Hamlet Meladze

In the present work the problems related to the construction of a probabilistic model of the Cartesian product of canonically conjugated fuzzy subsets is investigated. The case of the Cartesian product of two fuzzy subsets with commuting colors is considered in detail. It is shown that the model most fully reflects the special "additional" nature of the connection between two canonically conjugated colors in a form of the uncertainty principle for the corresponding dispersions.

In this paper, factorized difference schemes for a two-dimensional system of hyperbolic type equations with mixed derivatives are considered. When constructing a difference scheme, the method of regularization of difference schemes, proposed by A. A. Samarsky, is used. The convergence of the scheme is proved. The factorized difference schemes for a second-order general hyperbolic system are used to solve numerically a system of equations of elasticity theory, in the case of two spatial variables. The constructed algorithm can be effectively used for parallel computing systems.

The present work is devoted to the formulation and investigation of a non-local contact problem for a parabolic-type linear differential equation with partial derivatives. In the first part of the work, the linear parabolic equation with constant coefficients is considered. To solve a non-local contact problem, the variable separation method (Fourier method) is used. Analytic solutions are built for this problem. One then elaborates on the non-local contact problem for parabolic equations with variable coefficients. Using the iterative method, the existence and uniqueness of the classical solution to the problem is proved. The proof of the existence and uniqueness of the solution is based on the use of the generalized Harnack theorem, which also is valid for linear differential equations with partial derivatives of parabolic type. The effectiveness of the method is confirmed by numerical

calculations.

In the present paper the nonlocal contact problems for heat equation with constant, as well as variable coefficients are investigated. A method of separation of variables (Fourier method) is implemented for solving the problem in case of constant coefficients. Existence and uniqueness of regular solution is proved. In case of variable coefficients the iterative procedure is constructed, by means of which the solution of an initial problem is reduced to the solution of the sequence of classical initialboundary problems. 

In the present work is considered an approach, according to which canonically conjugate colors in the theory of fuzzy sets are related to the properties of information functions and noncommutative linear operators in Gilbert's space: each information state corresponds to the estimation of compatibility function, every color – to the operator. It is supposed that color, as some property (attribute), characterizing a condition of a system, can receive different values called by eigenvalues of this color. The cases of discrete and continuous spectrum of eigenvalues of color are considered. The example of calculation of conditional computable values of color is given.

In the present paper, a multi-unit redundant system with unreliable, repairable units is considered. Two types of maintenance operations - the replacement of the failed main unit by the redundant one and the repair of the failed unit - are performed. The case of the system with one replacement server with arbitrary replacement time distribution function and repair server with an exponential distribution of repair time is considered. For this system mixed-type semi-Markov queuing model with the bifurcation of arrivals is constructed. It represents a non-classical boundary value problem of mathematical physics with non-local boundary conditions.

The present paper deals with optimal control problems whose behavior is described by elliptic equations with m-point Bitsadze–Samarskiǐ [3] boundary conditions. Necessary optimality conditions are established by using the approach worked out in [2] for controlled systems of general type. To investigate the conjugate problem, we use the algorithm reducing nonlocal boundary value problems to a sequence of Dirichlet problems. Such a method makes it possible to solve the problem numerically. In paper [4], for the numerical solution of the Dirichlet boundary value problem, the relaxation method is used. Modified Explicit Decoupled Group (MEDG) method uses a skewed difference formula which leads to lower computational complexities since the iterative procedure need only involve nodes on half of the total grid points in the solution domain and thus a reduced system of linear equations is attained. A MEDG method is presented for numerical solving an optimal control problem for elliptic equations by means of the Mathcad.

The different processes in networks of electrical power systems, gas transmission and distribution pipelines, other pipelines carrying material such as water, etc. can be described using mathematical models with nonstationary systems of nonlinear partial differential equations given on graphs. But for practical realisation, the linear models are used, which do not depend on the time. In the present work, the boundary value problem is considered for the system of linear second order ordinary differential equations, given on graphs. The existence and uniqueness of the solution of the formulated problem are proved. The numerical method for solving this problems is proposed. In the case of constant coefficients, the analytical solution of the problem is constructed.

В представленном докладе исследуются краевые и начально-краевые за дачи с нело кальными контактными условиями для линейных уравнений эл липтического и параболического типов с переменными коэффициентами.

В докладе построена двумерная математическая модель формированная прорывной волны при разрушении плотин. Для численного решения соответствующих уравнений в частных произ водных построены и обоснованы двухслойные линеаризованные разностные схемы с нелинейным регуляризатором. Конкретные численные расчеты параметров оползневых волн проведены на примере оползня Ток, обрушение которого произошло в водохранилище Вайонт (Италия) в 1963 г. Результаты численных расчетов представлены в виде двумерных и трех мерных графиков. Расчеты показали удовлетворительное совпадение с известными результа тами натурных наблюдений.

