Reducing and monitoring round-off error propagation for symplectic implicit Runge-Kutta schemes

We propose an implementation of symplectic implicit Runge-Kutta schemes for highly accurate numerical integration of non-stiff Hamiltonian systems based on fixed point iteration. Provided that the computations are done in a given floating point arithmetic, the precision of the results is limited by round-off error propagation. We claim that our implementation with fixed point iteration is near-optimal with respect to round-off error propagation under the assumption that the function that evaluates the right-hand side of the differential equations is implemented with machine numbers (of the prescribed floating point arithmetic) as input and output. In addition, we present a simple procedure to estimate the round-off error propagation by means of a slightly less precise second numerical integration. Some numerical experiments are reported to illustrate the round-off error propagation properties of the proposed implementation.     

基于符号数值混合计算的混成系统Lyapunov函数构造

基于平方和松弛和有理向量恢复,提出了一种符号数值混合计算方法来构造多项式Lyapunov函数以判定非线性混成系统的稳定性,首先,为Lyapunov函数预定一个给定次数的多项式模板,则Lyapunov函数构造问题可转化为相应的带参数的多项式优化问题,然后运用平方和松弛方法求得一个近似的数值多项式Lyapunov函数,再应用高斯-牛顿精化和有理向量恢复将数值多项式转化为验证的有理多项式Lyapunov函数.     

Monotonicity of solutions converging to a saddle point equilibrium

If a difference equation has a saddle point equilibrium, then there are solutions which converge to this equilibrium. For second order equations, conditions are given which imply that these solutions are monotone. These results are used to analyze a rational difference equation which possesses a saddle point equilibrium.     

平行六边形区域上的快速离散傅立叶变换

In this paper, we propose a fast algorithm for computing the DGFT (Discrete Generalized Fourier Transforms) on hexagon domains [6], based on the geometric properties of the domain. Our fast algorithm (FDGFT) reduces the computation complexity of DGFT from O(N4) to O(N2 log N). In particulary, for N =2^P23^P34^P45^P56^P6, the floating point computation working amount equals to(17/2P2 16p3 135/8p4 2424/25p5 201/2P6)3N^2. Numerical examples are given to access our analysis.     

Global dynamics of a competitive system of rational difference equations

In this paper, we study the qualitative behavior of a competitive system of second‐order rational difference equations. More precisely, we investigate the boundedness character, existence and uniqueness of positive equilibrium point, local asymptotic stability and global stability of the unique positive equilibrium point and rate of convergence of positive solutions of the system. Some numerical examples are given to verify our theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.     

二元向量值有理插值存在性的一种判别方法

文章给出了对于矩形网格上基于二元Newton插值公式的二元向量值有理插值存在性的充要条件.在存在的情况下,建立了具有显式表达式的不同于向量连分式的二元向量值有理插值函数,并且这种方法具有承袭性.最后给出的实例说明了这种算法的有效性.     

Rational ODEs with Movable Algebraic Singularities

A class of second-order rational ordinary differential equations, admitting certain families of formal algebraic series solutions, is considered. For all solutions of these equations, it is shown that any movable singularity that can be reached by analytic continuation along a finite-length curve is an algebraic branch point. The existence of these formal series expansions is straightforward to determine for any given equation in the class considered. We apply the theorem to a family of equations, admitting different kinds of algebraic singularities. As a further application we recover the known fact for generic values of parameters that the only movable singularities of solutions of the Painlevé equations   P_II – P   _VI   are poles.     

Inclusion Theorems for Non-linear Difference Equations with Applications

For non-linear difference equations an inclusion theorem is set up which guarantees the existence of solutions in a given asymptotic stripe. This theorem is the limit case of an inclusion theorem for a boundary value problem in a finite interval. Different applications to rational difference equations of order two are given.     

Solution of Lane–Emden type equations using rational Bernoulli functions

In this paper, a numerical method for solving Lane‐Emden type equations, which are nonlinear ordinary differential equations on the semi‐infinite domain, is presented. The method is based upon the modified rational Bernoulli functions; these functions are first introduced. Operational matrices of derivative and product of modified rational Bernoulli functions are then given and are utilized to reduce the solution of the Lane‐Emden type equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Copyright © 2015 John Wiley & Sons, Ltd.     

