"常微分方程"课程建设探讨

根据常微分方程课程特点及常微分方程的发展,探讨了常微分方程精品课程建设中师资队伍建设和教材改革与建设的重要性,提出了对改革有关教学内容以及编写具有本课程特色的优秀教材的具体设想.     

Sell’s conjecture for non-autonomous dynamical systems

This paper is dedicated to the study of the G. Sell’s conjecture for general non-autonomous dynamical systems. We give a positive answer for this conjecture and we apply this result to different classes of non-autonomous evolution equations: Ordinary Differential Equations, Functional Differential Equations and Semi-linear Parabolic Equations.     

The ODEs forms of linearly elastic shell model and its numerical computing

In this paper, we successfully use 1D model to approximate the 3D problems. Firstly the PDEs (Partial Differential Equations) forms of Koiter’s Model for 2D linear elastic shell is proposed on special curvature coordinate system, i.e., spherical-coordinate system. Then the ODEs (Ordinary Differential Equations) forms of Koiter’s Model for 2D linear elastic shell is proposed under the assumption that the shell is axis-symmetric. Finally, we do numerical experiments to verify validity and accuracy of 1D models.     

Convergence results for general linear methods on singular perturbation problems

Many numerical methods used to solve Ordinary Differential Equations, or Differential Algebraic Equations can be written as general linear methods. The B-convergence results for general linear methods are for algebraically stable methods, and therefore useless for nearly A-stable methods. The purpose of this paper is to show convergence for singular perturbation problems for the class of general linear methods without assuming A-stability.     

Existence and uniqueness of a steady state for an OTC market with several assets

We introduce and study a class of over-the-counter market models specified by systems of Ordinary Differential Equations (ODE’s), in the spirit of Duffie-Gârleanu-Pedersen Duffie et al. (Econometrica 73(1):1815–1847, 2005). The key innovation is allowing for multiple assets. We show the existence and uniqueness of a steady state for these ODE’s.     

Systematic dynamic modelling of mechanical systems containing kinematic loops

In this tutorial paper a systematic procedure is presented to obtain the dynamic models of mechanical systems containing kinematic loops, with a main emphasis on efficiency and with particular regard to robotic systems. The procedure retains a minimal set of generalized coordinates for the corresponding open loop structure, obtained by removing some additional constraints closing loops in the original structure, while adopting an efficient Newton-Euler formulation of the equations of motion. Two methods for including the loop closure equations are then discussed: the Lagrange multipliers method and the method based on an explicit solution of the constraint equations. In the first case the dynamic model is given in the form of an implicit Differential Algebraic Equations (DAE) system, while in the second case an Ordinary Differential Equations (ODE) system could be obtained.     

A note on topological methods for a class of differential-algebraic equations

We study a particular class of autonomous Differential-Algebraic Equations that are equivalent to Ordinary Differential Equations on manifolds. Under appropriate assumptions we determine a straightforward formula for the computation of the degree of the associated tangent vector field that does not require any explicit knowledge of the manifold. We use this formula to study the set of harmonic solutions to periodic perturbations of our equations. Two different classes of applications are provided.     

The existence of almost periodic solutions of certain perturbation systems

Certain almost periodic perturbation systems are considered in this paper. By using the roughness theory of exponential dichotomies and the contraction mapping principle, some sufficient conditions are obtained for the existence and uniqueness of almost periodic solution of the above systems. Our results generalize those in [J.K. Hale, Ordinary Differential Equations, Krieger, Huntington, 1980; C. He, Existence of almost periodic solutions of perturbation systems, Ann. Differential Equations 9 (1992) 173-181; M. Lin, The existence of almost periodic solution and bounded solution of perturbation systems, Acta Math. Sinica 22A (2002) 61-70 (in Chinese); W.A. Coppel, Almost periodic properties of ordinary differential equations, Ann. Math. Pura Appl. 76 (1967) 27-50; A.M. Fink, Almost Periodic Differential Equations, Lecture Notes in Math., vol. 377, Springer-Verlag, New York, 1974; Y. Xia, F. Chen, A. Chen, J. Cao, Existence and global attractivity of an almost periodic ecological model, Appl. Math. Comput. 157 (2004) 449-475].     

Book Reviews

Book reviewed in this article: It's All Done by Numbers, by Douglas St. Paul Barnard. Introduction to Probability and Statistics, Fourth Edition, by Henry L. Alder and Edward B. Roessler Ordinary Differential Equations and Stability Theory: An Introduction, by David A Sanchez     

On a Method of Mathematical Modeling of Electrical Synapses

Differential Equations - We introduce a new mathematical model of a neural network with electrical connections. The model is a singularly perturbed system of delay differential–difference...     

Evolving Systems of Stochastic Differential Equations

Journal of Theoretical Probability - We introduce Evolving Systems of Stochastic Differential Equations. This model generalizes the well-known stochastic differential equations with Markovian...     

