含三角函数的一般形式复杂对偶积分方程组的理论解

本文基于Gopson法，进行研究，改进，推广，应用于一般形式，复杂的对偶积分方程组的求解，首先引入函数进行方程组变换，其次引入未知函数的积分变换实现退耦，应用Abel反演变换，使方程组正则化为Fredholm第二类积分方程组，并由此给出对偶积分方程组的一般性解，本文给出的解法和理论解，可供求解复杂的数学，物理，力学中的混合边值问题参考，选用．同时也提供求解复杂的对偶积分方程组另一种有效的解法。     

State space collapse with application to heavy traffic limits for multiclass queueing networks

Heavy traffic limits for multiclass queueing networks are a topic of continuing interest. Presently, the class of networks for which these limits have been rigorously derived is restricted. An important ingredient in such work is the demonstration of state space collapse. Here, we demonstrate state space collapse for two families of networks, first-in first-out (FIFO) queueing networks of Kelly type and head-of-the-line proportional processor sharing (HLPPS) queueing networks. We then apply our techniques to more general networks. To demonstrate state space collapse for FIFO networks of Kelly type and HLPPS networks, we employ law of large number estimates to show a form of compactness for appropriately scaled solutions. The limits of these solutions are next shown to satisfy fluid model equations corresponding to the above queueing networks. Results from Bramson [4,5] on the asymptotic behavior of these limits then imply state space collapse. The desired heavy traffic limits for FIFO networks of Kelly type and HLPPS networks follow from this and the general criteria set forth in the companion paper Williams [41]. State space collapse and the ensuing heavy traffic limits also hold for more general queueing networks, provided the solutions of their fluid model equations converge. Partial results are given for such networks, which include the static priority disciplines. This revised version was published online in June 2006 with corrections to the Cover Date.     

一般形式的奇异对偶积分方程组正交多项式求解法

基于Jacobi正交多项式法,直接求解一般形式的对偶积分方程组,将对偶积分方程组中的未知函数,表示成n次Jacobi正交多项式级数,用正交多项式将奇异对偶积分方程组,化成线性代数方程组,通过求解级数中的各项系数,由此给出奇异对偶积分方程组的一般性解,并严格证明了奇异对偶积分方程组和由它化成的线性代数方程组的等价性,解的存在性和解的表示形式不唯一性．本文给出的理论解和解法,可供求解复杂的数学、物理、软科学中的混合边值问题应用．     

Entire solutions for some Fermat type functional equations concerning difference and partial differential in ℂ2

The main purpose of this paper is concerned with the existence and the forms of transcendental entire solutions of several Fermat type functional equations concerning difference and partial differential in ?², by utilizing the Nevanlinna theory of meromorphic functions in several complex variables. Some results are obtained to give the forms of entire solutions for such equations, which are some improvements and generalizations of the previous theorems given by Xu and Cao, Liu and Dong. Moreover, some examples are given to show that there are great differences in the forms of transcendental entire solutions with finite order of Fermat type functional equations between in several complex variables and in a single complex variable.

     

一类对偶积分方程组正则化为Cauchy奇异积分方程组解法

本文基于Mellin变换法求解复杂更一般形式的对偶积分方程组．通过积分变换，由实数域化成复数域上的方程组，引入未知函数的积分变换，移动积分路径，应用Cauchy积分定理，实现退耦正则化为Cauchy奇异积分方程组，由此给出一般性解，并严格证明了对偶积分方程组退耦正则化为Cauchy奇异积分方程组与原对偶积分方程组等价性，以及对偶积分方程组解的存在性和唯一性．给出的解法和理论解，作为求解复杂对偶积分方程组一种有效解法，可供求解复杂的数学、物理、力学中的混合边值问题应用．     

A package for symbolic solution of real functional equations of real variables

Summary This paper discusses the problems associated with the symbolic treatment of functional equations and presents a Mathematica package for the solution of real functional equations of real variables. The package includes a minimal basic database which contains a reduced set of functional equations with its four components: equation, domain, class and the corresponding solution. The word minimal is used in the sense that any equation that is solvable by the system using non-searching methods is excluded from the database. The package incorporates a searching algorithm which can solve functional equations independently of their notation and their algebraic representation. Not only general solutions but particular and candidate solutions are dealt with. This implies a careful analysis of domains and classes. The package includes some methods for solving functional equations, which are used when the input functional equations are not found in the database. Some methods have been implemented internally and some are in an external package. Finally, some examples illustrate the use of the package.     

