Mixed boundary value problems with a shift for a pair of metaanalytic and analytic functions

In this article, we reconsider the mixed boundary value problem on the unit circle for a pair of metaanalytic and analytic functions as in Du and Wang (2008) [9]. By adopting appropriate transformations, we convert the problem into two independent boundary value problems for analytic functions. We then obtain expressions of solution and condition of solvability for the mixed boundary value problem. The forms of the solutions and the condition of solvability here are rather dissimilar to those in Du and Wang (2008) [9]. But the equivalence is established at the end of this article.     

广义解析函数的带位移的非线性边值问题

考虑多复变广义解析函数的一个带位移的非线性边值问题,首先将其化为奇异积分方程问题,然后利用Schauder不动点原理及压缩映象原理,证明了解的存在性并给出解的积分表达式.     

The Schwarz problem for analytic functions in torus-related domains

The Cauchy kernel is one of the two significant tools for solving the Riemann boundary value problem for analytic functions. For poly-domains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in poly-domains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. It is also decisive for the establishment of connection between the Riemann–Hilbert problem and the Riemann problem. Thus, not only the Schwarz problem for holomorphic functions in poly-domains is solved, but also the basis is established for solving some other problems. The boundary values of functions, holomorphic in poly-domains, are classified in the Wiener algebra. The general integral representation formulas for these functions, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary poly-domains is given. At the end, well-posed formulations of the torus-related problems are considered.     

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.     

Functional-differential equations in a class of analytic functions and its application to elastic composites

According to Muskhelishvili’s approach, two-dimensional elastic problems for media with non-overlapping inclusions are reduced to boundary value problems for analytic functions in multiply connected domains. Using a method of functional equations developed by Mityushev, we reduce such a problem for a circular multiply connected domain to functional-differential equations. It is proved that the operator corresponding to the functional-differential equations is compact in the Hardy–Sobolev space. Moreover, these equations can be solved by the method of successive approximation under some natural conditions.     

一类Riemann-Hilbert边值逆问题

给出解析函数的一类R iem ann-H ilbert边值逆问题的数学提法,依据解析函数R iem ann-H ilbert边值问题的经典理论,讨论了此边值问题的可解性,给出了该边值问题的可解条件和解的表示式.     

Pseudodifferential operators with real analytic symbols and approximation methods for pseudodifferential equations

This paper introduces some methods (including an approximation method) for investigating pseudodifferential equations and related problems (Cauchy problems, boundary value problems,…) based on the technique of pseudodifferential operators with real analytic symbols.     

Sufficient conditions for the finite-valence of analytic functions and their applications

In the first part of the paper there is given a survey of the contemporary state of the theory of the sufficient univalence and p-valence conditions for analytic and meromorphic functions of a complex variable. The second part is devoted to a survey of the results on the conditions for the univalent solvability of applied inverse boundary value problems.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 25, pp. 3–121, 1987.     

An analytic series method for Laplacian problems with mixed boundary conditions

Mixed boundary value problems are characterised by a combination of Dirichlet and Neumann conditions along at least one boundary. Historically, only a very small subset of these problems could be solved using analytic series methods (“analytic” is taken here to mean a series whose terms are analytic in the complex plane). In the past, series solutions were obtained by using an appropriate choice of axes, or a co-ordinate transformation to suitable axes where the boundaries are parallel to the abscissa and the boundary conditions are separated into pure Dirichlet or Neumann form. In this paper, I will consider the more general problem where the mixed boundary conditions cannot be resolved by a co-ordinate transformation. That is, a Dirichlet condition applies on part of the boundary and a Neumann condition applies along the remaining section. I will present a general method for obtaining analytic series solutions for the classic problem where the boundary is parallel to the abscissa. In addition, I will extend this technique to the general mixed boundary value problem, defined on an arbitrary boundary, where the boundary is not parallel to the abscissa. I will demonstrate the efficacy of the method on a well known seepage problem.     

Koch曲线上的齐次Riemann边值问题

当L为典型的分形曲线一Koch曲线时,提出了Riemann边值问题,但在一般情况下,在Koch曲线上所做的Cauchy型积分无意义.当对已知函数G(z),g(z)增加一定的解析条件,同时利用一列Cauchy型积分的极限函数,对定义在Koch曲线上的齐次Riemann边值问题进行了讨论,并得到与经典解析函数边值问题相类似的结果.     