Nonlocal boundary and initial-boundary problems represent very interesting generalizations of classical problems. At the same time, they quite often arise during the creation of mathematical models of real processes and the phenomena in physics, engineering, ecology, etc.

In the present report, the initial-boundary problems with nonlocal contact condition is investigated for non-stationary linear partial differential equations with variable coefficients. For the solution of these problems a method of separation of variables (also known as the Fourier method) is considered. Existence and uniqueness of regular solution is proved.

In the present report, the boundary and initial-boundary problems with nonlocal contact conditions are investigated for the linear partial differential equations of elliptic and parabolic types with variable coefficients. Existence and uniqueness of regular solution is proved. The iterative procedure is constructed, by means of which the solution of an initial problem is reduced to the solution of sequence of classical Dirichlet problems (for the elliptic equations) and CauchyDirichlet problems (for the parabolic equations). The parallel algorithms for the solution of these problems are considered. Numerical results of the solution of some specific problems for the elliptic and parabolic equations are given. In the second part of the report, a method of separation of variables (also known as the Fourier method) for some stationary and non-stationary problems with nonlocal contact conditions is considered.

In the present work, the non-local initial-boundary contact problems for one dimensional parabolic type equation is considered. For the stated problem, the existence and uniqueness of the solution is proved. The iteration process is constructed, which allows one to reduce the solution of the initial non-classical problem to the solution of a sequence of classical

Cauchy-Dirichlet problems. The convergence of the proposed iterative process is proved; the speed of convergence is estimated. On the basis of this algorithm the method for numerical solution of the initial problem is described.

In the present work, the non-local initial-boundary contact problems for one dimensional parabolic type equation is considered. For the stated problem, the existence and uniqueness of the solution is proved. The iteration process is constructed, which allows one to reduce the solution of the initial non-classical problem to the solution of a sequence of classical

Cauchy-Dirichlet problems. The convergence of the proposed iterative process is proved; the speed of convergence is estimated. On the basis of this algorithm the method for numerical solution of the initial problem is described.

The present talk is devoted to the investigation of special decomposition methods for stationary and nonstationary problems of partial differential equations: the decomposition of the basic area or the basic operator of the initial problem. These methods are based on the reduction of the solution of initial problem to the solution of some more "simple" sub-problems and open the great possibilities in designing algorithms of parallel implementation and creation the program products for computers. We consider also the parallel version of the Schwarz alternating method, based on area decomposition. The independent problem is the solution of difference problems representing itself the system of linear or nonlinear algebraic equations. The parallel iterative methods for the numerical solution of nonlinear equations and systems of equations will be considered as well. In the talk the primary attention will be inverted on the works, conducted in the Tbilisi State University and I.Vekua Institute of Applied Mathematics.

The present work is devoted to the review of the articles, where for some equations of the mathematical physics the boundary and initial-boundary problems with nonlocal contact conditions are considered. For these problems, the existence and uniqueness of the solution is proved. The algorithms for numerical solution are constructed and investigated.

In the present work, the initial-boundary problem with nonlocal contact condition for heat (diffusion) equation is considered. For the stated problem, the existence and uniqueness of the solution is proved. The convergence of the proposed iterative process is proved; the speed of convergence is estimated. The algorithm is suitable for parallel implementation. The specific problem is considered as an example and solved numerically.

The nonlocal contact problem for elliptic equation and its numerical solution is cosidered. Existance and uniqueness of the solution is proved.

In the present work the iterative algorithm for solving the systems of nonlinear algebraic equations is constructed, taking into account the features of parallel calculations. Speed of convergence of the offered iterative method is estimated.

The present work is devoted to the specific nonlocal statement and analysis of one contact problem for Poisson’s equation in two-dimensional domain. For numerical solution the iteration process is constructed, which allows one to reduce the solution of the initial problem to the solution of a sequence of classical Dirichlet problems. The algorithm is suitable for parallel realization. The specific problem is considered as example and solved numerically.

In this paper one generalization of contact problem for Poisson's equation in rectangular area is

considered, when nonlocal conditions are stated for the finite number of segments. The existence and

uniqueness of a regular solution is proved. The iteration procedure is constructed and investigated.

The results of numerical calculations are given.

