A new application of Lagrange-Bürmann expansions. I. General principle

A new scheme is developed for improving the convergence of slowly convergent series solutions. The method is based on a transformation of variables of similarity form whereby the resulting composite function is constructed by its Lagrange-Bürmann expansion. It is the improved convergence of the new expansion that we take most advantage of in this method. The convergence of the Lagrange-Bürmann expansion as well as its inversion scheme is proved for analytic (object) functions. The inversion is required to recover from the Lagrange-Bürmann expansion the object function which is imbedded in the mapping functions. Several numerical examples demonstrate the improved convergence of the new method. The improvement owes much to the invariance properties of the mapping function under a group and the “built-in” feature of analytic continuation of the method. These features are elucidated in detail.     

Integral equations and the interior Dirichlet potential problem

The two standard approaches for reformulating the interior Dirichlet potential problem as a boundary integral equation of the second kind are discussed. The integral equation derived from the representation of the solution as a double layer is shown to be more general than the one derived from Green's theorem. The boundary integral equation of the latter method, however, has definite analytical and numerical value. From it a new integral equation is derived whose solution can be represented as a convergent Neumann series and it is shown that the Green's function of the first kind can be obtained from it. An example is supplied to illustrate the method.     

旋波媒质中谱域电并矢Green函数的分解

介绍了一种推导无耗、互易和无界旋波媒质中谱域并矢Green函数表达式的新方法·　这种方法以Hemholtz定理以及并矢Diracδ函数的无散和无旋分解为基础,首先将电矢量的并矢Green函数方程分解成无散电矢量的并矢Green函数方程和无旋电矢量的并矢Green函数方程,然后经Fourier变换导出了旋波媒质中谱域电并矢Green函数的无散分解表达式和无旋分解表达式·　用这种方法推导旋波媒质中并矢Green函数就可以避免必须用波场的分解法和并矢Green函数的本征函数展开法·     

Ursell's approach to obtaining an a priori estimate for the solution of the Neumann problem for the Helmholtz equation

The geometric optics asymptotic expression for Green's function for the Helmholtz equation in the exterior of a bounded convex region in R³ is rigorously justified (it is assumed that the Neumann boundary condition is satisfied). An integral equation constructed by a three-dimensional analogue of Ursell's method from Green's function for a paraboloid of revolution is the basis of all considerations. Analysis of this integral equation also makes it possible to prove the exponential decay of Green's function for the Helmholtz equation in the zone of deep shadow.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 85–154, 1974.     

A new approach to the representation of trigonometric and hyperbolic functions by infinite products

An innovative technique is developed for obtaining infinite product representations for some elementary functions. The technique is based on the comparison of alternative expressions of Green's functions constructed by two different methods. Some standard boundary value problems are considered posed for two-dimensional Laplace equation on regions of a regular configuration. Classical closed analytic form of Green's functions for such problems are compared against those obtained by the method of images in the form of infinite products. This yields a number of new infinite product representations for trigonometric and hyperbolic functions.     

On variations in the solutions of integro-differential equations of mixed problems of elasticity theory for domain variations

The behaviour of the solution of the boundary value problem for a pseudodifferential equation (PDE), Green's function of this problem, and also some of their local and global characteristics, during variation of the domain is investigated. Formulas are proposed that enable the solution of a broad class of PDE in a domain to be expressed in terms of the solution in the near domain. Local characteristics of the solution are expressed in terms of the local characteristics of the solution in the near domain. A double asymptotic form of Green's function for both arguments tending to the domain boundary occurs in the variation formula. The variation of this double asymptotic form as the domain varies is expressed in terms of this same asymptotic form. The system of variation formulas obtained is closed. It enables the PDE solution in the domain to be reduced to the solution of an ordinary differential equation in functional space. The local characteristics of the solution can also be found by this method without calculating the solution itself. If there is sufficient symmetry in the initial operator, then conservation laws in the Noether sense are obtained for its Green's function and its asymptotic form. The behaviour of the quantities under investigation is studied under inversion.

The investigation of variations of the solutions of problems for the variation of the domain occurs in the paper by Hadamard /1/, who studied the variation in conformal mapping and obtained a formula similar to (1.4). The formula for the variation of the solution of the boundary value problem for an elliptic differential equation is obtained in /2/. Variation formulas for the case of the operator of the problem about a crack and a circular domain are obtained in /3, 4/. The Irwin formula /5/ is obtained from formulas (1.4) and (1.21) by substitution.     

