Boundary Value Problems in the Theory of Thermoelasticity for Triple Porosity Materials

2016 ◽  
Author(s):  
Merab Svanadze
Keyword(s):  
Boundary Value Problems ◽  
Darcy’S Law ◽  
Existence Theorems ◽  
Displacement Vector ◽  
Boundary Value ◽  
Darcy's Law ◽  
External Boundary ◽  
Classical Solutions ◽  
Governing Equations ◽  
Equilibrium Equations

This paper concerns with the quasi static linear theory of thermoelasticity for triple porosity materials. The system of governing equations based on the equilibrium equations, conservation of fluid mass, the constitutive equations, Darcy’s law for materials with triple porosity and Fourier’s law of heat conduction. The cross-coupled terms are included in the equations of conservation of mass for the fluids of the three levels of porosity (macro-, meso- and micropores) and in the Darcy’s law for materials with triple porosity. The system of general governing equations is expressed in terms of the displacement vector field, the pressures in the three pore systems and the temperature. The basic internal and external boundary value problems (BVPs) are formulated and on the basis of Green’s identities the uniqueness theorems for the regular (classical) solutions of the BVPs are proved. The surface (single-layer and double-layer) and volume potentials are constructed and their basic properties are established. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method and the theory of singular integral equations.

EXISTENCE THEOREMS OF POSITIVE SOLUTIONS FOR FOURTH-ORDER FOUR-POINT BOUNDARY VALUE PROBLEMS

2004 ◽  
Vol 02 (01) ◽  
pp. 71-85 ◽  
Author(s):  
YUJI LIU ◽  
WEIGAO GE
Keyword(s):  
Differential Equation ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fourth Order ◽  
Existence Theorems ◽  
Boundary Value ◽  
Growth Conditions ◽  
Point Boundary ◽  
Order Ordinary Differential Equation ◽  
Order Four

In this paper, we study four-point boundary value problems for a fourth-order ordinary differential equation of the form [Formula: see text] with one of the following boundary conditions: [Formula: see text] or [Formula: see text] Growth conditions on f which guarantee existence of at least three positive solutions for the problems (E)–(B1) and (E)–(B2) are imposed.

scholarly journals Existence theorems for a second orderm-point boundary value problem at resonance

1995 ◽  
Vol 18 (4) ◽  
pp. 705-710 ◽  
Author(s):  
Chaitan P. Gupta
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Existence Theorems ◽  
Boundary Value ◽  
Continuation Theorem ◽  
Point Boundary ◽  
Existence Of A Solution ◽  
At Resonance ◽  
Problem At Resonance

Letf:[0,1]×R2→Rbe function satisfying Caratheodory's conditions ande(t)∈L1[0,1]. Letη∈(0,1),ξi∈(0,1),ai≥0,i=1,2,…,m−2, with∑i=1m−2ai=1,0<ξ1<ξ2<…<ξm−2<1be given. This paper is concerned with the problem of existence of a solution for the following boundary value problemsx″(t)=f(t,x(t),x′(t))+e(t),0<t<1,x′(0)=0,x(1)=x(η),x″(t)=f(t,x(t),x′(t))+e(t),0<t<1,x′(0)=0,x(1)=∑i=1m−2aix(ξi).Conditions for the existence of a solution for the above boundary value problems are given using Leray Schauder Continuation theorem.

Basic Boundary Value Problems of Thermoelasticity for Anisotropic Bodies with Cuts. I

1995 ◽  
Vol 2 (2) ◽  
pp. 123-140
Author(s):  
R. Duduchava ◽  
D. Natroshvili ◽  
E. Shargorodsky
Keyword(s):  
Boundary Value Problems ◽  
Displacement Vector ◽  
Boundary Value ◽  
Three Dimensional ◽  
Manifolds With Boundary ◽  
Hölder Continuous ◽  
Arbitrary Configuration ◽  
Anisotropic Bodies ◽  
Potential Methods ◽  
Basic Boundary

Abstract The three-dimensional problems of the mathematical theory of thermoelasticity are considered for homogeneous anisotropic bodies with cuts. It is assumed that the two-dimensional surface of a cut is a smooth manifold of an arbitrary configuration with a smooth boundary. The existence and uniqueness theorems for boundary value problems of statics and pseudo-oscillations are proved in the Besov and Bessel-potential spaces by means of the classical potential methods and the theory of pseudodifferential equations on manifolds with boundary. Using the embedding theorems, it is proved that the solutions of the considered problems are Hölder continuous. It is shown that the displacement vector and the temperature distribution function are Cα -regular with any exponent α < 1/2. This paper consists of two parts. In this part all the principal results are formulated. The forthcoming second part will deal with the auxiliary results and proofs.

scholarly journals Three-Dimensional Generalized Dynamics of Soft-Matter Quasicrystals

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Tian-You Fan ◽  
Zhi-Yi Tang
Keyword(s):  
Boundary Value Problems ◽  
Soft Matter ◽  
Boundary Value ◽  
Three Dimensional ◽  
Governing Equations

The three-dimensional generalized dynamics of soft-matter quasicrystals was investigated, in which the governing equations of the dynamics are derived for observed 12-fold symmetry quasicrystals and possibly observed 8- and 10-fold symmetry ones in soft matter. The solving methods, possible solutions for some initial- and boundary-value problems of the equations, and possible applications are discussed as well.

Linear second order elliptic equations with Venttsel boundary conditions

1991 ◽  
Vol 118 (3-4) ◽  
pp. 193-207 ◽  
Author(s):  
Yousong Luo ◽  
Neil S. Trudinger
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Elliptic Equations ◽  
Existence Result ◽  
Boundary Value ◽  
Second Order ◽  
Classical Solutions ◽  
Schauder Estimate ◽  
Second Order Elliptic Equations

SynopsisWe prove a Schauder estimate for solutions of linear second order elliptic equations with linear Venttsel boundary conditions, and establish an existence result for classical solutions for such boundary value problems.

Boundary Integral Equations Method in the Coupled Theory of Thermoelasticity for Porous Materials

2019 ◽  
Author(s):  
Merab Svanadze
Keyword(s):  
Porous Materials ◽  
Integral Equations ◽  
Volume Fraction ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Darcy's Law ◽  
Classical Solutions ◽  
Governing Equations ◽  
Basic Properties ◽  
Steady Vibrations

Abstract This paper concerns with the coupled linear theory of thermoelasticity for porous materials and the coupled phenomena of the concepts of Darcy’s law and the volume fraction is considered. The system of governing equations based on the equations of motion, the constitutive equations, the equation of fluid mass conservation, Darcy’s law for porous materials, Fourier’s law of heat conduction and the heat transfer equation. The system of general governing equations is expressed in terms of the displacement vector field, the change of volume fraction of pores, the change of fluid pressure in pore network and the variation of temperature of porous material. The fundamental solution of the system of steady vibration equations is constructed explicitly by means of elementary functions and its basic properties are presented. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated and on the basis of Green’s identities the uniqueness theorems for the regular (classical) solutions of the BVPs are proved. The surface (single-layer and double-layer) and volume potentials are constructed and their basic properties are established. Finally, the existence theorems for classical solutions of the BVPs of steady vibrations are proved by means of the boundary integral equations method (potential method) and the theory of singular integral equations.

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