Nonlocal problems represent quite interesting generalization of classical problems of mathematical physics and at the same time they are naturally raised at construction of mathematical models of real processes and the phenomenon. The present report is devoted to statement and the analysis of nonlocal contact boundary problems for linear elliptic equations of second order in two-dimensional domains. The existence and uniqueness of a regular solution is proved. The iteration process is constructed, which allows one to reduce the solution of the initial problem to the solution

of a sequence of classical Dirichlet problems. In the report the results of numerical calculations of nonlocal contact problem for Poisson’s equation in two-dimensional domain are given.

In this paper one generalization of contact problem for poisson's equation in rectangular area is considered. Existance and uniqueness of the solution is proved.

The present work is devoted to the statement and analysis of one nonlocal contact problem for Poisson's equation in twodimensional domain. For numerical solution the iteration process is constructed, which allows one to reduce the solution of the initial problem to the solution of a sequence of classical Dirichlet problems. The algorithm is suitable for parallel realization. The specific problem is considered as example and solved numerically by using Wolfram Mathematica.

The paper discusses the possibility of verifying programs for certain classes of computational tasks using the Model checking method, taking into account the specificity of parallelism in programs. The validation of the program differs from the traditional one and is reduced not only to the verification of individual branches, but also to the analysis of the interaction of branches by considering their parallel structure.

The proposed work has developed a technique that explores the problem of constructing a synchronous iterative method for solving systems of nonlinear algebraic equations, which can be effectively implemented on parallel computer systems. The speed of convergence of the proposed iterative method is estimated.

The given paper deals with the problem of structural control for a wide class of any territorially distributed standby systems consisting of unreliable repairable elements. Mathematical models for interaction of degradation and its compensation processes in the above mentioned systems are proposed and their possible applications are partially analyzed. These models represent open and closed special type queuing ystems for two parallel maintenance operations-replacements and repairs. The problem for optimization of said system comic criterion is stated. The possible ways of its solution are discussed.

The given paper deals with the problem of structural control for a wide class of any territorially distributed standby systems consisting of unreliable repairable elements. Mathematical models for interaction of degradation and its compensation processes in the above mentioned systems are proposed and their possible applications are partially analyzed. These models represent open and closed special type queuing ystems for two parallel maintenance operations-replacements and repairs. The problem for optimization of said system comic criterion is stated. The possible ways of its solution are discussed.

A nonlocal contact boundary problem for Poisson equation is stated and investigated in rectangle area. The uniqueness of a solution is proved. The iteration process is constructed, which allows one to reduce the solution of the initial nonlocal contact problem to the solution of a sequence of classical Dirichlet problems. The difference scheme for numerical solution of stated problem is considered.

Parallel Algorithm of the Solution of Boundary Problem for System of the First Order Ordinary Differential Equations were considered: Problem formulation; Description of the iterative method; Algorithm for solving the problem; Implementation of the algorithm for a parallel system; Results of numerical experiments.

In the presented work, for some equations of mathematical physics, boundary and initial-boundary problems with non-local contact conditions are considered. Using an iterative procedure, solving the original problem is reduced to solving the of sequence of Dirichlet problems.

The paper deals with optimal control problems whose behavior is described by an elliptic equations with Bitsadze–Samarski nonlocal boundary conditions. The theorem about a necessary and sufficient optimality condition is given. The existence and uniqueness of a solution of the conjugate problem are proved. A numerical method of the solution of an optimal problem by means of the Mathcad package is presented.

The mixed problem with first order boundary conditions for systems of quasilinear equations, which describes dynamics of homogeneous and isotropic elastic body in case of flat deformation is considered.

The numerical solution of this problem, as a rule, requires essential computing resources. One of methods of abbreviation of time of the solution is use of parallel computing systems and parallel algorithms. In this paper for solving of stated problem is constructed three-layer factorized difference scheme.

For the numerical solution of the received difference equations the algorithm, which can be used effectively for parallel computing systems, is offered. The pseudocode of this algorithm is given.

The mixed problem with first order boundary conditions for systems of quasilinear equations, which describes dynamics of homogeneous and isotropic elastic body in case of flat deformation is considered.

The numerical solution of this problem, as a rule, requires essential computing resources. One of methods of abbreviation of time of the solution is use of parallel computing systems and parallel algorithms. In this paper for solving of stated problem is constructed three-layer factorized difference scheme.

For the numerical solution of the received difference equations the algorithm, which can be used effectively for parallel computing systems, is offered. The pseudocode of this algorithm is given.

