G-Brown运动驱动的中立型随机时滞微分方程的指数稳定性

研究了一类G-Brown运动驱动的中立型随机时滞微分方程的指数稳定性.在G-框架意义下,运用合适的Lyapunov-Krasovskii泛函,中立型时滞微分方程理论以及随机分析技巧,证明了所研究方程平凡解的p-阶矩指数稳定性,得到了所研究方程平凡解是p-阶矩指数稳定的充分条件.最后通过例子说明所得的结果.     

Moment equation methods for nonlinear stochastic systems

One method of approaching models represented by systems of stochastic ordinary differential equations is to consider the moment equations. This approach can be far more efficient than a Monte Carlo simulation or a finite-difference solution of the associated Fokker-Plank equation. However, a nonlinear system generates an infinite hierarchy of moment equations, which requires the adoption of some hierarchy truncation technique to facilitate solution. This paper considers a method of hierarchy truncation, based on the quasi-moments of the state-variables.     

The th moment stability for the stochastic pantograph equation

In this paper, we investigate the αth moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the Razumikhin technique. Especially the linear stochastic pantograph equations and the semi-implicit Euler method applying them are considered. The convergence result of the semi-implicit Euler method is obtained. The stability conditions of the analytic solution of those equations and the numerical method are given. Finally, some experiments are given.     

Scenario tree generation and multi-asset financial optimization problems

We compare two popular scenario tree generation methods in the context of financial optimization: moment matching and scenario reduction. Using a simple problem with a known analytic solution, moment matching–when ensuring absence of arbitrage–replicates this solution precisely. On the other hand, even if the scenario trees generated by scenario reduction are arbitrage-free, the solutions are biased and highly variable. These results hold for correlated and uncorrelated asset returns, as well as for normal and non-normal returns.     

无限时滞的随机泛函微分方程解的渐近性质

本文研究了无限时滞随机泛函微分方程解的存在唯一性,矩有界性的问题.利用Lyapunov函数法以及概率测度的引入得到了确保方程解在唯一、矩有界、时间平均矩有界同时成立的一个新的条件.推广了Khasminskii-Mao定理的相关结果.     

Stability of neutral stochastic functional differential equations with Markovian switching driven by G-Brownian motion

Little seems to be known about stability results on the neutral stochastic function differential equations with Markovian switching driven by G-Brownian (G-NSFDEwMSs). This paper aims at investigating the pth moment exponential stability for G-NSFDEwMSs to fill this gap. Some sufficient conditions on the pth moment exponential stability of the trivial solution are derived by employing the Razumikhin-type method, stochastic analysis, and algebraic inequality technique. Moreover, an example is provided to illustrate the effectiveness of the obtained results.     

Stability in mean square of a linear stochastic differential equation

The stability of the zero solution of a first-order linear differential equation with a random right-hand side is investigated using moment equations. Transformations of moment equations are considered. Conditions for reducing the order of the moment equations are derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 119–126, 1987.     

Bifurcating periodic solution for a class of first-order nonlinear delay differential equation after Hopf bifurcation

In this paper, we predict the accurate bifurcating periodic solution for a general class of first-order nonlinear delay differential equation with reflectional symmetry by constructing an approximate technique, named residue harmonic balance. This technique combines the features of the homotopy concept with harmonic balance which leads to easy computation and gives accurate prediction on the periodic solution to the desired accuracy. The zeroth-order solution using just one Fourier term is applied by solving a set of nonlinear algebraic equations containing the delay term. The unbalanced residues due to Fourier truncation are considered iteratively by solving linear equations to improve the accuracy and increase the number of Fourier terms of the solutions successively. It is shown that the solutions are valid for a wide range of variation of the parameters by two examples. The second-order approximations of the periodic solutions are found to be in excellent agreement with those obtained by direct numerical integration. Moreover, the residue harmonic balance method works not only in determining the amplitude but also the frequency of the bifurcating periodic solution. The method can be easily extended to other delay differential equations.     