Time-stepping algorithms for semidiscretized linear parabolic PDEs based on rational approximants with distinct real poles

Time dependent problems in Partial Differential Equations (PDEs) are often solved by the Method Of Lines (MOL). For linear parabolic PDEs, the exact solution of the resulting system of first order Ordinary Differential Equations (ODEs) satisfies a recurrence relation involving the matrix exponential function. In this paper, we consider the development of a fourth order rational approximant to the matrix exponential function possessing real and distinct poles which, consequently, readily admits a partial fraction expansion, thereby allowing the distribution of the work in solving the corresponding linear algebraic systems in essentially Backward Euler-like solves on concurrent processors. The resulting parallel algorithm possesses appropriate stability properties, and is implemented on various parabolic PDEs from the literature including the forced heat equation and the advection-diffusion equation.Dedicated to Professor J. Crank on the occasion of his 80th birthday     

Positivity inheritance for linear problems

We discuss the numerical solution of positive Differential-Algebraic-Equations (DAEs). For Ordinary Differential Equations (ODEs) where the system matrix is a -M-Matrix, Runge-Kutta- or Multistep-Method are positive if the stepsize is chosen within the absolutely monotonicity radius of the considered method. We extend this concept to matrix pairs and present conditions for positivity preserving discretizations of linear, time-invariant DAEs. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A modified Runge-Kutta method for the numerical solution of ODE''s with oscillation solutions

A modified Runge-Kutta method with minimal phase-lag is developed for the numerical solution of Ordinary Differential Equations with oscillating solutions. The method is based on the accurate Runge-Kutta method of Sharp and Smart RK4SS(5) (see [1]) of order five. Numerical and theoretical results show that this new approach is more efficient, compared with the fifth order Runge-Kutta Sharp and Smart method.     

Forward Feynman-Kac type representation for semilinear non-conservative partial differential equations

ABSTRACT

We propose a non-linear forward Feynman-Kac type equation, which represents the solution of a nonconservative semilinear parabolic Partial Differential Equations (PDE). We show in particular existence and uniqueness. The solution of that type of equation can be approached via a weighted particle system.     

Lie-group method for unsteady flows in a semi-infinite expanding or contracting pipe with injection or suction through a porous wall

The unsteady incompressible laminar flow in a semi-infinite porous circular pipe with injection or suction through the pipe wall whose radius varies with time is considered. The present analysis simulates the flow field by the burning of inner surface of cylindrical grain in a solid rocket motor, in which the burning surface regresses with time. We apply Lie-group method for determining symmetry reductions of partial differential equations. Lie-group method starts out with a general infinitesimal group of transformations under which given partial differential equations are invariant, then, the determining equations are derived [Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, Wiley, New York, 1999; Hydon, Symmetry Methods for Differential Equations, Cambridge University Press, Cambridge, 2000; Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986; Seshadri, Na, Group invariance in engineering boundary value problems, Springer, New York, 1985; Yi, Fengxiang, Lie symmetries of mechanical systems with unilateral holonomic constraints, Chinese Sci. Bull. 45 (2000) 1354–1358; Moritz, Schwalm, Uherka, Finding Lie groups that reduce the order of discrete dynamical systems, J. Phys. A: Math. 31 (1998) 7379–7402; Nucci, Clarkson, The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh–Nagumo equation, Phys. Lett. A 164 (1992) 49–56; Basarab, Lahno, Group classification of nonlinear partial differential equations: a new approach to resolving the problem, Proceedings of Institute of Mathematics of NAS of Ukraine, vol. 43, 2002, pp. 86–92; Burde, Expanded Lie group transformations and similarity reductions of differential equations, Proceedings of Institute of Mathematics of NAS of Ukraine, vol. 43, 2002, pp. 93–101; Gandarias, Bruzon, Classical and nonclassical symmetries of a generalized Boussinesq equation, J. Nonlinear Math. Phys. 5 (1998) 8–12; Hill, Solution of Differential Equations by Means of One-Parameter Groups, Pitman Publishing Co., 1982]. The determining equations are a set of linear differential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equation may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables in the system. Effect of the cross-flow Reynolds number Re and the dimensionless wall expansion ratio $\alpha$ on velocity, flow streamlines, axial and radial pressure drop, and wall shear stress has been studied both analytically and numerically and the results are plotted.     

金融危机传染的非线性动力学模型

越来越多的证据表明全球金融市场是高度复杂,广泛联接的非线性动力系统网络,由于系统的非线性和复杂性特征使得金融危机极易通过系统间的耦合作用而发生传染.针对金融危机传染所表现出的非线性动力学特性,本文构建了以传染源国家和受传染国家股票收益率为变量的微分动力学传染模型.利用微分方程定性理论对模型的奇点进行讨论,得出极限环存在的条件及形式,并进而得出金融危机传染的三种情况:轻度传染、可控传染和强烈传染.     

Reflected Backward Stochastic Differential Equation with Locally Monotone Coefficient

Abstract

We study the existence and uniqueness of Reflected Backward Stochastic Differential Equation (RBSDE for short) with both monotone and locally monotone coefficient and squared integrable terminal data. This is done with a polynomial growth condition on the coefficient. An application to the homogenization of multivalued Partial Differential Equations (PDEs for short) is given.     

Grid approximations of the pseudospectral type with an improved condition number for second-order differential operators

Schemes that are free of the disadvantages of both the finite-difference and finite-element methods and retain the advantages of the saturation-free grid methods are proposed and investigated. The asymptotic behavior of their maximal N th eigenvalue is the same as the behavior of the N the eigenvalue of a differential operator, and it is not difficult to apply these discretizations to nonstationary problems. In contrast to polynomial pseudospectral approximations, the schemes of this paper, as well as of E. B. Karpilovskaya, “Convergence of the collocation method,” Sov. Math. Dokl.,4, No. 2, 1070–1073 (1963) and I. P. Gavrilyuk and L. D. Grekov, On Algorithms for the Realization of Grid Schemes without the Accuracy Staturation for Second-Order Ordinary Differential Equations [in Russian], Deposited at UkrNIINTI 16.08.1991, utilize uniform grids. Bibliography: 27 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 78, 1994, pp. 1–27.     

Asymptotic Stability of Solutions of the Cauchy Problem for Models of Nonequilibrium Thermodynamics. Stable Hyperbolic Pencils

The paper distinguishes a class of stable polynomial pencils reproduced by first moments in the Grad hierarchy for the kinetic Boltzmann and Fokker-Planck equations. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 12, Partial Differential Equations, 2004.