PADE APPROXIMANTS AS LIMITS OF RATIONAL FUNCTIONS OF BEST APPROXIMATION IN ORLICZ SPACE

In this paper, we prove that the best rational approximation of a given analytic function in Orlicz space L~*(G), where G = {|z|≤∈}, converges to the Pade approximants of the function as the measure of G approaches zero.

CONVERGENCE IN MEASURE OF MULTIVARIATE PADE APPROXIMANTS

Convergence conclusions of Pade approximants in the univariate case can be found in various papers. However,resuhs in the multivariate case are few.A.Cuyt seems to be the only one who discusses convergence for multivariate Pade approximants,she gives in[2]a de Montessus de Bollore type theorem.In this paper,we will discuss the zero set of a real multivariate polynomial,and present a convergence theorem in measure of multivariate Pade approximant.The proof technique used in this paper is quite different from that used in the univariate case.

Forward Scattering Series and Pade Approximants for Acoustic Wavefield Propagation in a Vertically Varying Medium

We present the application of the theory of Pade approximants to extending the perturbative solutions of acoustic wave equation for a three dimensional vertically varying medium with one interface.These type of solutions have limited convergence properties depending on either the degree of contrast between the actual and the reference medium or the angle of incidence of a plane wave component.We show that the sequence of Pade approximants to the partial sums in the forward scattering series for the 3D wave equation is convergent for any contrast and any incidence angle.This allows the construction of any reflected waves including phase-shifted post-critical plane waves and,for a point-source problem,refracted events or headwaves,and it also provides interesting interpretations of these solutions in the scattering theory formalism.

Adomian decomposition method and Padé approximants for solving the Blaszak-Marciniak lattice

The Adomian decomposition method （ADM） and Pade approximants are combined to solve the well-known Blaszak-Marciniak lattice, which has rich mathematical structures and many important applications in physics and mathematics. In some cases, the truncated series solution of ADM is adequate only in a small region when the exact solution is not reached. To overcome the drawback, the Pade approximants, which have the advantage in turning the polynomials approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. By using the ADM-Pade technique, the soliton solutions of the Blaszak-Marciniak lattice are constructed with better accuracy and better convergence than by using the ADM alone. Numerical and figurative illustrations show that it is a promising tool for solving nonlinear problems.

The construction of homoclinic and heteroclinic orbitals in asymmetric strongly nonlinear systems based on the Pad'e approximant

In this paper, the extended Pade approximant is used to construct the homoclinic and the heteroclinic trajectories in nonlinear dynamical systems that are asymmetric at origin. Meanwhile, the conservative system, the autonomous system, and the nonautonomous system equations with quadratic and cubic nonlinearities are considered. The disturbance parameter ~ is not limited to being small. The ranges of the values of the linear and the nonlinear term parameters, which are variables, can be determined when the boundary values are satisfied. New conditions for the potentiality and the convergence are posed to make it possible to solve the boundary-value problems formulated for the orbitals and to evaluate the initial amplitude values.

COMPUTATION FORMULAS OF GENERALIZED INVERSE PAD APPROXIMANTS USING FOR SOLUTION OF INTEGRAL EQUATIONS

For the generalized inverse function-valued Pade approximants, its intact computation formulas are given. The explicit determinantal formulas for the denominator scalar polynomials and the numerator function-valued polynomials are first established. A useful existence condition is given by means of determinant form.

分子阻障内旋转配分函数的二点pade’逼近

一、分子阻障内旋转配分函数的计算在理论化学中一直为人们所研究,早在五十年代Pitzer就用数值方法进行了计算,但求得这一配分函数的近似解析表示式仍有重要意义,在文献中作者曾用Wigner-Kirkwood微扰展开法求得阻障内旋转配分函数的近似解析表示式,根据此式所作的数值计算结果表明:在温度较高,内旋转动惯量较大时和Pitzer的数值计算结果符合得甚好,但随着温度的降低偏离也就增大,计算Wigner-Kirkwood展开式的的高次项固然可以改善所得之配分函数,但此时计算烦杂,困难较大,本文则采用二点Pade’逼近法将Wigner-Kirkwood微扰展开式（包括一级量子改正项）作为高温展开式,并选用配分函数Q=Σe-tEi的前三项作为其低温展开式,然后用二点Pade’逼近进行内插,求得阻障内旋转配分函数的[5/5]Pade’逼近式,它在整个温度范围内均能给出良好的数值结果。

Numerical Solution of the Diffusion Equation with Restrictive Pade Approximation

The problem of solving the linear diffusion equation by a method related to the Restrictive Pade Approximation (RPA) is considered. The advantage is that it has the exact value at certain r. This method will exhibit several advantages for example highly accurate, fast and with good results, etc. The absolutely error is still very small. The obtained results are compared with the exact solution and the other methods. The numerical results are in agreement with the exact solution.