An analytic shooting-like approach for the solution of nonlinear boundary value problems

In this paper a novel approach is presented for an analytic approximate solution of nonlinear differential equations with boundary conditions. By converting the nonlinear problem into an initial value form, a shooting-like procedure is introduced based on the powerful homotopy analysis technique. The proposed methodology is shown to work adequately for solving single or multiple solutions of some sample nonlinear boundary value problems.     

Mixed Elastico-Plasticity Problems with Partially Unknown Boundaries

In this paper,we study mixed elastico-plasticity problems in which part of the boundary is known,while the other part of the boundary is unknown and is a free boundary.Under certain conditions,this problemcan be transformed into a Riemann-Hilbert boundary value problem for analytic functions and a mixed boundaryvalue problem for complex equations.Using the theory of generalized analytic functions,the solvability of theproblem is discussed.     

Analyticity of solutions of analytic non-linear general elliptic boundary value problems, and some results about linear problems

The main aim of this paper is to discuss the problem concerning the analyticity of the solutions of analytic non-linear elliptic boundary value problems. It is proved that if the corresponding first variation is regular in Lopatinskiĭ sense, then the solution is analytic up to the boundary. The method of proof really covers the case that the corresponding first variation is regularly elliptic in the sense of Douglis-Nirenberg-Volevich, and hence completely generalize the previous result of C. B. Morrey. The author also discusses linear elliptic boundary value problems for systems of elliptic partial differential equations where the boundary operators are allowed to have singular integral operators as their coefficients. Combining the standard Fourier transform technique with analytic continuation argument, the author constructs the Poisson and Green’s kernel matrices related to the problems discussed and hence obtain some representation formulae to the solutions. Some a priori estimates of Schauder type and L _p type are obtained. __________ Translated from Acta Sci. Natur. Univ. Jilin, 1963, (2): 403–447 by GAO Wenjie.     

The structure of instantaneous frequencies of periodic analytic signals

In this paper, the structure of analytic signals is investigated by means of the relation between analytic signals and functions in the Hardy space. It is shown that an analytic signal is made up of two parts, one depending on the amplitude of the signal and another on the boundary value of an inner function. Based on this result, properties of the instantaneous frequencies of these two parts are studied, and it is found that negative instantaneous frequencies are caused by the amplitude of a signal. Finall...     

实轴上的双解析函数Riemann边值逆问题

提出了一类实轴上的双解析函数Riemann边值逆问题.先消去参变未知函数,再采用易于推广的矩阵形式记法,可把问题转化为两个实轴上的解析函数Riemann边值问题.利用经典的Riemann边值问题理论,讨论了该问题正则型情况的解法,得到了它的可解性定理.     

Lp-spaces of functions analytic in a region with piecewise analytic boundary

In this paper we study classes of analytic functions which are p-th power summable in a region with piecewise analytic boundary. We also study the spaces conjugate to these spaces and the question of the existence of a basis in these spaces.Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 537–548, October, 1976.     

Mixed boundary value problem for some pairs of metaanalytic function and analytic function

In this article, some mixed boundary value problems (BVPs) on the unit circumference for some pairs of a metaanalytic function and an analytic function are discussed. Using the relationship between metaanalytic function and polyanalytic function, the expression of solution and the condition of solvability for the problem are obtained by reducing the problem to an equivalent system of a Haseman BVP for analytic function and a Hilbert BVP for analytic function. Copyright © 2008 John Wiley & Sons, Ltd.     

Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane–Emden–Fowler type

The particular motivation of this work is to develop a computational method to calculate exact and analytic approximate solutions to singular strongly nonlinear initial or boundary value problems of Lane–Emden–Fowler type which model many phenomena in mathematical physics and astrophysics. A powerful algorithm is proposed based on the series representation of the solution via suitable base functions. The utilization of such functions converts the solution of a given nonlinear differential equation to the solution of algebraic equations. Error analysis and convergence of the method is presented. Comparisons with the other methods reveal validity, applicability and great potential of the method. Several physical problems are treated to illustrative the good performance and high accuracy of the technique.     

Best approximation of the differentiation operator on the class of functions analytic in a strip

On the class of functions analytic in a strip, we study a number of related extremal problems for the differentiation operator: the best approximation of the operator, computation of its modulus of continuity, and optimal recovery of the operator from boundary values of a function given with error on a straight line.