Schrodinger方程的变分迭代解法

该文以Ｓｃｈｒｏｄｉｎｇｅｒ方程为例,分析变分迭代法的一些基本特点．在该方法中引进了一广义拉氏乘子构造了一迭代格式,拉氏乘子可由变分理论最佳识别．由于在识别拉氏乘子是应用了限制变分的概念,所以只能通过迭代才能得到收敛解．为了加快收敛速度,可以在初始近似引入待定常数,而待定常数又可用各种方式最佳识别．文中初步分析了该方法的收敛性,对于Ｓｃｈｒｏｄｉｎｇｅｒ方程,其一阶近似即可得到Ｊｏｓｔ解．     

无粘、可压、绝热流体的Euler方程初值问题的适定性

根据分层理论提供的基本方法,讨论Euler方程的初值问题的适定性,给出了方程的典型初边值问题适定性的判别条件,确定了Euler方程的局部(准确)解的解空间构造,对适定问题给出了解析解的计算公式．     

Variational and reciprocal principles in thermoelasticity without energy dissipation

In the context of the linear theory of thermoelasticity without energy dissipation for homogeneous and isotropic materials, variational principles of Biot-and Hamilton-types and a reciprocal principle of Betti-Rayleigh-type are presented.     

Continuous and Discrete Clebsch Variational Principles

The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group via a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity map is a Lie algebra action, thereby producing the Euler–Poincaré (EP) equation for the vector space variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff) arise as momentum maps in the Clebsch approach. In the case of finite-dimensional Lie groups, the Clebsch variational principle is discretized to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space. We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretize infinite-dimensional Clebsch systems, so as to produce conservative numerical methods for fluid dynamics.      

Modified Variational Iteration Method for Heat and Wave-Like Equations

In this paper, we apply the modified variational iteration method (MVIM) for solving the heat and wave-like equations. The proposed modification is made by introducing He’s polynomials in the correction functional. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that proposed technique solves nonlinear problems without using the Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.     

Variational approach to impulsive evolution equations

This article uses variational method for studying existence and uniqueness of solutions for impulsive evolution equations. The main techniques include Hilbert triple, Sobolev embedding theorem, Galerkin approximation and weak convergence for passing to the limit.     

An alternative approach to establishment of a variational principle for the torsional problem of piezoelastic beams

The semi-inverse method is adopted to search for a variational principle for an unelectroded piezoelastic beam. A trial variational formulation with energy integral is constructed with an unknown function, which is identified so that the Euler–Lagrange equations are equivalent to the governing equations.     

Applications of He’s principles to partial differential equations

This paper applies the variational iteration method (VIM) and semi-inverse variational principle to obtain solutions of linear and nonlinear partial differential equations. The nonlinear model is considered from gas dynamics, fluid dynamics and Burgers equation. The linear model is the heat transfer (diffusion) equation. Results show that variational iteration method is a powerful mathematical tool for solving linear and nonlinear partial differential equations, and therefore, can be widely applied to engineering problems.     

New Approach to Stochastic Optimal Control

This paper provides new insights into the solution of optimal stochastic control problems by means of a system of partial differential equations, which characterize directly the optimal control. This new system is obtained by the application of the stochastic maximum principle at every initial condition, assuming that the optimal controls are smooth enough. The type of problems considered are those where the diffusion coefficient is independent of the control variables, which are supposed to be interior to the control region. Two referees provided useful suggestions. Both authors gratefully acknowledge financial support from the regional Government of Castilla y León (Spain) under Project VA099/04, the Spanish Ministry of Education and Science and FEDER funds under Project MTM2005-06534.     

A remark on a paper of F. Luca and A. Sankaranarayanan

We generalize a result of F. Luca and A. Sankaranarayanan by proving that the set of n for which ϕ(1) + ⃛ + ϕ(n) is squareful is of zero density. A similar statement holds for σ (n) instead of ϕ(n) and for some other multiplicative functions. Dedicated to Professor Eugenijus Manstavičius on his 60th anniversary     

Variational method to differential equations with instantaneous and non-instantaneous impulses

The aim of this paper is to study the existence of solutions for second-order differential equations with instantaneous and non-instantaneous impulses. Applying variational method, the existence result is obtained.     

