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.     

Green functions of a quasi-static problem

In this paper Green functions are constructed in analytic form for a deformable half-plane of a quasi-static problem of thermoelasticity when the heat flow on the boundary x₂=0 of the half-plane is zero. To construct the Green functions, certain integral representations are used whose kernels are known Green functions of the corresponding problems of elasticity theory. The functions constructed make it possible to obtain a wide class of new solutions of boundary-value problems of thermoelasticity, in particular solutions for a piecewise homogeneous half-plane. Bibliography: 6 titles. Translated fromObchyslyuwval’na ta Pryklandna Matematyka, No. 77, 1993, pp. 97–104.     

Generalized solutions of initial-boundary value problems for second-order hyperbolic systems

The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.     

Non-convex self-dual Lagrangians: New variational principles of symmetric boundary value problems

We study the concept and the calculus of Non-convex self-dual (Nc-SD) Lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems. They indeed provide new representations and formulations for the superposition of convex functions and symmetric operators. They yield new variational resolutions for large class of Hamiltonian partial differential equations with variety of linear and nonlinear boundary conditions including many of the standard ones. This approach seems to offer several useful advantages: It associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of Euler-Lagrange functions. These potential functions are quite flexible, and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems. Additionally, in most cases the solutions generated using this new method have greater regularity than the solutions obtained using the standard Euler-Lagrange function. Perhaps most remarkable, however, are the permanence properties of Nc-SD Lagrangians; their calculus is relatively manageable, and their applications are quite broad.     

基于李群方法的贝塞尔函数数学实现

基于李群的表示理论,首先讨论了欧拉群的表示及其性质;然后,从该群的表示理论出发,分别导出了第一类贝塞尔函数的积分形式和幂级数形式.该研究表明了群方法可以求解对称边界问题的解析波函数,并为用群方法求解电磁场问题创造了条件.     

Analysis solution method for 3D planar crack problems of two-dimensional hexagonal quasicrystals with thermal effects

An analysis solution method (ASM) is proposed for analyzing arbitrarily shaped planar cracks in two-dimensional (2D) hexagonal quasicrystal (QC) media. The extended displacement discontinuity (EDD) boundary integral equations governing three-dimensional (3D) crack problems are transferred to simplified integral-differential forms by introducing some complex quantities. The proposed ASM is based on the analogy between these EDD boundary equations for 3D planar cracks problems of 2D hexagonal QCs and those in isotropic thermoelastic materials. Mixed model crack problems under combined normal, tangential and thermal loadings are considered in 2D hexagonal QC media. By virtue of ASM, the solutions to 3D planar crack problems under various types of loadings for 2D hexagonal QCs are formulated through comparison to the corresponding solutions of isotropic thermoelastic materials which have been studied intensively and extensively. As an application, analytical solutions of a penny-shaped crack subjected uniform distributed combined loadings are obtained. Especially, the analytical solutions to a penny-shaped crack subjected to the anti-symmetric uniform thermal loading are first derived for 2D hexagonal QCs. Numerical solutions obtained by EDD boundary element method provide a way to verify the validity of the presented formulation. The influences of phonon-phason coupling effect on fracture parameters of 2D hexagonal QCs are assessed.     

Representations of super-holomorphic functions and their applications

In this paper, we give three kinds of Almansi representations for super-polyharmonic functions. As applications of these representations, we investigate two kinds of boundary value problems in superspace and construct solutions to the super-Poisson equation and the inhomogeneous super-polyharmonic equation.     

关于C~n中凸域上解析函数的积分表示

<正> 本文目的是建立C~n中凸域D={z|Φ(z)<0}上解析函数f(z)的一类积分表示     

Ⅲ型界面裂纹的非对称动态扩展问题

采用复变函数论的方法,对Ⅲ型界面裂纹的非对称动态扩展问题进行了研究．通过自相似函数的方法可以获得解析解的一般表达式．应用该法可以迅速地将所论问题转化为Riemann-Hilbert问题,并求得了非对称扩展裂纹分别在集中载荷、阶跃载荷作用下的解析解．利用这些解并采用叠加原理,就可以求得任意复杂问题的解．     

BOUNDARY PENALTY FINITE ELEMENT METHODS FOR BLENDING SURFACES,I BASIC THEORY

1.IntroductionBlendingsurfacesissaidifwhentwoframesurfaces(orbodies)arelocatedalreadyjasmoothlytransferringsurfaceissoughttoconnectthetwoframesurfacesalongcertainboundary.Usually,theterminology"smoothness"meansthattheblendingsurfacebelongstogeometriccontinuityCI(Foleyetal.(90)[14),i.e.,theblendingsurfaceanditstangentplanearecontinuousuntilthejointboundary.Manyliteraturesarereportedonthissubject.Wemerelymentionafewofthemrelevanttothispaper.Uniforml)Thi.workwassupportedbytheresearchgrantsfromt…     

Integral equations requiring small numbers of Krylov-subspace iterations for two-dimensional smooth penetrable scattering problems

This paper presents a class of boundary integral equations for the solution of problems of electromagnetic and acoustic scattering by two-dimensional homogeneous penetrable scatterers with smooth boundaries. The new integral equations, which, as is established in this paper, are uniquely solvable Fredholm equations of the second kind, result from representations of fields as combinations of single and double layer potentials acting on appropriately chosen regularizing operators. As demonstrated in this text by means of a variety of numerical examples (that resulted from a high-order Nyström computational implementation of the new equations), these “regularized combined equations” can give rise to important reductions in computational costs, for a given accuracy, over those resulting from previous iterative boundary integral equation solvers for transmission problems.     

