Solution of boundary-value problems for a degenerating elliptic equation of the second kind by the method of potentials

In this paper we study basic boundary value problems for one multidimensional degenerating elliptic equation of the second kind. Using the method of potentials we prove the unique solvability of the mentioned problems. We construct a fundamental solution and obtain an integral representation for the solution to the equation. Using this representation we study properties of solutions, in particular, the principle of maximum. We state the basic boundary value problems and prove their unique solvability. We introduce potentials of single and double layers and study their properties. With the help of these potentials we reduce the boundary value problems to the Fredholm integral equations of the second kind and prove their unique solvability.     

Antiplane shear deformations in a linear theory of elasticity with microstructure

In this paper, we solve fundamental boundary value problems in a theory of antiplane elasticity which includes the effects of material microstructure. Using the real boundary integral equation method, we reduce the fundamental problems to systems of singular integral equations and construct exact solutions in the form of integral potentials.     

Antiplane shear deformations in a linear theory of elasticity with microstructure

In this paper, we solve fundamental boundary value problems in a theory of antiplane elasticity which includes the effects of material microstructure. Using the real boundary integral equation method, we reduce the fundamental problems to systems of singular integral equations and construct exact solutions in the form of integral potentials.Received: March 25, 2002     

Boundary value problems in the theory of binary mixtures

In this paper, the boundary value problems of steady oscillation (vibration) of the linear theory of thermoelasticity for binary mixtures are investigated by means of the boundary integral equation method (potential method). The uniqueness and existence theorems of solutions of the exterior boundary value problems by means potential method and multidimensional singular integral equations are proved. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamical problems of the theory of elasticity for solids with double porosity

In this paper the dynamical theory of elasticity for solids with double porosity is presented. The single-layer and double-layer potentials are constructed and basic properties are established. The uniqueness theorems of the internal and external boundary value problems (BVPs) of steady vibrations are proved. The existence theorems of classical solution of the external BVPs by means of the boundary integral method and the theory of multidimensional singular integral equations are proved. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary value problems of the theory of viscoelasticity for Kelvin-Voight materials

In this paper, the classical Kelvin-Voight model of the linear theory of viscoelasticity is considered and the following results are obtained: the fundamental solution of the equation of steady vibrations is constructed, the basic properties of plane waves and elastopotentials are established, the uniqueness theorem of the internal and external boundary value problems (BVPs) are proved, the existence theorems for classical solutions of the BVPs by means of the potential method and the theory of two-dimensional singular integral equations are proved. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

重调和椭圆边值问题的正则积分方程

我们熟知,利用位势理论或由Green公式及基本解出发区域内调和及重调和边值问题可归化为边界上的积分方程。近年来冯康又提出一种更自然而直接的归化,即从Green公式及Green函数出发将微分方程边值问题化为边界上的含有广义函数意义下发散积分有限部分的奇异积分方程,这种归化在各种边界归化中占有特殊地位,被称为正则边界归化,本文将这一理论应用于重调和椭圆边值问题,研究了其正则归化的性质,并通过利用Green函数、Fourier分析及复变函数论方法等不同途径求出了在上半平面、单位圆内部、单位圆外部三种区域的Poisson积分公式及正则积分方程,其离散化可用于实际计算。 本文是在导师冯康教授指导下完成的,作者谨在此对他表示衷心的感谢。     

Schwarz problem for higher‐order complex partial differential equations in the upper half plane

Linear and nonlinear elliptic complex partial differential equations of higher‐order are considered under Schwarz conditions in the upper‐half plane. Firstly, using the integral representations for the solutions of the inhomogeneous polyanalytic equation with Schwarz conditions, a class of integral operators is introduced together with some of their properties. Then, these operators are used to transform the problem for linear equations into singular integral equations. In the case of nonlinear equations such a transformation yields a system of integro‐differential equations. Existence of the solutions of the relevant boundary value problems for linear and nonlinear equations are discussed via Fredholm theory and fixed point theorems, respectively.     

On the extraction technique in boundary integral equations

In this paper we develop and analyze a bootstrapping algorithm for the extraction of potentials and arbitrary derivatives of the Cauchy data of regular three-dimensional second order elliptic boundary value problems in connection with corresponding boundary integral equations. The method rests on the derivatives of the generalized Green's representation formula, which are expressed in terms of singular boundary integrals as Hadamard's finite parts. Their regularization, together with asymptotic pseudohomogeneous kernel expansions, yields a constructive method for obtaining generalized jump relations. These expansions are obtained via composition of Taylor expansions of the local surface representation, the density functions, differential operators and the fundamental solution of the original problem, together with the use of local polar coordinates in the parameter domain. For boundary integral equations obtained by the direct method, this method allows the recursive numerical extraction of potentials and their derivatives near and up to the boundary surface.

     

The Three-Dimensional Problem of Statics of the Elastic Mixture Theory with Displacements Given on the Boundary

The first three-dimensional boundary value problem is considered for the basic equations of statics of the elastic mixture theory in the finite and infinite domains bounded by the closed surfaces. It is proved that this problem splits into two problems whose investigation is reduced to the first boundary value problem for an elliptic equation which structurally coincides with an equation of statics of an isotropic elastic body. Using the potential method and the theory of Fredholm integral equations of second kind, the existence and uniqueness of the solution of the first boundary value problem is proved for the split equation.     

