Wedging of an elastic wedge by a rigid plate under conditions of contact with detaching

We consider a problem of wedging of an elastic wedge by a rigid plate along an edge crack that is located on the axis of symmetry of the wedge and reaches its vertex. The detachment of the crack faces from the surfaces of the plate is taken into account. Using the Wiener–Hopf method, we obtain an analytic solution of the problem. The size of the detachment zone, the stress intensity factor, the distribution of stresses on the line of continuation of the crack and in the contact domain, and circular displacements of the crack faces are determined.     

Convergence analysis of an hp finite element method for singularly perturbed transmission problems in smooth domains

We consider a two‐dimensional singularly perturbed transmission problem with two different diffusion coefficients, in a domain with smooth (analytic) boundary. The solution will contain boundary layers only in the part of the domain where the diffusion coefficient is high and interface layers along the interface. Utilizing existing and newly derived regularity results for the exact solution, we prove the robustness of an hp finite element method for its approximation. Under the assumption of analytic input data, we show that the method converges at an “exponential” rate, provided the mesh and polynomial degree distribution are chosen appropriately. Numerical results illustrating our theoretical findings are also included. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

圆柱壳开孔的应力集中──非圆孔问题的一般解

本文从Donnell型圆柱壳的基本方程出发,利用复变函数方法和保角映射技术,对圆柱壳开非圆形孔的问题进行了研究.首先给出了逼近具有非圆形孔的圆柱壳开孔问题一般解的完备函数序列,构造出了问题的一般解;其次利用有关圆柱壳开小孔的假设概念,给出了圆柱壳开非圆孔时边界条件的一般表达式.进而利用正交函数展开的方法,将待解的问题归结为一组无穷代数方程组的求解问题,并进行直接求解.在本文最后,对圆柱壳开圆孔.椭圆孔附近的应力集中问题进行了数值计算,给出了分析结果.     

Solution of a singular homogeneous Hilbert boundary-value problem for analytic function in multiply connected circular domain

We offer a new approach for solving the homogeneous Riemann-Hilbert boundary-value problem for analytic function in multiply connected circular domains. The approach is based on determination of analytic function in terms of known boundary values of its argument in a special case.     

Approximate analytic solutions to Radulet equations for the analysis of electromagnetic transients in single-phase power transmission lines

In this work, an analytic development for a transmission line with a corona effect for simulating an electromagnetic transient is presented. The asymptotic solution for the Radulet equations in which a nonlinear term is presented is obtained. The study is carried out for a single-phase transmission line. The electrical parameters for an overhead line are defined and several formulations for their calculation are presented. The frequency dependence of the electrical parameters is considered. In the first part, the linear problem solution is found; the Fourier and Laplace transforms are applied with respect to distance and time respectively. After finding the solution in the Fourier–Laplace domain, which is expressed in terms of a Green’s function integral, an approximate analytical solution is obtained in the distance–time domain by means of asymptotic methods. Finally, the nonlinear solution is found using as a first approach the linear solution. The results obtained show an attenuation in the voltage wave due to the corona effect.     

Regularization of the solution of the Cauchy problem for the system of Maxwell equations in an unbounded domain

We study the analytic continuation of the solution of the system of Maxwell equations in a spatial unbounded domain from its values on a part of the boundary of this domain. We construct an approximate solution of this problem based on the Carleman matrix method.     

An interpolation problem in tube domains

Some recent results concerning theL-problem of moments in two variables are related via the Fourier-Laplace transform to an interpolation problem in the tube domain over a quadrant inR ². The class of analytic functions for which the interpolation problem is posed is identified with the symbols of all bounded analytic Wiener-Hopf operators acting on theH ²-Hardy space of the tube domain. The extremal solutions of the corresponding truncated problem are computed and the related uniqueness phenomenon is also discussed.     

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.     

厚壁圆柱壳开孔应力集中问题的复变函数解法

本文基于考虑横向剪切变形影响的厚壳理论建立了求解圆柱壳开孔应力集中问题的复变函数方法，得到了此种问题的一般解和满足任意形开孔边界条件的表达式·该应力集中问题可以简化为求解无穷代数方程组的问题·用复变函数方法可以规范地求解应力集中问题·文中给出了圆柱壳开小圆孔和椭圆孔时应力集中系数的数值结果·     

A Nonhomogeneous Boundary-Value Problem for the Korteweg–de Vries Equation Posed on a Finite Domain

Abstract

Studied here is an initial- and boundary-value problem for the Korteweg–de Vries equation posed on a bounded interval with nonhomogeneous boundary conditions. This particular problem arises naturally in certain circumstances when the equation is used as a model for waves and a numerical scheme is needed. It is shown here that this initial-boundary-value problem is globally well-posed in the L ₂-based Sobolev space H  ^s (0, 1) for any s ≥ 0. In addition, the mapping that associates to appropriate initial- and boundary-data the corresponding solution is shown to be analytic as a function between appropriate Banach spaces.     

Projections of convex bodies and the fourier transform

The Fourier analytic approach to sections of convex bodies has recently been developed and has led to several results, including a complete analytic solution to the Busemann-Petty problem, characterizations of intersection bodies, extremal sections ofl _p-balls. In this article, we extend this approach to projections of convex bodies and show that the projection counterparts of the results mentioned above can be proved using similar methods. In particular, we present a Fourier analytic proof of the recent result of Barthe and Naor on extremal projections ofl _p-balls, and give a Fourier analytic solution to Shephard’s problem, originally solved by Petty and Schneider and asking whether symmetric convex bodies with smaller hyperplane projections necessarily have smaller volume. The proofs are based on a formula expressing the volume of hyperplane projections in terms of the Fourier transform of the curvature function.     