Algorithms for the Laplace–Stieltjes Transforms of First Return Times for Stochastic Fluid Flows

We derive several algorithms, including quadratically convergent algorithms, which can be used to calculate the Laplace–Stieltjes transforms of the time taken to return to the initial level in the Markovian stochastic fluid flow model. We give physical interpretations of the algorithms and consider their numerical analysis. The numerical performance of the algorithms, which depends on the physical properties of the process, is discussed and illustrated with simple examples. Besides the powerful algorithms, this paper contributes interesting theoretical results. In particular, the methodology for constructing these algorithms is a valuable contribution to the theory of fluid flow models. Moreover, useful physical interpretations of the algorithms, and related expressions, given in terms of the fluid flow model, can assist in further analysis and help in a better understanding of the model. The authors would like to thank the Australian Research Council for funding this research through Discovery Project DP0770388.     

Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper

The aim of the paper is to relate computational and arithmetic questions about Euler’s constant γ with properties of the values of the q-logarithm function, with natural choice of~q. By these means, we generalize a classical formula for γ due to Ramanujan, together with Vacca’s and Gosper’s series for γ, as well as deduce irrationality criteria and tests and new asymptotic formulas for computing Euler’s constant. The main tools are Euler-type integrals and hypergeometric series. 2000 Mathematics Subject Classification Primary—11Y60; Secondary—11J72, 33C20, 33D15 The work of the second author is supported by an Alexander von Humboldt research fellowship Dedication: To Leonhard Euler on his 300th birthday.     

Fluctuations in the mean of Euler’s phi function

We consider the error term in the mean value estimate of Euler’s phi function ψ(n), and show that it is Ω_+- (x(log log x)^1/2). This improves on the earlier results of Pillai and Chowla, and of Erdös and Shapiro.     

There Are Not Too Many Magic Configurations

A finite planar point set P is called a magic configuration if there is an assignment of positive weights to the points of P such that, for every line l determined by P, the sum of the weights of all points of P on l equals 1. We prove a conjecture of Murty from 1971 and show that if a set of n points P is a magic configuration, then P is in general position, or P contains n−1 collinear points, or P is a special configuration of 7 points. The research by Rom Pinchasi was supported by a Grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

A Variational Approach to Dynamical Systems and Its Numerical Simulation

We propose and describe an alternative perspective to the study and numerical approximation of dynamical systems. It is based on a variational approach that seeks to minimize the quadratic error understood as a deviation of paths from being a solution of the corresponding system. Although this philosophy has been examined recently from the point of view of the direct method, we exploit optimality conditions and steepest descent strategies to establish precise and easy-to-implement numerical schemes for the approximation. We show the practical performance in a number of selected examples and indicate how this strategy, with minor changes, may also be used to deal with boundary value problems. Our emphasis is placed more so on relevant results that justify the numerical implementation and less on abstract theoretical results under optimal sets of assumptions.     

A superadditive property of Hadamard’s gamma function

Hadamard’s gamma function is defined by

Variational iteration method for the Burgers’ flow with fractional derivatives—New Lagrange multipliers

The flow through porous media can be better described by fractional models than the classical ones since they include inherently memory effects caused by obstacles in the structures. The variational iteration method was extended to find approximate solutions of fractional differential equations with the Caputo derivatives, but the Lagrange multipliers of the method were not identified explicitly. In this paper, the Lagrange multiplier is determined in a more accurate way and some new variational iteration formulae are presented.     

Another approach to asymptotic expansions for Euler’s approximations of semigroups

In [2], optimal bounds for the remainder terms in asymptotic expansions for Euler’s approximations of semigroups were derived. The approach was based on applications of the Fourier-Laplace transforms, which allowed one to reduce the problem to estimation of error terms in the Law of Large Numbers. In this paper, we propose an alternative (direct) approach based on application of certain integro-differential identities (the so-called multiplicative representations of differences). Such identities were introduced by Bentkus [3] and applied (see Bentkus and Paulauskas [4]) to derive the optimal convergence rates in Chernoff-type lemmas and Euler’s approximations of semigroups. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 267–284, April–June, 2006.