Discrete weighted least-squares method for the Poisson and biharmonic problems on domains with smooth boundary

In this article a discrete weighted least-squares method for the numerical solution of elliptic partial differential equations exhibiting smooth solution is presented. It is shown how to create well-conditioned matrices of the resulting system of linear equations using algebraic polynomials, carefully selected matching points and weight factors. Two simple algorithms generating suitable matching points, the Chebyshev matching points for standard two-dimensional domains and the approximate Fekete points of Sommariva and Vianello for general domains, are described. The efficiency of the presented method is demonstrated by solving the Poisson and biharmonic problems with the homogeneous Dirichlet boundary conditions defined on circular and annular domains using basis functions in the form satisfying and in the form not satisfying the prescribed boundary conditions.     

Hausdorff moment problem and fractional moments

Hausdorff moment problem is considered and a solution, consisting of the use of fractional moments, is proposed. More precisely, in this work a stable algorithm to obtain centered moments from integer moments is found. The algorithm transforms a direct method into an iterative Jacobi method which converges in a finite number of steps, as the iteration Jacobi matrix has null spectral radius. The centered moments are needed to calculate fractional moments from integer moments. As an application few fractional moments are used to solve finite Hausdorff moment problem via maximum entropy technique. Fractional moments represent a remedy to ill-conditioning coming from an high number of integer moments involved in recovering procedure.     

Efficient numerical methods for radiation in gas turbines

Numerical methods for radiative heat transfer equations coupled to a temperature equation are considered. Efficient solution methods and approximate equations for this system are investigated and a comparative numerical study of the different approximations is given. The approximate equations considered in this paper include moment methods and diffusive approximations. Fast iterative solvers for the problem like multilevel methods with suitable preconditioning are considered in detail. Numerical experiments and comparisons in different space dimensions and for various physical situations are presented.     

A moment problem for one-dimensional nonlinear pseudoparabolic equation

We are concerned with a moment problem for a nonlinear pseudoparabolic equation with one space dimension on an interval. The boundary conditions are imposed in terms of the zero-order moment and the first-order moment. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in the usual Sobolev space. We are able to get regularity of the solution so that both solution and its derivative with respect to the time variable belong to the same Sobolev space with respect to the space variable. This feature is different from problems with parabolic equations, where the regularity order of solution is higher than that of the time derivative with respect to the space variable. Previous results reflected only this parabolic nature for the pseudoparabolic equation.     

Forced oscillations of an unbalanced rotor in nonisotropic bearings

The influence of anisotropy of elastic bearings on forced oscillations of a rotor with the static and moment unbalance is studied for the cases of its fastening on a rigid shaft and on a flexible one. The rotor with four degrees of freedom is considered. It is suggested that the shaft is fixed in linear elastic nonisotropic bearings. The differential equations of rotation of the rotor are written in complex variables, and an exact solution to the equation system is found that corresponds to the elliptical synchronous precession. The exact solution is a sum of two vectors, one of which parameterizes a forward precession, while another parameterizes a reverse precession. Amplitude-frequency characteristics of forward and reverse precessions and elliptical trajectories of the rotor axis ends are constructed. It is shown that, in case of nonisotropic bearings, both the forward and reverse precession, as well as the axis motion of nonsimple type (when its one end is moving forward, while another is moving in the reverse direction), can take place. The influence of anisotropy of elastic bearings also manifests itself by change in critical frequencies towards their reduction and by arising of additional critical frequencies in the bottom part of the spectrum, which significantly complicates dynamics of the high-speed rotor at the moment when it reaches the working angular speed.     

Existence results and the moment estimate for nonlocal stochastic differential equations with time-varying delay

This paper considers a class of nonlocal stochastic differential equations with time-varying delay whose coefficients are dependent on the pth moment. By applying the fixed point theorem, the existence and uniqueness of the solution of nonlocal stochastic differential delay equations is studied. Also, a class of moment estimates of solutions is considered. The results are a generalization and continuation of the recent results on this issue. An example is provided to illustrate the effectiveness of our results.     