Further calculations for Israeli options

Recently Kifer introduced the concept of an Israeli (or Game) option. That is a general American-type option with the added possibility that the writer may terminate the contract early inducing a payment not less than the holder's claim had they exercised at that moment. Kifer shows that pricing and hedging of these options reduces to evaluating a stochastic saddle point problem associated with Dynkin games. Kyprianou, A.E. (2004) "Some calculations for Israeli options", Fin. Stoch. 8, 73-86 gives two examples of perpetual Israeli options where the value function and optimal strategies may be calculated explicity. In this article, we give a third example of a perpetual Israeli option where the contingent claim is based on the integral of the price process. This time the value function is shown to be the unique solution to a (two sided) free boundary value problem on (0, ∞) which is solved by taking an appropriately rescaled linear combination of Kummer functions. The probabilistic methods we appeal to in this paper centre around the interaction between the analytic boundary conditions in the free boundary problem, Itô's formula with local time and the martingale, supermartingle and submartingale properties associated with the solution to the stochastic saddle point problem.     

The Cauchy problem for the system of thermoelasticity equations in space

We consider the problem of analytic continuation of the solution of the system of thermoelasticity equations in a bounded three-dimensional domain on the basis of known values of the solution and the corresponding stress on a part of the boundary, i.e., the Cauchy problem. We construct an approximate solution of the problem based on the method of Carleman's function.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 212–217, August, 1998.In conclusion, the authors wish to thank Professor M. M. Lavrent'ev and Professor Sh. Ya. Yarmukhamedov for setting the problem and for discussions in the course of the solution.     

On Analytic Interpolation Manifolds in Boundaries of Weakly Pseudoconvex Domains

Let Ω be a bounded, weakly pseudoconvex domain in C ⁿ , n ≤ 2, with real-analytic boundary. A real-analytic submanifold M ? ?Ω is called an analytic interpolation manifold if every real-analytic function on M extends to a function belonging to (Ω¯). We provide sufficient conditions for M to be an analytic interpolation manifold. We give examples showing that neither of these conditions can be relaxed, as well as examples of analytic interpolation manifolds lying entirely within the set of weakly pseudoconvex points of ?Ω.     

Transformation of an axialsymmetric disk problem for the helmholtz equation into an ordinary differential equation

The problem of solving the three-dimensional Helmholtz equation in the exterior of a circular disk is considered where radially symmetric Dirichlet data on the disk are assumed to be prescribed. This problem for example arises in the scattering of plane (sound) waves at an infinite plane screen with a circular aperture if the direction of the incident wave is normal to the screen, as well as in the process of diffusion through a circular hole. By applying the factorization technique developed in [N. GORENFLO, M. WERNER,Solution of a finite convolution equation with a Hankel kernel by matrix factorization, SIAM J. Math. Anal., 28 (1997), pp. 434–451] to the disk problem an equivalent ordinary differential equation is derived, whose solution leads directly to the solution of the disk problem. This differential equation belongs to a class of ordinary differential equations which are of higher complexity than the standard ordinary differential equations of mathematical physics. The examination of this new class of differential equations therefore is motivated.     

On the solution of Dirichlet’s problem of the complex Monge–Ampère equation for the Cartan–Hartogs domain of the first type

The complex Monge–Ampère equation is a nonlinear equation with high degree; therefore getting its solution is very difficult. In the present paper how to get the solution of Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type is discussed, using an analytic method. Firstly, the complex Monge–Ampère equation is reduced to a nonlinear ordinary differential equation, then the solution of Dirichlet’s problem of the complex Monge–Ampère equation is reduced to the solution of a two-point boundary value problem for a nonlinear second-order ordinary differential equation. Secondly, the solution of Dirichlet’s problem is given as a semi-explicit formula, and in a special case the exact solution is obtained. These results may be helpful for a numerical method approach to Dirichlet’s problem of the complex Monge–Ampère equation on the Cartan–Hartogs domain of the first type.     

A criterion for the fundamental principle to hold for invariant subspaces on bounded convex domains in the complex plane

Let D be a bounded convex domain of the complex plane. We study the problem of whether the fundamental principle holds for analytic function spaces on D invariant with respect to the differentiation operator and admitting spectral synthesis. Earlier this problem was solved under a restriction on the multiplicities of the eigenvalues of the differentiation operator. In the present paper, we lift this restriction. Thus, we present a complete solution of the fundamental principle problem for arbitrary nontrivial closed invariant subspaces admitting spectral synthesis on arbitrary bounded convex domains.     

Canonical dual approach to solving the maximum cut problem

This paper presents a canonical dual approach for finding either an optimal or approximate solution to the maximum cut problem (MAX CUT). We show that, by introducing a linear perturbation term to the objective function, the maximum cut problem is perturbed to have a dual problem which is a concave maximization problem over a convex feasible domain under certain conditions. Consequently, some global optimality conditions are derived for finding an optimal or approximate solution. A gradient decent algorithm is proposed for this purpose and computational examples are provided to illustrate the proposed approach.     

Uniqueness of analytic solutions for stationary plate oscillations in an annulus

The equations governing the harmonic oscillations of a plate with transverse shear deformation are considered in an annular domain. It is shown that under nonstandard boundary conditions where both the displacements and tractions are zero on the internal boundary curve, the corresponding analytic solution is zero in the entire domain. This property is then used to prove that a boundary value problem with Dirichlet or Neumann conditions on the external boundary and Robin conditions on the internal boundary has at most one analytic solution.