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.
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 ...
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...
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, gavalish@yahoo.com 2 School of Informatics, Engineering and Mathematics, University of Georgia, 77a M. Kostava Str., Tbilisi 0175, Georgia, mavalish@yahoo.com 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, dgord37@hotmail.com
The present paper is devoted to the construction and investigation of algorithms of spectral appr... more The present paper is devoted to the construction and investigation of algorithms of spectral approximation of three-dimensional problems for nonhomogeneous anisotropic elastic multistructures in curvilinear coordinates, which are junctions of three-dimensional body, shell clamped in it and curvilinear rod clamped in shell. Sequences of boundary and initial-boundary value problems defined on the union of three, two and one-dimensional space domains are obtained and investigated in suitable Sobolev spaces. Moreover, the convergence of the sequence of vector-functions of three space variables restored from the solutions of the reduced problems to the solutions of the original three-dimensional problems in corresponding spaces is proved and under additional conditions, the rate of approximation is estimated.
In the present paper dynamical three-dimensional model for linearly elastic shells in curvilinear... more In the present paper dynamical three-dimensional model for linearly elastic shells in curvilinear coordinates is considered and a hierarchy of two-dimensional models of the corresponding initial-boundary value problem is constructed. The well-posedness of the two-dimensional problems is investigated in suitable spaces and the accuracy of approximation of the solution to the original problem by the vector functions restored from the solutions of the reduced problems is estimated.
In the present paper thermoelastic solid is considered within the framework of Lord- Shulman nonÂ... more In the present paper thermoelastic solid is considered within the framework of Lord- Shulman nonÂclassical theory of thermoelasticity. Applying variational approach initialÂboundary 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...
This paper is devoted to the construction and investigation of a hierarchy of two-dimensional mod... more This paper is devoted to the construction and investigation of a hierarchy of two-dimensional models for thermoelastic piezoelectric plate with variable thickness, which may vanish on a part of the lateral boundary. The hierarchical two-dimensional models are constructed for plate consisting of inhomogeneous anisotropic thermoelastic piezoelectric material with regard to magnetic field, when density of surface force, and normal components of electric displacement, magnetic induction and heat flux vectors are given along the upper and the lower face surfaces of the plate. The boundary value problems corresponding to the constructed static two-dimensional models are investigated in suitable weighted Sobolev spaces. The relationship between the constructed two-dimensional models and the original three-dimensional one is investigated, and the convergence of the sequence of vector-functions of three variables restored from the solutions of the constructed two-dimensional problems to the ...
Piezoelectric materials are widely used to build engineering smart structures, because of the eas... more Piezoelectric materials are widely used to build engineering smart structures, because of the ease of controlling by voltage, low weight and low power requirements. Inhomogeneous and, in particular, functionally graded materials are used to increase the durability and efficiency of engineering constructions undergoing high mechanical and thermal loads. Various parts of smart devices consisting of piezoelectric materials are often plate or shell like structures, and, therefore, construction and investigation of mathematical models of inhomogeneous thermoelastic piezoelectric plates and shells are important from both theoretical and practical viewpoints. In this paper, linear dynamical three-dimensional model [1, 2] of thermoelastic piezoelectric shell with variable thickness, which may vanish on a part of the lateral boundary, consisting of several inhomogeneous anisotropic thermoelastic piezoelectric layers with regard to magnetic field is considered. In order to construct dynamical...
This paper is devoted to the investigation of three-dimensional models of thermo-electro-magneto-... more This paper is devoted to the investigation of three-dimensional models of thermo-electro-magneto-elastic solids made of a multidomain inhomogeneous anisotropic material. General boundary and initial boundary value problems corresponding to the static and dynamic models are studied where, on certain parts of the boundary, mechanical displacement, electric and magnetic potentials and temperature vanish and, on the corresponding remaining parts of the boundary, the mechanical stress vector and components of the electric displacement, magnetic induction and heat flux along the outward normal vector of the boundary are given. Variational formulations of the boundary and initial boundary value problems are obtained and, applying them, existence and uniqueness results and the continuous dependence of solutions on given data, in suitable factor spaces of Sobolev spaces or spaces of vector-valued distributions, are proved.
This paper deals with nonclassical initial boundary value problems for string oscillation and tel... more This paper deals with nonclassical initial boundary value problems for string oscillation and telegraph equations. Algorithms for a direct construction of solutions are proposed. The existence and uniqueness of a solution of a nonlocal boundary value problem is proved in the case of a general one-dimensional hyperbolic equation.
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.
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 ...