Artificial boundary condition for Klein-Gordon equation by constructing mechanics structure

An innovative local artificial boundary condition is proposed to numerically solve the Cauchy problem of the Klein-Gordon equation in an unbounded domain.Initially,the equation is considered as the axial wave prop-agation in a bar supported on a spring foundation.The numerical model is then truncated by replacing the half-infinitely long bar with an equivalent mechanical structure.The effective frequency-dependent stiffness of the half-infinitely long bar is expressed as the sum of rational terms using Pade approximation.For each term,a corresponding substructure composed of dampers and masses is constructed.Finally,the equivalent mechan-ical structure is obtained by parallelly connecting these substructures.The proposed approach can be easily implemented within a standard finite element framework by incorporating additional mass points and damper elements.Numerical examples show that with just a few extra degrees of freedom,the proposed approach effec-tively suppresses artificial reflections at the truncation boundary and exhibits first-order convergence.

Mean-field and high temperature series expansion calculations of some magnetic properties of Ising and XY antiferromagnetic thin-films

In this work, the magnetic properties of Ising and XY antiferromagnetic thin-films are investigated each as a function of Neel temperature and thickness for layers （n = 2, 3, 4, 5, 6, and bulk （∞） by means of a mean-field and high temperature series expansion （HTSE） combined with Pade approximant calculations. The scaling law of magnetic susceptibility and magnetization is used to determine the critical exponent γ, veff （mean）, ratio of the critical exponents γ/v, and magnetic properties of Ising and XY antiferromagnetic thin-films for different thickness layers n = 2, 3, 4, 5, 6, and bulk （∞）.

Numerical Treatment of Initial Value Problems of Nonlinear Ordinary Differential Equations by Duan-Rach-Wazwaz Modified Adomian Decomposition Method

We employ the Duan-Rach-Wazwaz modified Adomian decomposition method for solving initial value problems for the systems of nonlinear ordinary differential equations numerically. In order to confirm practicality, robustness and reliability of the method, we compare the results from the modified Adomian decomposition method with those from the MATHEMATICA solutions and also from the fourth-order Runge Kutta method solutions in some cases. Furthermore, we apply Padé approximants technique to improve the solutions of the modified decomposition method whenever the exact solutions exist.

Free Convection of a Viscous Electrically Conducting Fluid Past a Stretching Surface

Free convection of a viscous electrically conducting liquid past a vertical stretching surface is investigated in the presence of a transverse magnetic field.Natural convection is driven by both thermal and solutal buoyancy.The original partial differential equations governing the problem are turned into a set of ordinary differential equations through a similar variables transformation.This alternate set of equations is solved through a Differential Transform Method(DTM)and the Pade approximation.The response of the considered physical system to the non-dimensional parameters accounting for the relative importance of different effects is assessed considering different situations.

Delay-dependent stability and added damping of SDOF real-time dynamic hybrid testing

It is well-recognized that a transfer system response delay that reduces the test stability inevitably exists in real-time dynamic hybrid testing （RTDHT）. This paper focuses on the delay-dependent stability and added damping of SDOF systems in RTDHT. The exponential delay term is transferred into a rational fraction by the Pad6 approximation, and the delay-dependent stability conditions and instability mechanism of SDOF RTDHT systems are investigated by the root locus technique. First, the stability conditions are discussed separately for the cases of stiffness, mass, and damping experimental substructure. The use of root locus plots shows that the added damping effect and instability mechanism for mass are different from those for stiffness. For the stiffness experimental substructure case, the instability results from the inherent mode because of an obvious negative damping effect of the delay. For the mass case, the delay introduces an equivalent positive damping into the inherent mode, and instability occurs at an added high frequency mode. Then, the compound stability condition is investigated for a general case and the results show that the mass ratio may have both upper and lower limits to remain stable. Finally, a high-emulational virtual shaking table model is built to validate the stability conclusions.