STRONG CONVERGENCE OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH MULTIPLICATIVE NOISE

This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations(SPDEs)with multiplicative noise.The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-sided Lipschitz condition.These assumptions are the same ones as the cases where numerical methods for general nonlinear stochastic ordinary differential equations(SODEs)under"minimum assumptions"were studied.As a result,the semilinear SPDEs considered in this paper are a direct generalization of these nonlinear SODEs.There are several difficulties which need to be overcome for this generalization.First,obviously the spatial discretization,which does not appear in the SODE case,adds an extra layer of difficulty.It turns out a spatial discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix.In this paper we use a finite element interpolation technique to discretize the nonlinear drift term.Second,in order to prove the strong convergence of the proposed fully discrete finite element method,stability estimates for higher order moments of the H1-seminorm of the numerical solution must be established,which are difficult and delicate.A judicious combination of the properties of the drift and diffusion terms and some nontrivial techniques is used in this paper to achieve the goal.Finally,stability estimates for the second and higher order moments of the L2-norm of the numerical solution are also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant.This is done by utilizing the interpolation theory and the higher moment estimates for the H1-seminorm of the numerical solution.After overcoming these difficulties,it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence.Numerical experiment results are also presented to validate the theoretical results and to demonstrate the efficiency of the proposed numerical method.     

The local boundedness of the perturbed motions of a gyroscope in gimbals with dissipative and accelerating forces

The motion of an unbalanced gyroscope in gimbals in a central Newtonian field of forces is considered, taking the masses of the suspension rings into account. It is assumed that there is a moment of forces of viscous friction acting on the axis of rotation of one of the rings, and there is an accelerating (electromagnetic) moment applied to the axis of rotation axis of the other ring. The equations of motion have a partial solution such that the mean velocity of the outer ring is perpendicular to the direction from the centre of gravitation S to the stationary point O, the middle plane of the inner ring contains this direction, and the gyroscope rotates about SO with an arbitrary constant angular velocity.     

一类非线性方程的激波解

该文是利用匹配条件讨论一类非线性方程激波解。得出了对应的激波解与边界条件的关系。     

Matched Asymptotic Expansions in the Two-body Problem with Quick Loss of Mass

In studying models for the two-body problem with quick lossof mass a boundary layer problem arises for a third-order systemof non-linear ordinary differential equations. The models areidentified by a real parameter n with n ? 1. It turns out thatfor n = 1 asymptotic approximations of the solutions can beobtained by applying the method of matched asymptotic expansionsaccouonding to Vasil'eva or a multiple time scales method developedby O'Malley. For n> 1 these methods break down and it isshown that this is due to the occurrence of "unexpected" orderfunctions in the asymptotic expansions. The expansions for n> 1 are obtained by constructing an inner and outer expansionof the solution and matching these by the process of takingintermediate limits. The asymptotic validity of the matched expansions is provedby using an iteration technique; the proof is constructive sothat it provides us at the same time with an alternative wayof constructing approximations without using a matching technique.     

A note on exact travelling wave solutions for the modified Camassa-Holm and Degasperis-Procesi equations

Exact solutions for the modified Camassa-Holm and Degasperis-Procesi equations by Liu et al. (2010) [Y.F. Liu, X.Y. Zhu, J.X. He, Factorization technique and new exact solutions for the modified Camassa-Holm and Degasperis-Procesi equations, Appl. Math. Comput. 217 (2010) 1658-1665] are investigated. Liu et al. has used the factorization technique to reduce the modified Camassa-Holm and Degasperis-Procesi equations to first-order ordinary differential equations, and then derived some exact travelling wave solutions by direct integral method. In this note, we will explain that the implementation of the so-called factorization technique is completely unnecessary. Moreover, based on the method of complete discrimination system for polynomial, we shall demonstrate that the general explicit exact solution and its classification for the above two types of equations can be obtained directly and many exact solutions by Liu et al. are our special cases. Besides, some known results in previously relevant literatures are extended and some simple remarks are also made.     

Modelling two-dimensional flow past arbitrary cylindrical bodies using boundary element formulations

This paper presents an efficient method of solving Queen's linearized equations for steady plane flow of an incompressible, viscous Newtonian fluid past a cylindrical body of arbitrary cross-section. The numerical solution technique is the well known direct boundary element method. Use of a fundamental solution of Oseen's equations, the ‘Oseenlet’, allows the problem to be reduced to boundary integrals and numerical solution then only requires boundary discretization. The formulation and solution method are validated by computing the net forces acting on a single circular cylinder, two equal but separated circular cylinders and a single elliptic cylinder, and comparing these with other published results. A boundary element representation of the full Navier-Stokes equations is also used to evaluate the drag acting on a single circular cylinder by matching with the numerical Oseen solution in the far field.