Papers by Gia Avalishvili
Bull. Georg. Natl. Acad. Sci, 2010
ABSTRACT. In the present paper Green-Lindsay nonclassical three-dimensional model for thermoelast... more ABSTRACT. In the present paper Green-Lindsay nonclassical three-dimensional model for thermoelastic plates with variable thickness is considered. Applying variational approach initial-boundary value problem corresponding to the three-dimensional model is ...
Bull. Georg. Natl. Acad. Sci, 2008
ABSTRACT. In the present paper an initial boundary value problem for thermoelastic prismatic shel... more ABSTRACT. In the present paper an initial boundary value problem for thermoelastic prismatic shells is considered. Three-dimensional dynamical problem for prismatic shell with surface forces given along the upper and the lower faces of the shell is reduced to ...
International Congress of Mathematicians
... [4] Cabré, X. and Chanillo S.: Stable solutions of semilinear elliptic problems ... show that... more ... [4] Cabré, X. and Chanillo S.: Stable solutions of semilinear elliptic problems ... show that a set of all initial data in D (A1 2) with which global strong solutions ... 65R20 We consider interior source methods for solving acoustic or electromagnetic scattering problems, which are usually ...
Journal of Applied Analysis, 2002
In this paper, nonlocal in time problem for abstract evolution equation of second order is studie... more In this paper, nonlocal in time problem for abstract evolution equation of second order is studied and theorem on existence and uniqueness of its solution is proved. Some applications of this theorem for hyperbolic partial differential equations and systems are considered and it is proved, that well-posedness of the mentioned problems depends on algebraic properties of ratios between the dimensions of the spatial boundary and the times appearing in the nonlocal in time initial conditions.
Differential Equations, May 1, 2005
The first systematic study of nonlocal boundary value problems was carried out in [1], where nonl... more The first systematic study of nonlocal boundary value problems was carried out in [1], where nonlocal problems were posed and analyzed for a certain class of elliptic equations. In [2, 3], the problem stated in [1] was named the Bitsadze-Samarskii problem, and iterative methods were suggested for solving such problems in the case of quite general elliptic equations. Note that the analysis of nonlocal boundary value problems is motivated by theoretical interest as well as practical needs, and numerous interesting papers [4-11] deal with these issues. Another type of nonclassical initial-boundary value problems for parabolic equations with nonlocal initial conditions was considered in [12], and the papers [13-15] deal with problems of this type arising in mathematical modeling of applied problems. Papers in which time-nonlocal problems are considered mainly deal with specific nonlocal initial conditions. In the present paper, we consider time-nonlocal problems for an abstract Schrödinger equation with various nonlocal operators and construct an algorithm approximating such problems of a certain class by a sequence of classical problems.
Mathematical Methods in The Applied Sciences, 2005
ABSTRACT The present paper is devoted to the design of a hierarchy of two-dimensional models for ... more ABSTRACT The present paper is devoted to the design of a hierarchy of two-dimensional models for dynamical problems within the theory of multicomponent linearly elastic mixtures in the case of prismatic shells with thickness which may vanish on some part of its boundary. The hierarchical model is obtained by a semidiscretization of the three-dimensional problem in the transverse direction. In suitable weighted Sobolev spaces we investigate the well-posedness of the two-dimensional problems, prove pointwise convergence of the sequence of approximate solutions restored from the solutions of the reduced problems to the exact solution of the original problem and estimate the rate of convergence. Copyright © 2005 John Wiley & Sons, Ltd.
Hiroshima Mathematical Journal, Nov 1, 2001
In the present article we are interested in the analysis of nonlocal initial boundary value probl... more In the present article we are interested in the analysis of nonlocal initial boundary value problems for some medium oscillation equations. More precisely, we investigate di¤erent types of nonlocal problems for one-dimensional oscillation equations and prove existence and uniqueness theorems. In some cases algorithms for direct construction of the solution are given. We also consider nonlocal problem for multidimensional hyperbolic equation and prove the uniqueness theorem for the formulated initial boundary value problem applying the theory of characteristics under rather general assumptions.
gmj
The present paper deals with nonclassical initial-boundary value problems for parabolic equations... more The present paper deals with nonclassical initial-boundary value problems for parabolic equations and systems and their generalizations in abstract spaces. Nonclassical problems with nonlocal initial conditions for an abstract first-order evolution equation with time-dependent operator are considered, the existence and uniqueness results are proved and the algorithm of approximation of nonlocal problems by a sequence of classical problems is constructed. Applications of the obtained general results to initial-boundary value problems for parabolic equations and systems are considered.