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...
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, gavalish@yahoo.com 2 School of Informatics, Engineering and Mathematics, University of Georgia, 77a M. Kostava Str., Tbilisi 0175, Georgia, mavalish@yahoo.com 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, dgord37@hotmail.com
The present paper is devoted to the construction and investigation of algorithms of spectral appr... more The present paper is devoted to the construction and investigation of algorithms of spectral approximation of three-dimensional problems for nonhomogeneous anisotropic elastic multistructures in curvilinear coordinates, which are junctions of three-dimensional body, shell clamped in it and curvilinear rod clamped in shell. Sequences of boundary and initial-boundary value problems defined on the union of three, two and one-dimensional space domains are obtained and investigated in suitable Sobolev spaces. Moreover, the convergence of the sequence of vector-functions of three space variables restored from the solutions of the reduced problems to the solutions of the original three-dimensional problems in corresponding spaces is proved and under additional conditions, the rate of approximation is estimated.
In the present paper dynamical three-dimensional model for linearly elastic shells in curvilinear... more In the present paper dynamical three-dimensional model for linearly elastic shells in curvilinear coordinates is considered and a hierarchy of two-dimensional models of the corresponding initial-boundary value problem is constructed. The well-posedness of the two-dimensional problems is investigated in suitable spaces and the accuracy of approximation of the solution to the original problem by the vector functions restored from the solutions of the reduced problems is estimated.
In the present paper thermoelastic solid is considered within the framework of Lord- Shulman nonÂ... more In the present paper thermoelastic solid is considered within the framework of Lord- Shulman nonÂclassical theory of thermoelasticity. Applying variational approach initialÂboundary 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...
This paper is devoted to the construction and investigation of a hierarchy of two-dimensional mod... more This paper is devoted to the construction and investigation of a hierarchy of two-dimensional models for thermoelastic piezoelectric plate with variable thickness, which may vanish on a part of the lateral boundary. The hierarchical two-dimensional models are constructed for plate consisting of inhomogeneous anisotropic thermoelastic piezoelectric material with regard to magnetic field, when density of surface force, and normal components of electric displacement, magnetic induction and heat flux vectors are given along the upper and the lower face surfaces of the plate. The boundary value problems corresponding to the constructed static two-dimensional models are investigated in suitable weighted Sobolev spaces. The relationship between the constructed two-dimensional models and the original three-dimensional one is investigated, and the convergence of the sequence of vector-functions of three variables restored from the solutions of the constructed two-dimensional problems to the ...
Piezoelectric materials are widely used to build engineering smart structures, because of the eas... more Piezoelectric materials are widely used to build engineering smart structures, because of the ease of controlling by voltage, low weight and low power requirements. Inhomogeneous and, in particular, functionally graded materials are used to increase the durability and efficiency of engineering constructions undergoing high mechanical and thermal loads. Various parts of smart devices consisting of piezoelectric materials are often plate or shell like structures, and, therefore, construction and investigation of mathematical models of inhomogeneous thermoelastic piezoelectric plates and shells are important from both theoretical and practical viewpoints. In this paper, linear dynamical three-dimensional model [1, 2] of thermoelastic piezoelectric shell with variable thickness, which may vanish on a part of the lateral boundary, consisting of several inhomogeneous anisotropic thermoelastic piezoelectric layers with regard to magnetic field is considered. In order to construct dynamical...
This paper is devoted to the investigation of three-dimensional models of thermo-electro-magneto-... more This paper is devoted to the investigation of three-dimensional models of thermo-electro-magneto-elastic solids made of a multidomain inhomogeneous anisotropic material. General boundary and initial boundary value problems corresponding to the static and dynamic models are studied where, on certain parts of the boundary, mechanical displacement, electric and magnetic potentials and temperature vanish and, on the corresponding remaining parts of the boundary, the mechanical stress vector and components of the electric displacement, magnetic induction and heat flux along the outward normal vector of the boundary are given. Variational formulations of the boundary and initial boundary value problems are obtained and, applying them, existence and uniqueness results and the continuous dependence of solutions on given data, in suitable factor spaces of Sobolev spaces or spaces of vector-valued distributions, are proved.
This paper deals with nonclassical initial boundary value problems for string oscillation and tel... more This paper deals with nonclassical initial boundary value problems for string oscillation and telegraph equations. Algorithms for a direct construction of solutions are proposed. The existence and uniqueness of a solution of a nonlocal boundary value problem is proved in the case of a general one-dimensional hyperbolic equation.
Uploads
Papers by Gia Avalishvili