Numerical analysis for bulk-arrival queueing systems: Root-finding and steady-state probabilities inGI r /M/1 queues

Queueing theorists have presented, as solutions to many queueing models, probability generating functions in which state probabilities are expressed as functions of the roots of characteristic equations, evaluation of the roots in particular cases being left to the reader. Many users have complained that such solutions are inadequate. Some queueing theorists, in particular Neuts [6], rather than use Rouché's theorem to count roots and an equation-solver to find them, have developed new algorithms to solve queueing problems numerically, without explicit calculation of roots. Powell [7] has shown that in many bulk service queues arising in transportation models, characteristic equations can be solved and state probabilities can be found without serious difficulty, even when the number of roots to be found is large. We have slightly modified Powell's method, and have extended his work to cover a number of bulk-service queues discussed by Chaudhry et al. [1] and a number of bulk-arrival queues discussed in the present paper.     

The cyclic reduction algorithm: from Poisson equation to stochastic processes and beyond

Cyclic reduction is an algorithm invented by G. H. Golub and R. W. Hockney in the mid 1960s for solving linear systems related to the finite differences discretization of the Poisson equation over a rectangle. Among the algorithms of Gene Golub, it is one of the most versatile and powerful ever created. Recently, it has been applied to solve different problems from different applicative areas. In this paper we survey the main features of cyclic reduction, relate it to properties of analytic functions, recall its extension to solving more general finite and infinite linear systems, and different kinds of nonlinear matrix equations, including algebraic Riccati equations, with applications to Markov chains, queueing models and transport theory. Some new results concerning the convergence properties of cyclic reduction and its applicability are proved under very weak assumptions. New formulae for overcoming breakdown are provided.     

An integral equation method for the electromagnetic scattering from cavities

Consider a time‐harmonic electromagnetic plane wave incident on a cavity in a ground plane. The physical process is modelled by Maxwell's equations. In this paper, integral representations of the solutions to the model problem in both fundamental polarizations are derived and studied. Existence and uniqueness of the solutions for the integral equations are established. The integral equations approach forms a basis for numerical solution of the model problem. In particular, for each fundamental polarization, an integral formulation with Gårding‐type estimates is derived. These formulations provide a basis for variational boundary element methods for solving the cavity problem. The Gårding‐type estimates imply convergence results for conforming boundary element methods. Copyright © 2000 John Wiley & Sons, Ltd     

The similarity forms and invariant solutions of two-layer shallow-water equations

In this study symmetry group properties and general similarity forms of the two-layer shallow-water equations are discussed by Lie group theory. We represent that Lie group theory can be used as an effective approach for investigation of the self-similar solutions for the shallow-water equations with variable inflow as the generalization of dimensional analysis that was used so far for a regular approach in the literature. We also represent that the results obtained by dimensional analysis are just a special case of the results obtained by Lie group theory and it is possible to obtain the new similarity forms and the new variable inflow functions for the study of gravity currents in two-layer flow under shallow-water approximations based on Lie group theory. The symmetry groups of the system of nonlinear partial differential equations are found and the corresponding similarity and reduced forms are obtained. Some similarity solutions of the reduced equations are investigated. It is shown that reduced equations and similarity forms of the system depend on the group parameters. We show that an analytic similarity solution for the system of equations can be found for some special values of them. For other values of the group parameters, the similarity solutions of the two-layer shallow-water equations representing the gravity currents with a variable inflow are found by the numeric integration.     

Solving evolutionary-type differential equations and physical problems using the operator method

We present a general operator method based on the advanced technique of the inverse derivative operator for solving a wide range of problems described by some classes of differential equations. We construct and use inverse differential operators to solve several differential equations. We obtain operator identities involving an inverse derivative operator, integral transformations, and generalized forms of orthogonal polynomials and special functions. We present examples of using the operator method to construct solutions of equations containing linear and quadratic forms of a pair of operators satisfying Heisenberg-type relations and solutions of various modifications of partial differential equations of the Fourier heat conduction type, Fokker–Planck type, Black–Scholes type, etc. We demonstrate using the operator technique to solve several physical problems related to the charge motion in quantum mechanics, heat propagation, and the dynamics of the beams in accelerators.     

A Critical Survey of Queueing Theory — Part 2

In this part of the paper, an elementary method (Erlang's) of setting up equations which describe queueing situations is discussed. Methods of solving these equations are given by reference to a simple example, and then to a more complex one.     