Green's functions for Laplace's equation in multiply connected domains

Analytical formulae for the first-type Green's function forLaplace's equation in multiply connected circular domains arepresented. The method is constructive and relies on the useof a special function known as the Schottky–Klein primefunction associated with multiply connected circular domains.It is shown that all the important functions of potential theory,including the first-type Green's function, the modified Green'sfunctions and the harmonic measures of a domain, can be writtenin terms of this prime function. A broad range of representativeexamples are given to demonstrate the efficacy of the methodas well as a quantitative comparison, in special cases, withthe results obtained using other independent methods.     

A Green’s function model for the analysis of laser heating of materials

An analytical model based on Green’s function method is developed to analyze the temperature distribution and heated regions in a material irradiated by a high-energy laser beam. The model is multi-dimensional, transient and incorporates different types of beam characteristics and boundary conditions. The multi-dimensional integration formulas in the Green’s function solution equation are evaluated using an adaptive numerical integration algorithm. A parametric study is conducted to show the effect of various laser beam parameters and material properties on the laser heating process.     

A generalized Green's function

Summary We present a method in which the given boundary conditions are replaced by integral boundary conditions, so chosen that the Green's function can be constructed without changing the principal spectral properties of the operator.Translated from Litovskii Matematicheskii Sbornik, Vol. 13, No. 1, pp. 109–114, January–March, 1973.     

A one-sweep method for linear elliptic equations over irregular domains

The determination of the configuration of equilibrium in a number of problems in mechanics and structures such as torsion, deflection of elastic membranes,etc., involve the solution of variational problems defined over irregular regions. This problem, in turn, may be reduced to the solution of elliptic differential equations subject to boundary conditions. In this paper, we study a method for the solution of such a problem when the region is of irregular shape. The method consists in solving the problem over a larger, imbedding, rectangular domain subject to appropriate constraints such as to satisfy the conditions of the original problem at the boundary. In this paper, we introduce the constraints by considering appropriate factors on the Green's function of the auxiliary problem. A conveniently discretized version of the problem is then treated by invariant imbedding, yielding some earlier results plus some new ones, namely, a direct one-sweep procedure that minimizes storage requirements. In addition, the present solution appears to be very convenient when the solution is required at a limited number of points. The derivations are specialized to Laplace's equation, but the method can be applied readily to general systems of second-order elliptic equations with no essential modifications. Finally, the existence of the necessary matrices in the imbedding equations is established.     

Molecular hydrodynamics of degenerate weakly nonideal bose gas. I. Generalized equations of two-fluid hydrodynamics

A degenerate weakly nonideal Bose gas is investigated at temperatures near zero. Relations connecting the irreducible Green's functions are used to obtain exact expressions for the two-time temperature-dependent Green's functions. In the case of weak nonideality an expression possessing interpolation properties with respect to the frequency and momentum is obtained for the density—density Green's function. At low frequencies, the results of two-fluid hydrodynamics are reproduced. At wavelengths less than the mean free path the energy of the elementary excitations and the damping obtained in the paper agree with the results of perturbation theory with respect to the coupling constant. An expression for the operator of the superfluid velocity is obtained.In memory of Nikolai Nikolaevich BogolyubovV. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 3, pp. 412–465, December, 1992.     

An initial-value method for the determination of Green's functions

The convergence of an initial-value method for computing the Green's function of a class of second-order differential operators is established. The proof relies on an interpolation procedure which is shown to generalize the Nyström method for Fredholm integral equations. The approximate Green's function is related to the solution of a discrete summation equation. The results of Anselone and Moore on collectively compact operators are then applied.This research was partially supported by UNLV Grant No. 001-060-4573.     

Three-dimensional Green's functions in anisotropic magneto-electro-elastic bimaterials

In this paper, we derive three-dimensional Green's functions in anisotropic magneto-electro-elastic full space, half space, and bimaterials based on the extended Stroh formalism. While in the full space, the Green's functions are obtained in an explicit form, those in the half space and bimaterials are expressed as a sum of the full-space Green's function and a Mindlin- type complementary part, with the latter being evaluated in terms of a regular line integral over [0, p][0, \pi]. Despite the complexity involved, the current Green's function expressions are surprisingly simple. Furthermore, the piezoelectric, piezomagnetic, and purely elastic Green's functions can all be obtained from the current Green's functions by setting simply the appropriate material coefficients to zero. A special material case, to which the extended Stroh formalism cannot be applied directly, has also been identified.¶Simple numerical examples are presented for Green's functions in full space, half space, and bimaterials with fully coupled and uncoupled anisotropic magneto-electro-elastic material properties.For given material properties and fixed source and field points, the effect of magneto-electro-elastic coupling on the Green's function is discussed. In particular, we observed that magneto-electro-elastic coupling could significantly alter the magnitude of certain Green's displacement and stress components, with difference as high as 45% being noticed. This result is remarkable and should be of great interest in the material analysis and design.     