带裂纹十次对称二维准晶平面弹性的无摩擦接触问题

借助经典平面弹性复变函数方法，研究了单个刚性凸基底压头作用下，带任意形状裂纹十次对称二维准晶半平面弹性的无摩擦接触问题.利用十次对称二维准晶位移、应力的复变函数表达式, 带任意形状裂纹的准晶半平面弹性无摩擦接触问题被转换为可解的解析函数复合边值问题，进而简化成一类可解的Riemann边值问题.通过求解Riemann边值问题，得到了应力函数的封闭解, 并给出了裂纹端点处应力强度因子和压头下方准晶体表面任意点处接触应力的显式表达式.从压头下方接触应力的表达式可以看出, 接触应力在压头边缘和裂纹端点处具有奇异性.当忽略相位子场影响时, 该文所得结论与弹性材料对应结果一致.数值算例分别给出了单个平底刚性压头无摩擦压入带单个垂直裂纹和水平裂纹的十次对称二维准晶下半平面的结果.该文所得结论为准晶材料的应用提供了理论参考.     

Periodic traveling wave solution for non-homogeneous BBM and KdVB equations

In this paper the Green’s function method and results about fixed point are used to get existence results on periodic traveling wave solution for non-homogeneous problems of generalized versions of the BBM and KdVB equations. It is shown through the constructions of explicit Green’s functions that the periodic boundary value problems for the traveling wave solutions of the BBM and KdVB equations are equivalent to integral equations which generate compact operators in the space of periodic functions. These integral representations allowed us to prove that if the speed of the wave propagation is suitably chosen, then the BBM and KdVB equations will admit periodic traveling wave solution.     

The hp-Version of the Boundary Element Method in ℝ3 The Basic Approximation Results

This paper deals with the basic approximation properties of the h–p version of the boundary element method (BEM) in ℝ³. We extend the results on the exponential convergence of the h–p version of the boundary element method on geometric meshes from problems in polygonal domains to problems in polyhedral domains. In 2D elliptic boundary value problems the solutions have only corner singularities whereas in 3D problems they contain additional edge and corner-edge singularities. The solutions of the corresponding boundary integral equations inherit those singularities. The detailed investigations in our analysis take care of the various types of those singularities. While edge singularities can be analysed using standard one-dimensional approximation results the corner-edge singularities demand a new analysis. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Well‐posedness and approximation of solutions of linear divergence‐form elliptic problems on exterior regions

This paper describes well‐posedness, spectral representations, and approximations of solutions of uniformly elliptic, second‐order, divergence form elliptic boundary value problems on exterior regions U in when N ≥ 3. Inhomogeneous Dirichlet, Neumann, and Robin boundary conditions are treated. These problems are first shown to be well‐posed in the space E¹(U) of finite‐energy functions on U using variational methods. Spectral representations of these solutions involving Steklov eigenfunctions and solutions subject to zero Dirichlet boundary conditions are described. Some approximation results for the A‐harmonic components are obtained. Positivity and comparison results for these solutions are given. Copyright © 2014 John Wiley & Sons, Ltd.     

双圆柱区域上的高阶四元数函数的Dirichlet边值问题

本文讨论了四元数分析中多圆柱区域上方程∂_zⁿu(z) = f 的Dirichlet 边值问题. 获得问题的可解条件, 并当可解条件满足时, 给出了解的积分表示.     

Solving boundary-value problems ofx k -analytic functions for a semicircle

Integral representations of x^k-analytic functions (k=const>0) in terms of boundary values of analytic functions are given. Using these integral representations, three boundary-value problems of x^k-analytic functions for a semicircle are solved by quadratures. Bibliography: 4 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 3–12.     

声波方程吸收边界条件的稳定性分析

引言对于无界区域中波动现象的数值模拟，必需引进人工边界将计算限制在一个有界区域上．为了确定解，需要在人工边界上加适当的边界条件．对于声波和弹性波方程，这样的一组人工边界条件，也叫吸收边界条件，在［1，2］中被系统地构造出来．对于声波方程，这些吸收边界条件恰好是单程波方程的近似．如山中所指出，减少边界反射，便于在计算中应用和稳定性是构造吸收边界条件的三点关键．Ellgqllist和Maid。用模态分析方法15］证明，带有［IJ中构造的吸收边界条件的波动方程初边值问题是适定的，并且估计了人工边界所产生的误差．对于更广…     

一类Dirichlet边值逆问题

给出解析函数的一类Dirichlet边值逆问题的数学提法.依据解析函数Dirichlet边值问题和广义Dirichlet边值问题的理论,讨论了此边值逆问题的可解性.利用解析函数Dirichlet边值问题的Schwarz公式,给出了该边值逆问题的可解条件和解的表示式.     

Free vibration multiquadric boundary elements applied to plane elasticity

This paper presents a new boundary element formulation for free vibration analysis of two-dimensional elastic structures. The dual reciprocity method (DRM) is introduced using the multiquadric radial basis functions (MQ). The required particular solution kernels for displacement and traction are derived and smoothed using simple mathematical trick; hence the limiting values of these kernels are computed. The eigen-problem of displacement is formulated and solved to obtain the required frequencies of different vibration modes. Finally, several numerical problems are studied to demonstrate the validity and accuracy of the developed formulation. The results are compared to those obtained from other interpolation functions, and finite element analysis to demonstrate the validity and superiority of the present formulation.