The compass (star) identity for vector-valued rational interpolants

The compass identity (Wynn's five point star identity) for Padé approximants connects neighbouring elements called N, S, E, W and C in the Padé table. Its form has been extended to the cases of rational interpolation of ordinary (scalar) data and interpolation of vector-valued data. In this paper, full specifications of the associated five point identity for the scalar denominator polynomials and the vector numerator polynomials of the vector-valued rational interpolants on real data points are given, as well as the related generalisations of Frobenius' identities. Unique minimal forms of the polynomials constituting the interpolants and results about unattainable points correspond closely to their counterparts in the scalar case. This revised version was published online in June 2006 with corrections to the Cover Date.     

Existence of asymptotically stable solutions for a mixed functional nonlinear integral equation in N variables

Motivated by recent known results about the solvability of nonlinear functional integral equations in one, two or N variables, this paper proves the existence of asymptotically stable solutions for a mixed functional integral equation in N variables with values in a general Banach space via a fixed point theorem of Krasnosels'ki?  type. In order to illustrate the results obtained here, an example is given.     

Differential Galois theory and non-integrability of planar polynomial vector fields

We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the “Risch algorithm”. In this way we point out the connection of the non integrability with some higher transcendent functions, like the error function.     

Solution of the equations associated with rational Runge-Kutta methods of orders up to four

The definition of rational Runge-Kutta methods for systems of equations is given. The equations associated with those methods are solved for the second, third and fourth order. The many free parameters in the solutions can be used to derive A- and L-stable explicit methods.     

An extension of Tikhonov's theorem in singular perturbations for the planar case

A classical error estimate for singularly perturbed ordinary differential equations due to A. N. Tikhonov is generalized in the neighbourhood of a so-called leaving point where a certain stability assumption ceases to be valid. This is done for the planar case, and an application to the Van der Pol relaxation oscillator is given.     

多孔胃管洗胃的数学方程

提出了多孔胃管洗胃的方程及其拟合方程.把多孔胃管的终端划分为连续的含孔微元,用离散分析研究了各微元流体在洗胃冲吸中的方程,从模拟洗胃实验得出各孔流速的约束条件,为胃管的工程估算与合理开孔提供数学依据.     

On the computation of resolvents and Galois groups

A large class of algorithms for computing resolvents of algebraic equations — so called rational transformations — is investigated and characterized group theoretically. The concept of rational transformations implies a program how to develop good methods to determine the Galois group of an equation. It is shown that some known methods are special cases of rational transformations, and a new procedure to find the group of a sextic equation is given. Moreover, all cases in which Galois resolvents can be found by means of rational transformations are classified.     

Wedge dislocations and three-dimensional gravity

The expression for the free energy of arbitrary static distribution of wedge dislocations in elastic media is proposed. In the framework of geometric theory of defects, the free energy is given by the Euclidean action for (1+2)-dimensional gravity interacting with N point particles. Relative movement of particles in gravity corresponds to bending of dislocations. The equations of equilibrium are obtained and analyzed. For two dislocations, the solution is found explicitly through hypergeometric functions.     

Solitons and Other Solutions for the Generalized KdV–mKdV Equation with Higher-order Nonlinear Terms

The generalized sub-ODEmethod, the rational (G' ⁄ G)-expansionmethod, the exp-function method and the sine-cosine method are applied for constructing many traveling wave solutions of nonlinear partial differential equations (PDEs). Some illustrative equations are investigated by these methods and many hyperbolic, trigonometric and rational function solutions are found. We apply these methods to obtain the exact solutions for the generalized KdV-mKdV (GKdV-mKdV) equation with higher-order nonlinear terms. The obtained results confirm that the proposed methods are efficient techniques for analytic treatment of a wide variety of nonlinear partial differential equations in mathematical physics. We compare between the results yielding from these methods. Also, a comparison between our new results in this paper and the well-known results are given.     

Convergence properties of materially and geometrically non-linear finite-element spatial beam analysis

The way the non-linear constitutive equations in the spatial beam formulations are solved, influences the rate of convergence and the computational cost. Three different approaches are studied: (i) the direct global approach, where the constitutive equations are taken to be the iterative part of the global governing equations, (ii) the local (or indirect global) approach, where the constitutive equations are solved separately in each step of the global iteration, and (iii) the partly reduced approach, which is the combination of (i) and (ii). The approaches are compared with regard to the number of global iterations and the total number of floating point operations. The direct global approach is found to be the best choice.     

A nonlinear two-phase Stefan problem with melting point gradient: a constructive approach

We consider a one-dimensional two-phase Stefan problem, modeling a layer of solid material floating on liquid. The model includes internal heat sources, variable total mass (resulting e.g. from sedimentation or erosion), and a pressure-dependent melting point. The problem is reduced to a set of nonlinear integral equations, which provides the basis for an existence and uniqueness proof and a new numerical method. Numerical results are presented.