In the present paper static one-dimensional hierarchical model for elastic cusped rod is construc... more In the present paper static one-dimensional hierarchical model for elastic cusped rod is constructed. The corresponding boundary value problem is studied and the uniqueness and existence of its solution in suitable weighted Sobolev spaces is proved. The convergence of the sequence of approximate solutions restored from the solutions of one-dimensional problems to the solution of original three-dimensional problem is proved and under regularity conditions the rate of approximation is estimated. Key words and phrases: Mathematical modelling of linearly elastic cusped rods, a priori error estimation. AMS subject classification: 74K10, 65N30, 42C10. The construction and the intensive investigation of the lower-dimensio-nal mathematical models of bodies with negligible thickness or width in comparison with the other geometric dimensions arise with the wide use of structures of such type in the practice ([1, 2]). One of the methods of constructing hierarchic models for elastic prismatic s...
ABSTRACT. In the present paper an initial boundary value problem for thermoelastic prismatic shel... more ABSTRACT. In the present paper an initial boundary value problem for thermoelastic prismatic shells is consid-ered. Three-dimensional dynamical problem for prismatic shell with surface forces given along the upper and the lower faces of the shell is reduced to hierarchy of two-dimensional problems. The obtained problems are investigated in suitable function spaces, the convergence of the sequence of vector-functions of three space variables, restored from the solutions of two-dimensional problems to the solution of the original three-dimensional problem, is proved and the rate of approximation is estimated. © 2008 Bull. Georg. Natl. Acad. Sci. Key words: dynamical problems for thermoelastic prismatic shells, Fourier-Legendre series. Two-dimensional and one-dimensional approximations of three-dimensional boundary and initial-boundary value problems are often used when constructing mathematical models of complicated building and various engineering structures. One of the methods of co...
This paper is devoted to the construction and investigation of twodimensional models for anisotro... more This paper is devoted to the construction and investigation of twodimensional models for anisotropic inhomogeneous thermo-electro-magneto-elastic shells with variable thickness, which may vanish on a part of the lateral boundary. The variational formulation in curvilinear coordinates of the boundary value problem corresponding to the three-dimensional model of the shell, when density of surface force and components of electric displacement, magnetic induction and heat flux along the outward normal vector are given along the upper and lower face surfaces of the shell, is obtained and the well-posedness result in suitable factor space of Sobolev space is given. The subspaces with special structures of the spaces corresponding to the original three-dimensional problem are considered and applying variational formulation a hierarchy of static two-dimensional models is constructed. The boundary value problems corresponding to the obtained two-dimensional models are investigated in factor ...
Gia A. Avalishvili1, Mariam A. Avalishvili and David G. Gordeziani3 1* Faculty of Exact and Natur... more Gia A. Avalishvili1, Mariam A. Avalishvili and David G. Gordeziani3 1* Faculty of Exact and Natural Sciences, I. Javakhishvili Tbilisi State University, 3 I. Tchavtchavadze Ave., Tbilisi 0179, Georgia, [email protected] 2 School of Informatics, Engineering and Mathematics, University of Georgia, 77a M. Kostava Str., Tbilisi 0175, Georgia, [email protected] 3 Faculty of Exact and Natural Sciences and I. Vekua Institute of Applied Mathematics, I. Javakhishvili Tbilisi State University, 3 I. Tchavtchavadze Ave., Tbilisi 0179, Georgia, [email protected]
gmj, 2008
The present paper is devoted to the investigation of one dynamical three-dimensional mathematical... more The present paper is devoted to the investigation of one dynamical three-dimensional mathematical model of the fluid-solid interaction. The variational formulation of the corresponding initial boundary problem is considered and a problem for abstract second order evolution equation is formulated, which is a generalization of the three-dimensional initial boundary value problem. For the stated abstract problem the existence and uniqueness of solution, and the energy equality are proved, which yield the corresponding result for the dynamical three-dimensional problem of fluid-solid interaction.