Localized Boundary-Domain Singular Integral Equations Based on Harmonic Parametrix for Divergence-Form Elliptic PDEs with Variable Matrix Coefficients

Employing the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems (BVPs) for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original BVPs is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik–Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.     

The solvability of some kinds of singular integral equations of convolution type with variable integral limits

In this paper, we discuss several classes of convolution type singular integral equations with variable integral limits in class $ H^*_1 $. By means of the theory of complex analysis, Fourier analysis and integral transforms, we can transform singular integral equations with variable integral limits into the Riemann boundary value problems with discontinuous coefficients. Under the solvability conditions, the existence and uniqueness of the general solutions can be obtained. Further, we analyze the asymptotic properties of the solutions at the nodes. Our work improves the Noether theory of singular integral equations and boundary value problems, and develops the knowledge architecture of complex analysis.     

Critical points and positive solutions of singular elliptic boundary value problems

Usually we do not think there is variational structure for singular elliptic boundary value problems, so it cannot be considered by using critical points theory. In this paper, we use critical theory on certain convex closed sets to solve positive solutions for singular elliptic boundary value problems, especially use the ordinary differential equation theory of Banach spaces to obtain new results on the existence of multiple positive solutions. The method is useful for other singular problems.     

Boundary value problems for two types of degenerate elliptic systems in R~4

Firstly, the Riemann boundary value problem for a kind of degenerate elliptic system of the first order equations in R~4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system's solution, the boundary value problem as stated above is transformed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R~4 are derived.     

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.     

Potential method in the steady vibrations problems of the theory of thermoviscoelasticity for Kelvin-Voigt materials with voids

In this paper the linear theory of thermoviscoelasticity for Kelvin-Voigt materials with voids is considered. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for classical solutions of the above mentioned BVPs are proved by using the potential method (boundary integral equation method) and the theory of singular integral equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Difference Potentials Method and its Applications

The purpose of this article is to acquaint the reader with the general concepts and capabilities of the Difference Potentials Method (DPM). DPM is used for the numerical solution of boundary-value and some other problems in difference and differential formulations. Difference potentials and DPM play the same role in the theory of solutions of linear systems of difference equations on multi-dimensional non-regular meshes as the classical Cauchy integral and the method of singular integral equations do in the theory of analytical functions (solutions Cauchy-Riemann system). The application of DPM to the solution of problems in difference formulation forms the first aspect of the method. The second aspect of the DPM implementation is the discretization and numerical solution of the Calderon-Seeley boundary pseudo-differential equations. The latter are equivalent to elliptical differential equations with variable coefficients in the domain; they are written making no use of fundamental solutions and integrals. Because of this fact ordinary methods for discretization of integral equations cannot be applied in this case. Calderon-Seeley equations have probably not been used for computations before the theory of DPM appeared. This second aspect for the implementation of DPM is that it does not require difference approximation on the boundary conditions of the original problem. The latter circumstance is just the main advantage of the second aspect in comparison with the first one. To begin with, we put forward and justify the main constructions and applications of DPM for problems connected with the Laplace equation. Further, we also outline the general theory and applications: both those already realized and anticipated.     

On the solution of elliptic partial differential equations on regions with corners II: Detailed analysis

In this paper we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations. Previously, we observed that when the boundary value problems are formulated as boundary integral equations of classical potential theory, the solutions are representable by series of elementary functions, to arbitrary order, for all but finitely many values of the angles. Here, we extend this observation to all values of the angles. We show that the solutions near corners are representable, to arbitrary order, by linear combinations of certain non-integer powers and non-integer powers multiplied by logarithms.     

Null space distributions—A new approach to finite convolution equations with a Hankel kernel

In some earlier publications it has been shown that the solutions of the boundary integral equations for some mixed boundary value problems for the Helmholtz equation permit integral representations in terms of solutions of associated complicated singular algebraic ordinary differential equations. The solutions of these differential equations, however, are required to be known on some infinite interval on the real line, which is unsatisfactory from a practical point of view. In this paper, for the example of one specific boundary integral equation, the relevant solutions of the associated differential equation are expressed by integrals which contain only one unknown generalized function, the support of this generalized function is no longer unbounded but a compact subset of the real line. This generalized function is a distributional solution of the homogeneous boundary integral equation. By this null space distribution the boundary integral equation can be solved for arbitrary right-hand sides, this solution method can be considered of being analogous to the method of variation of parameters in the theory of ordinary differential equations. The nature of the singularities of the null space distribution is worked out and it is shown that the null space distribution itself can be expressed by solutions of the associated ordinary differential equation.     

On an integral equation for the two-dimensional exterior Stokes problem

This paper is concerned with solving the two-dimensional exterior Stokes problem by a combination of double- and simple-layer potentials. This leads to a boundary integral equation, which is shown to be uniquely solvable. The present approach is a variation of a technique due to Leis, Brakhage and Werner, and Panich for treating the scalar Helmholtz equation. However, it contains extra parameter which may be chosen in an optimal manner from a computational point of view. Some numerical results in this connection are also included.