Self-consistent cluster theory of the electron spectra of alloys

A method of self-consistent cluster expansion for the mass operator of the Green's function with allowance for short-and longrange order in alloys is proposed. A self-consistent system of equations for the coherent potential and the mass operator is obtained in the approximation in which the contributions of processes involving electron scattering by clusters of three or more atoms are ignored. The contributions of the scattering processes to the Green's function of the alloy decrease with increasing number of particles in a cluster and can be estimated by means of a certain small parameter. Analytic and numerical investigations of the energy dependence of the density of single-electron states of a binary alloy are made for different values of the parameter of the short-range order.Institute of Metal Physics, Ukrainian Academy of Sciences; Kiev University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 2, pp. 304–319, November, 1993.     

A modified Schwarz–Christoffel mapping for regions with piecewise smooth boundaries

A method where polygon corners in Schwarz–Christoffel mappings are rounded, is used to construct mappings from the upper half-plane to regions bounded by arbitrary piecewise smooth curves. From a given curve, a polygon is constructed by taking tangents to the curve in a number of carefully chosen so-called tangent points. The Schwarz–Christoffel mapping for that polygon is then constructed and modified to round the corners.     

Accurate solutions for steady plane flow in the driven cavity. I. Stokes flow

The problem considered here is the steady, incompressible plane Stokes flow in a rectangular cavity generated by uniform translation of the upper wall. An exact analytical solution of the governing biharmonic equation is derived which not only contains the leading term of the required singularities at the upper corners, but also approximately satisfies the boundary conditions at all four walls. A standard numerical algorithm is employed to correct the small deviations in the boundary conditions satisfied by the analytical solution. This technique enables accurate computation of the solution uniformly throughout the domain; in particular, near the singular corners and in those regions where the flow is extremely weak, for example, in the secondary vortex regions of the deep cavity. The method is illustrated for the square cavity and also for a deep cavity with a depth-to-width ratio of five, and the results are compared with previous analytical and numerical solutions.     

Green's function in plane anisotropic bimaterials with imperfect interface

^** Email: lsudak{at}ucalgary.ca The fundamental solutions or Green's functions for 2D or 3Danisotropic media with imperfect interface remain a challengingproblem. In this paper, a general method is presented for therigorous solution for the 2D Green's function in an anisotropicelastic bimaterial subject to a line force or a line dislocation.Most significant is the fact that the bonding along the bimaterialinterface is considered to be homogeneous imperfect. Specifically,the tractions are continuous but the displacements are discontinuousand proportional, in terms of interface stiffness parameters,to their respective traction components. Using complex variabletechniques, the basic boundary-value problem for two analyticvector functions is reduced to a coupled linear first-orderdifferential equation for a single analytic vector functiondefined in the lower half space. The coupled linear differentialequation for the single analytic vector function can be subsequentlydecoupled into three independent linear first-order differentialequations for three newly defined analytic functions. Closed-formsolutions for the 2D Green's function are derived in terms ofthe exponential integral. Unlike previous works which involvesome sort of inverse transform method to obtain the physicalquantities from the transform domain, the key feature of thepresent method is that the physical quantities can be readilycalculated without the need to perform any inverse transformoperations.     

求同伦映射的一个方法

利用待定连续函数的方法求出所需要的同伦映射 ,利用图形求出所需要的同伦映射 .介绍一种同伦映射的构造方法并给出具体表达式 .     

均布载荷作用下波纹扁壳的非线性振动

应用轴对称旋转扁壳的非线性大挠度动力学方程,研究了波纹扁壳在均布载荷作用下的非线性受迫振动问题．采用格林函数方法,将扁壳的非线性偏微分方程组化为非线性积分微分方程组．再使用展开法求出格林函数,即将格林函数展开为特征函数的级数形式,积分微分方程就成为具有退化核的形式,从而容易得到关于时间的非线性常微分方程组．针对单模态振形,得到了谐和激励作用下的幅频响应．作为算例,研究了正弦波纹扁球壳的非线性受迫振动现象．该文的解答可供波纹壳的设计参考．     

椭圆孔边裂纹对SH波的散射及其动应力强度因子

采用复变函数和Green函数方法求解具有任意有限长度的椭圆孔边上的径向裂纹对SH波的散射和裂纹尖端处的动应力强度因子．取含有半椭圆缺口的弹性半空间水平表面上任意一点承受时间谐和的出平面线源荷载作用时的位移解作为Green函数,采用裂纹“切割”方法,并根据连续条件建立起问题的定解积分方程,得到动应力强度因子的封闭解答．讨论了孔洞的存在对动应力强度因子的影响．