In the present paper the Chandrasekharaiah-Tzou dynamical nonclassical model for thermoelastic pr... more In the present paper the Chandrasekharaiah-Tzou dynamical nonclassical model for thermoelastic prismatic shell is studied. The initial-boundary value problem corresponding to the dynamical three-dimensional model is investigated applying variational approach in suitable spaces of vector-valued distributions. A hierarchy of two-dimensional models is constructed for thermoelastic prismatic shell, when surface forces and the normal component of heat flux are given along the upper and the lower faces of the prismatic shell. The two-dimensional initial-boundary value problems corresponding to the models of the hierarchy are investigated in suitable function spaces. Moreover, the convergence of the sequence of approximate solutions of three space variables, constructed by means of the solutions of the reduced two-dimensional problems, to the exact solution of the original three- dimensional problem is proved and under suitable regularity conditions the rate of convergence is estimated. © ...
The present paper deals with the results of investigation of nonclassical problems for elliptic a... more The present paper deals with the results of investigation of nonclassical problems for elliptic and hyperbolic partial differential equations with integral nonlocal boundary conditions. Boundary value problems for elliptic equation on multidimensional cylindrical domain with one and two integral boundary conditions are considered. The nonclassical problems for elliptic equation are investigated applying variational approach in suitable Sobolev spaces and the existence and uniqueness results are proved. Nonclassical problems for multidimensional hyperbolic equation with integral boundary conditions are studied and the uniqueness of classical solution is proved. © 2011 Bull. Georg. Natl. Acad. Sci.
In this paper a boundary value problem for thermoelastic prismatic shell with microtemperatures i... more In this paper a boundary value problem for thermoelastic prismatic shell with microtemperatures is considered. A hierarchy of two-dimensional models for a static threedimensional model for prismatic shell with surface force, the normal component of heat flux and the first heat flux moment given on the upper and the lower faces of the prismatic shell is constructed. The two-dimensional boundary value problems corresponding to the hierarchical models are investigated in suitable function spaces. The convergence of the sequence of vector-functions of three space variables, restored from the solutions of the two-dimensional boundary value problems of the constructed hierarchy to the exact solution of the original three-dimensional problem is proved and the rate of approximation is estimated provided that the solution satisfies additional regularity conditions. © 2013 Bull. Georg. Natl. Acad. Sci.
In the present paper thermoelastic solid is considered within the framework of Lord- Shulman non... more In the present paper thermoelastic solid is considered within the framework of Lord- Shulman nonclassical theory of thermoelasticity. Applying variational approach initialboundary value problem corresponding to the three-dimensional model is investigated in suitable spaces of vector-valued distributions with values in Sobolev spaces. An algorithm of approximation by two-dimensional problems of the three-dimensional dynamical model for plate with variable thickness is constructed, when densities of surface force and normal component of heat flux are given on the upper and the lower face surfaces of the plate. The obtained two-dimensional initial-boundary value problems are investigated in suitable function spaces. Moreover, convergence of the sequence of vector-functions of three space variables restored from the solutions of the constructed two-dimensional problems to the solution of the original three-dimensional initial-boundary value problem is proved and under additional condi...
Applications of Mathematics and Informatics in Natural Sciences and Engineering, 2020
In this paper nonclassical dynamical model for thermoelastic bodies with three phase-lags is stud... more In this paper nonclassical dynamical model for thermoelastic bodies with three phase-lags is studied. Applying variational formulation of the general three-dimensional initial-boundary value problem existence and uniqueness of solution in suitable spaces is proved. Spectral algorithm of approximation of the three-dimensional initial-boundary value problem for thermoelastic shell by a sequence of two-dimensional ones is constructed, convergence of the algorithm in corresponding spaces is proved and the rate of convergence is estimated.
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Papers by Gia Avalishvili