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)     

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 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)     

Explicit solution of the boundary value problems of the theory of elasticity for solids with double porosity

In this paper the Aifantis' theory of elasticity for solids with double porosity is considered and the 2D boundary value problem (BVP) of static is investigated. The uniqueness theorem of the internal BVP is proved. The explicit solution the BVP is constructed in the form of absolutely and uniform convergent series for a circle. The numerical solution of the BVP for a circle is obtained. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

External boundary value problems in the quasi static theory of elasticity for triple porosity materials

In this paper the quasi static linear theory of elasticity for materials with triple porosity is considered. Basic external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for regular (classical) solutions of these BVPs are established. (© 2016 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 of steady vibrations in the theory of rigid bodies with a double porosity structure

This paper concerns with the linear 3D theory of rigid solids with a double porosity structure. Basic external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for regular (classical) solutions of these BVPs are established. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary value problems in queueing theory

Recently complex function techniques have been developed for the analysis of queueing systems which need for their modelling a two dimensional state space. A variety of computer- and communication networks gives rise to such two-dimensional queueing systems and their analysis is needed for the performance evaluation of these aggregates. The present study reviews these developments     

Uniqueness and existence theorems of solutions of the boundary value problems of the theory of consolidation with double porosity

In this paper we consider the Aifantis' theory of consolidation with double porosity and we prove the uniqueness and existence theorems of solutions of basic boundary value problems (BVPs) of statics for the two-dimensional finite and infinite domains. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Boundary value problems occurring in plate deflection theory

The paper deals with several iterative methods for solving some boundary value problems occurring in plate deflection theory, making it possible to avoid the lengthy problem preparation which is required to solve these problems by finite difference methods.     

Boundary value problems in the theory of thermomicrostretch elastic solids

In this paper the linear theory of thermomicrostretch elastic solids is considered. The basic properties of wave numbers of the longitudinal and transverse plane waves are established.The existence of eigenfrequencies of the interior homogeneous boundary value problem (BVP) of steady oscillations is studied. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Boundary value problems in generalized biaxially symmetric potential theory

Interior boundary value problems are solved for the operator of generalized biaxially symmetric potential theory. The boundary conditions consist of Dirichlet data on the nonsingular part of the boundary and Dirichlet data or growth restrictions on the singular hyperplanes, depending on the values of parameters of the operator. Continuation of solutions beyond the singular hyperplanes is considered, yielding an improvement of a result of Huber. Potential theoretic methods are used for the investigation.     

Boundary value problems in the theory of binary mixtures of thermoelastic solids

In this paper the diffusion model of the linear theory of thermoelasticity of binary mixtures is considered. The basic properties of wave numbers of the longitudinal and transverse plane waves are established. The interior boundary value problem (BVP) of steady vibration of binary mixtures of thermoelastic solids is formulated. The existence theorems of eigenfrequencies of the BVP of steady vibrations are proved. The connection between plane waves and existence of eigenfrequencies is established. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fundamental solutions in the linear theory of consolidation for elastic solids with double porosity

This paper concerns the theory of consolidation for elastic solids with double porosity, and the governing fully coupled linear quasi-static equations are considered. The system of these equations is based on the equilibrium equations for a solid, conservation of fluid mass, the effective stress concept, and Darcy’s law for material with double porosity. Two levels of spatial cases of consolidation theory for a solid with double porosity are considered: equations of steady vibrations and equations of equilibrium. The fundamental solutions of these equations are constructed by means of elementary functions. Finally, the basic properties of these solutions are established.     

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)     

Boundary value problems for the Fitzhugh-Nagumo equations

This paper is concerned with the study of the FitzHugh-Nagumo equations. These equations arise in mathematical biology as a model of the transmission of electrical impulses through a nerve axon; they are a simplified version of the Hodgkin-Huxley equations. The FitzHugh-Nagumo equations consist of a non-linear diffusion equation coupled to an ordinary differential equation. v_t = v_xx + f(v) ? u, u_t = σv ? γu. We study these equations with either Dirichlet or Neumann boundary conditions, proving local and global existence, and uniqueness of the solutions. Furthermore, we obtain L_∞ estimates for the solutions in terms of the L₁ norm of the boundary data, when the boundary data vanish after a finite time and the initial data are zero. These estimates allow us to prove exponential decay of the solutions.     

Boundary value problems with causal operators

In this paper, we apply the monotone iterative method for nonlinear two-point boundary value problems with causal operators. We formulate sufficient conditions under which such problems have extremal or quasisolutions in a corresponding sector. We also investigate differential inequalities.     

Boundary value problems for differential-algebraic equations

This paper shows how certain regularization methods can be used to determine the solution structure of linear differential systems subject to linear constraints and boundary conditions. Attention is restricted to the relatively straightforward index-one problems and to certain index-two problems, where endpoint jumps are related to the underlying boundary layer structure of the corresponding singular perturbation problems.