Boundary value problems of steady vibrations in the theory of thermoelasticity with microtemperatures

In this paper the linear theory of steady vibrations of thermoelasticity with microtemperatures for isotropic solids with microstructure is considered. The uniqueness and existence theorems of solutions of the internal and external second boundary value problems (BVPs) by means of the boundary integral method (potential method) and the theory of singular integral equations are proved. The existence of eigenfrequencies of the internal homogeneous BVP of steady vibrations is studied. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the boundary transmission problem of thermoelastostatics in a plane domain with interface corners

This paper deals with bimetal problems of thermoelastostatics. By means of an explicit particular solution a reduction to problems of elastostatics is given. An indirect boundary integral method is applied for solving the traction boundary value problem. The solution is represented by a potential of single layer type having Green's contact tensor as the kernel. Thus, from the first the transmission conditions are satisfied. The Fredholm property of the boundary integral operator as well as the asymptotics of the potential density at an interface corner depend on the symbol of a Mellin convolution operator. The singular functions at corners can be obtained by calculating the potential for terms in the asymptotic expansion of the density.     

Boundary value problems in the theory of thermoporoelasticity for materials with double porosity

In this paper the linear quasi-static theory of thermoelasticity for solids with double porosity is considered. The system of equations of this theory is based on the equilibrium equations for solids with double porosity, conservation of fluid mass, constitutive equations, Darcy's law for materials with double porosity and Fourier's law for heat conduction. 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 means of the potential method (boundary integral equation method) and the theory of singular integral equations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Plane Waves and Boundary Value Problems in the Theory of Elasticity for Solids with Double Porosity

This paper concerns with the dynamical theory of elasticity for solids with double porosity. This theory unifies the earlier proposed quasi-static model of Aifantis of consolidation with double porosity. The basic properties of plane waves are established. The radiation conditions of regular vectors are given. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness theorems are proved. The basic properties of elastopotentials are given. The existence of regular (classical) solution of the external BVP by means of the potential method (boundary integral method) and the theory of singular integral equations are proved.     

Steady vibrations problem in the theory of viscoelasticity for Kelvin-Voigt materials with voids

In this paper the linear theory of viscoelasticity for Kelvin-Voigt materials with voids is considered. The uniqueness and existence theorems for internal boundary value problem (BVP) of steady vibrations are proved by means of the potential method (boundary integral method) and the theory of singular integral equations. The application of this method to the 3D BVP of the considered theory reduces this problem to 2D singular integral equation. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mathematical problems of the theory of elasticity of chiral materials for Lipschitz domains

By the potential method, we investigate the Dirichlet and Neumann boundary value problems of the elasticity theory of hemitropic (chiral) materials in the case of Lipschitz domains. We study properties of the single‐ and double‐layer potentials and of certain, generated by them, boundary integral operators. These results are applied to reduce the boundary value problems to the equivalent first and the second kind integral equations and the uniqueness and existence theorems are proved in various function spaces. Copyright © 2005 John Wiley & Sons, Ltd.     

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)     

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)     

External boundary value problems in the quasi static theory of viscoelasticity for Kelvin-Voigt materials with double porosity

In the present paper the linear quasi static theory of viscoelasticity for Kelvin-Voigt materials with double porosity is considered. The basic external boundary value problems (BVPs) of steady vibrations in this theory are formulated. The uniqueness and existence theorems for regular (classical) solutions of the BVPs are proved by using of the potential method (boundary integral equations method) and the theory of singular integral equations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary value problems of the theory of heat propagation in a binary mixture

In this paper we study the problems of heat propagation in a binary mixture. The fundamental solution of system of equations of steady vibrations is constructed in terms of elementary functions. The uniqueness and existence theorems of classical solution of the interior and exterior boundary value problems by means of the boundary integral method are proved. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The boundary value problems of the full coupled theory of poroelasticity for materials with double porosity