ON AXISYMMETRIC TRAVELING WAVES AND RADIAL SOLUTIONS OF SEMI‐LINEAR ELLIPTIC EQUATIONS

ABSTRACT. Combining analytical techniques from perturbation methods and dynamical systems theory, we present an elementaryapproach to the detailed construction of axisymmetric diffusive interfaces in semi‐linear elliptic equations. Solutions of the resulting non‐autonomous radial differential equations can be expressed in terms of a slowlyvarying phase plane system. Special analytical results for the phase plane system are used to produce closed‐form solutions for the asymptotic forms of the curved front solutions. These axisym‐metric solutions are fundamental examples of more general curved fronts that arise in a wide variety of scientific fields, and we extensivelydiscuss a number of them, with a particular emphasis on connections to geometric models for the motion of interfaces. Related classical results for traveling waves in one‐dimensional problems are also reviewed briefly. Manyof the results contained in this article are known, and in presenting known results, it is intended that this article be expositoryin nature, providing elementarydemonstrations of some of the central dynamical phenomena and mathematical techniques. It is hoped that the article serves as one possible avenue of entree to the literature on radiallysymmetric solutions of semilinear elliptic problems, especiallyto those articles in which more advanced mathematical theoryis developed.     

广义变系数Kdv方程的对称及其群不变解

利用经典李对称的方法对广义变系数Kdv方程进行研究,利用这种方法得到了该方程的一个新的精确解,这种方法的基本思路是通过对称约化将原来较难求解的偏微分方程转化为较易求解的常微分方程进行求解.实例证明这种方法具有一般性,适合于求一大类变系数的非线性演化方程.     

Analysis of the second order matrix Riccati equations

This paper is concerned with periodic solutions of 2x2 autonomous matrix Riccati differential equations. The author had given a necessary and sufficient condition for periodicity of solutions of matrix Riccati differential equations of general type and some examples. However, it is not so simple to verify whether this condition is satisfied or not. So this paper simplifies the verification by restricting to special cases. In particular, we show that there may exist periodic solutions for any case where the coefficient matrix of the linear part of the equation has complex eigenvalues if we choose an initial value suitably. Many examples having a periodic solution are also shown by systematic analysis; such examples are seldom seen in the literature.     

Generalized solutions of mixed problems for first-order partial functional differential equations

A theorem on the existence of solutions and their continuous dependence upon initial boundary conditions is proved. The method of bicharacteristics is used to transform the mixed problem into a system of integral functional equations of the Volterra type. The existence of solutions of this system is proved by the method of successive approximations using theorems on integral inequalities. Classical solutions of integral functional equations lead to generalized solutions of the original problem. Differential equations with deviated variables and differential integral problems can be obtained from the general model by specializing given operators. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 804–828, June, 2006.     

The equivalence of team theory's integral equations and a Cauchy system: sensitivity analysis of a variational problem

Team decision theory studies the problem of how a group of decision makers should use information to coordinate their actions. Mathematically, the task is to find functions that maximize an objective functional. The Euler equations take the form of a system of integral equations. In this paper, it will be shown that a class of such integral equations has solutions that are identical to the solutions of a system of initial-valued integrodifferential equations. This Cauchy system describes the sensitivity of the solutions to underlying parameters and provides an efficient technique for solving difficult team decision problems. An analysis of a profit maximizing firm demonstrates the usefulness of the Cauchy system.     

Analytic study of some differential equations related to problems of space dynamics

We present a survey of results of the study of differential equations whose solutions have singularities of a certain type, in particular movable singular points with fairly simple topology. New statements on the form of partial and general solutions of these equations are obtained. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 2, pp. 162–171, February, 2007.     

Solutions of Emden–Fowler equations by homotopy-perturbation method

In this paper, approximate and/or exact analytical solutions of the generalized Emden–Fowler type equations in the second-order ordinary differential equations (ODEs) are obtained by homotopy-perturbation method (HPM). The homotopy-perturbation method (HPM) is a coupling of the perturbation method and the homotopy method. The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve. In this work, HPM yields solutions in convergent series forms with easily computable terms, and in some cases, only one iteration leads to the high accuracy of the solutions. Comparisons with the exact solutions and the solutions obtained by the Adomian decomposition method (ADM) show the efficiency of HPM in solving equations with singularity.