A New Form of Boussineq Equation with Improved Linear Dispersion Characteristics

It has been shown that Boussinesq type equations, which include the lowest order effects of nonlinearity and frequency dispersion, can provide an accurate description of wave evolution in coastal regions. But different linear dispersion characteristics of the equation can be obtained by different integrating method. In this paper, a new form of the Boussinesq equation is derived by use of two different layer horizontal velocity variables instead of the commonly used depth-averaged velocity or an arbitrary layer velocity. This significantly improves the linear dispersion properties of the Boussinesq equation and enables it to be applied to a wider range of water depth.

应用双幂指数散射截面计算离子射程分布

应用作者本人最近提出的双幂指数散射截面求解了离子射程分布输运方程.尽管当ｍ＝０．２时单、双幂指数两种散射截面相差甚大，但只要利用一适当长度单位，两种截面所给出的分布函数却相差很小．另外，采用了Ｐａｄｅ＇逼近重新组合射程分布函数，计算精度都达到１０－8以上．

Solving multi-object radar cross section based on wide-angle parabolic equation method

Based on a Pade approximation, a wide-angle parabolic equation method is introduced for computing the multiobject radar cross section （RCS） for the first time. The method is a paraxial version of the scalar wave equation, which solves the field by marching them along the paraxial direction. Numerical results show that a single wide-angle parabofic equation run can compute multi-object RCS efficiently for angles up to 45 ° . The method provides anew and efficient numerical method for computation electromagnetics.

Numerical Simulation of Modified Kortweg-de Vries Equation by Linearized Implicit Schemes

In this paper, we are going to derive four numerical methods for solving the Modified Kortweg-de Vries (MKdV) equation using fourth Pade approximation for space direction and Crank Nicolson in the time direction. Two nonlinear schemes and two linearized schemes are presented. All resulting schemes will be analyzed for accuracy and stability. The exact solution and the conserved quantities are used to highlight the efficiency and the robustness of the proposed schemes. Interaction of two and three solitons will be also conducted. The numerical results show that the interaction behavior is elastic and the conserved quantities are conserved exactly, and this is a good indication of the reliability of the schemes which we derived. A comparison with some existing is presented as well.

Implications of Stahl's Theorems to Holomorphic Embedding Part II: Numerical Convergence

What has become known as Stahl's Theorem in power-engineering circles has been used to justify a convergence guarantee of the Holomorphic Embedding Method(HEM)as it applies to the power-flow problem.In this,the second part of a two-part paper,we examine implications to numerical convergence of the HEM and the numerical properties of a Pade approximant algorithm.We show that even if the point of interest is within the convergence domain,numerical convergence of the sequence of Pade approximants computed with finite precision is not guaranteed.We propose a convergence factor equation that can be used to both estimate the convergence rate and the capacity of the branch cut.We also show that the study of convergence properties of the Pade approximant is the study of the location of the branch-points of the function,which in turn dictate branch-cut topology and capacity and,therefore,convergence rate.

Implications of Stahl's Theorems to Holomorphic Embedding Part I: Theoretical Convergence

What is known as StahPs Theorem in power engineering circles is used to justify a convergence guarantee of the Holomorphic Embedding Method(HEM)as it applies to powerflow(PF)problem.In this two-part paper,we examine in more detail the implications of Stahl's theorems to both theoretical and numerical convergence for a wider range of problems to which these theorems are now being applied.We show that the difference between StahPs extremal domain and the function's domain is responsible for theoretical nonconvergence and that the fundamental cause of numerical nonconvergence is the magnitude of logarithmic capacity of the branch cut,a concept central to understanding nonconvergence.We introduce theorems using the necessary mathematical parlance and then translate the language to show its implications to convergence of nonlinear problems in general and specifically to the PF problem.We show that,among other possibilities,the existence of Chebotarev points,which are embedding specific,are a possible theoretical impediment to convergence・The theoretical foundation of Part I is necessary for understanding the numerical behavior of HEM discussed in Part II.

NUMERICAL SIMULATION FOR SOLITARY WAVES OF RLW EQUATION

A high order finite difference scheme for solving the Regularised Long Wave(RLW) equation, based on the high order Pade approximation and Lax-Wendroff type timediscretization, was developed in this paper. Some numerical examples, including the propagation of asingle soliton, the interaction of double solitary waves and the temporal evolution of a Maxwellianinitial pulse, were studied to test the accuracy, efficiency and conservation property of thescheme. Compared with the results based on finite element methods, the scheme is proven successful.