In this paper the full coupled quasi-static theory of poroelasticity for materials with double porosity is considered. The basic boundary value problems (BVPs) of the steady vibrations are investigated. The uniqueness theorems of the internal BVPs of steady vibrations are proved. The basic properties of elastopotentials are established. The existence of regular solutions of the BVPs by means of the boundary integral equations method and the theory of singular integral equations is proved. (© 2012 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)     

The boundary element method for stochastic potential problems

Stochastic Dirichlet and Neumann boundary value problems and stochastic mixed problems have been formulated. As a result the stochastic singular integral equations have been obtained. A way of solving these equations by means of discretization of a boundary using stochastic boundary elements has been presented, resulting in a set of random algebraic equations. It has been proved that for Dirichlet and Neumann problems probabilistic characteristics (i.e. moments: expected value and correlation function) fulfilled deterministic singular integral equations. A numerical method of evaluation of moments on a boundary and inside a domain has been presented.     

Boundary element method for spatially-periodic potential problems

In this paper, we propose a boundary element method for two-dimensional potential problems with one-dimensional spatial periodicity, which have been difficult to be solved by the ordinary boundary element method. In the presented method, we reduce the potential problems with Dirichlet and Neumann boundary conditions to integral equation problems with the periodic fundamental solution of the Laplace operator and, then, obtain approximate solutions by solving linear systems given by discretizing the integral equations. Numerical examples are also included. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary Integral Equation Method in the Steady State Oscillation Problems for Anisotropic Bodies

The three-dimensional steady state oscillation problems of the elasticity theory for homogeneous anisotropic bodies are studied. By means of the limiting absortion principle the fundamental matrices maximally decaying at infinity are constructed and the generalized Sommerfeld–Kupradze type radiation conditions are formulated. Special functional spaces are introduced in which the basic and mixed exterior boundary value problems of the steady state oscillation theory have unique solutions for arbitrary values of the oscillation parameter. Existence theorems are proved by reduction of the original boundary value problems to equivalent boundary integral (pseudodifferential) equations. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Axisymmetric Stokes' flow in a spherical shell revisited via the Fokas method. Part I: Irrotational flow

The axisymmetric irrotational Stokes' flow for a spherical shell is analysed by means of the recently developed Fokas method via the use of global relations. Alternative series and new integral representations concerning a system of concentric spheres, yielding, by a limiting procedure, the Dirichlet or Neumann problems for the interior and the exterior of a sphere, are presented. The boundary value problems considered can be classically solved using either the finite Gegenbauer transform or the Mellin transform. Application of the Gegenbauer transform yields a series representation which is uniformly convergent at the boundary, but not convenient for many applications. The Mellin transform, on the other hand, furnishes an integral representation which is not uniformly convergent at the boundary. Here, by algebraic manipulations of the global relation: (i) a Gegenbauer series representation is derived in a simpler manner, instead of solving ODEs and (ii) an alternative integral representation, different from the Mellin transform representation is derived which is uniformly convergent at the boundary. Copyright © 2011 John Wiley & Sons, Ltd.     

Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions

In this paper, we consider the properties of Green’s function for a class of nonlinear Caputo fractional differential equations with integral boundary conditions by constructing an available integral operator. By means of well-known fixed point theorems and lower and upper solutions method, some new existence and nonexistence criteria of single or multiple positive solutions for fractional differential equation boundary value problems are established. As applications, some interesting examples are presented to illustrate the main results.     

三维Helmholtz方程边值问题的新型边界积分-微分方程及其边界元解法

用双层位势表示的二维Neumann边值问题的边界归化方法,将原始问题归化为新型边界积分-微分方程,由此导出一种新的既能保持原始问题的自伴性,又具有可积弱奇性积分核的边界变分方程.本文将此法推广到三维Helmholtz方程Neumann边值问题,并给出最优能量模误差估计和内部最大模超收敛估计.     

On the Solutions of the Coupled Nonlinear Parabolic Equations

By means of the fixed point technique and integral estimation method, we study the solutions of periodic boundary value problem and initial value problem for the coupled nonlinear parabolic equations. The global classical solutions of the mentioned problems are shown to exists.     

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.