and heat equations as particular examples. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We develop a moment-equation-copula-closure method for the inexpensive approximation of the steady state statistical structure of strongly nonlinear systems which are subjected to correlated excitations. Our approach relies on the derivation of moment equations that describe the dynamics governing the two-time statistics. These are combined with a non-Gaussian pdf representation for the joint response-excitation statistics, based on copula functions that has (i) single time statistical structure consistent with the analytical solutions of the Fokker–Planck equation, and (ii) two-time statistical structure with Gaussian characteristics. Through the adopted pdf representation, we derive a closure scheme which we formulate in terms of a consistency condition involving the second order statistics of the response, the closure constraint . A similar condition, the dynamics constraint , is also derived directly through the moment equations. These two constraints are formulated as a low-dimensional minimization problem with respect to the unknown parameters of the representation, the minimization of which imposes an interplay between the dynamics and the adopted closure. The new method allows for the semi-analytical representation of the two-time, non-Gaussian structure of the solution as well as the joint statistical structure of the response-excitation over different time instants. We demonstrate its effectiveness through the application on bistable nonlinear single-degree-of-freedom energy harvesters with mechanical and electromagnetic damping, and we show that the results compare favorably with direct Monte-Carlo simulations. [ABSTRACT FROM AUTHOR]

The difference in the theoretical structure between monatomic and polyatomic gases in highly nonequilibrium states is discussed from the viewpoint of molecular extended thermodynamics (MET) of rarefied gases, which is free from the local equilibrium assumption. The MET theories of these two types of gases are based on the moment balance equations with different hierarchy structures due to whether the internal degrees of freedom of a molecule are incorporated in their distribution functions or not. In particular, the number of balance equations in the MET theory of polyatomic gases is greater than the number in the corresponding theory of monatomic gases. The closure procedure for the system of balance equations of polyatomic gases obtained in a recent paper (Arima et al., 2014) is adopted. We prove that the solutions for polyatomic gases converge, in the limit where the degrees of freedom of a molecule D tend to 3, to the ones for monatomic gases provided that we impose appropriate initial conditions compatible with monatomic gases. Thus a MET theory of rarefied monatomic gases can be identified as a singular limit of the corresponding MET theory of rarefied polyatomic gases. As illustrative examples, the asymptotic behaviors when D → 3 in the dispersion relation of ultrasonic waves and in the shock wave structure are shown. [ABSTRACT FROM AUTHOR]

We study an elliptic equation with stochastic coefficient modeled as a lognormal random field. A perturbation approach is adopted, expanding the solution in Taylor series around the nominal value of the coefficient. The resulting recursive deterministic problem satisfied by the expected value of the stochastic solution, the so called moment equation , is discretized with a full tensor product finite element technique. To overcome the incurred curse of dimensionality the solution is sought in a low-rank tensor format, the so called Tensor Train format. We develop an algorithm for solving the recursive first moment problem approximately in the Tensor Train format and show its effectiveness with numerical examples. [ABSTRACT FROM AUTHOR]

The strategy for computing the Boltzmann collision integrals for gaseous mixtures is presented and bestowed to compute the fully non-linear Boltzmann collision integrals for hard sphere gas-mixtures. The Boltzmann collision integrals associated with the first 26 moments of each constituent in a gas-mixture are presented. Moreover, the Boltzmann collision integrals are exploited to study the relaxation phenomena of diffusion velocities, stresses and heat fluxes in binary gas-mixtures of Maxwell molecules and hard spheres. [ABSTRACT FROM AUTHOR]

The stability analysis of a Beddington-DeAngelis (B-D) type predator-prey model driven by symmetric trichotomous noises is presented in this paper. Using the Shapiro-Loginov formula, the first-order and second-order solution moments of the system are obtained. The moment stability conditions of the B-D predator-prey model are given by using Routh-Hurwitz criterion. It is found that the stabilities of the first-order and second-order solution moments depend on the noise intensities and correlation time of noise. The first-order and second-order moments are stable when the correlation time of noise is increased. That is, the trichotomous noise plays a constructive role in stabilizing the solution moment with regard to Gaussian white noise. Finally, some numerical results are performed to support the theoretical analyses. [ABSTRACT FROM AUTHOR]