The mixed problem with first order boundary conditions for system of differential equations, which describes dynamics of homogeneous and isotropic elastic body in case of flat deformation is considered. The numerical solution of this problem, as a rule, requires essential computing resources. One of the methods of abbreviation of time of the solution is the use of parallel algorithms. In this paper for solving of stated problem is constructed three-layer factorized difference scheme. For the numerical solution of the received difference equations the algorithm, which can be used effectively for parallel computing systems, is offered. The pseudocode of this algorithm is given. 

In this report the problem of construction of three-layer factorized scheme for solving of mixed problem with first order boundary conditions for systems of linear equations of parabolic type B ∂u/ ∂t = Lu + f is considered, where B is positively defined and symmetric matrix, L is strong elliptic operator with variable coefficients, containing the mixed derivatives, u and f are n-dimensional vectors. The absolutely stable three-layer factorized scheme is constructed , whose solution requires no inversion of matrix B. Separately considered the case, when B is the unit matrix. In this case the absolutely stable three-layer factorized scheme is constructed. For difference scheme the aprioristic estimation on layer in norm of mesh space ◦ W (1) 2 is received, on which basis convergence of solution of the difference scheme to the solution of an initial problem is proved with the speed O(τ + h^ 2 ) and in the second case with the speed O(τ^ 2 + h^ 2 ), where τ - the step of time grid and h - the step of space grid. The received algorithms can be effectively used for multiprocessing computing systems.

 In the talk some methods of constructing computational algorithms are considered called the additive averaged schemes (AAS) of parallel calculation. For parabolic and hyperbolic problems the construction of such AAS is based on the decomposition of the operator of the initial problem; simultaneously with this are proposed and investigated AAS for solving the specific problems of thermoelasticity, shell theory, problems of distribution of pollution in water substances and etc. In the talk we also consider some versions of the method of summary approximation (MSA) for the multidimensional equations of parabolic and hyperbolic types; the

questions of convergence of the solutions of models MSA to generalized and classical solution of initial problem are investigated.

Mathematical modeling of various processes in the nets of gas pipeline, system of submission and distribution of water, drainpipe, also long current lines and different types of engineering constructions quite naturally leads to the consideration of differential equations on graphs. In this paper we consider the mathematical model of electro power system, which is the boundary value problem for ordinary differential equations, defined on graphs. Correctness of the prob lem is investigated. Constructed and investigated the corresponding finite-difference scheme. Double sweep type formulas for finding solutions of finite difference scheme are offered. This al gorithm is essentially a parallel algorithm and efficiently implemented on computers with parallel processors.

The present paper devoted to investigation of special decomposition methods for stationary and nonstationary problems in the case of partial differential equations. Based on the proposed decomposition method are constructed parallel computing algorithms. We consider also the parallel version of the Schwarz alternating method, based on area decomposition. The independent problem is the solution of difference problems representing itself the system of linear or nonlinear algebraic equations. In this paper is considered both synchronous and asynchronous parallel iterative methods for the numerical solution of nonlinear equations and systems of equations. © 2012 by Nova Science Publishers, Inc. All rights reserved.

A conditional characteristic function of color and conditional computable values of color are considered. When calculating the compatibility function of the Cartesian product of two fuzzy canonically conjugate subsets, we will proceed from the corresponding characteristic function.

Cases of Cartesian product of two fuzzy subsets with both commuting and non-commuting colors are considered.

It is shown that from the probability model of the Cartesian product of two fuzzy subsets, a relation follows, establishing a relationship between the variances of canonically conjugate colors.

In this paper, we construct and analyze the family of synchronous iterative methods for solving the systems of nonlinear equations. These methods can be effectively realised on parallel computing systems. At minimum restrictions on the operator the local convergence theorems of these iterative methods are proved and the quadratic convergence is shown. Numerical results of applying this method to some test problems show the efficiently and reliability of these methods.

For the Modeling of the American Option Pricing we consider the financial ( B, S) -market consisting only of two assets: a bank account (bonds) B = (Bn) and a stocks S = (Sn) , where n changes from zero to N, n = 0,1,..., N . 

The generalized information theory is constructed on the base of chromotheory of canonicaly conjugate fuzzy subsets. This allows us to process the objective and subjective information simultaneously and carry informational functions in the Hilbert space, where the colours will be presented as linear operator. Using this approaches we can establish the commutation conditions for operators that are corresponding to various colours, i. e. to receive the analogues of Heisenberg principle for knowledge presentation and construction of arithmetic of fuzzy numbers, which allows us development of fuzzy information processing methods and decision making algorithms.

On the base of canonically conjugate fuzzy subsets we construct the optimal arithmetic of canonically conjugate fuzzy real numbers, which allows consideration of the combined statistics (probabilistic and possibilistic) of natural language. Constructed formalism permits to investigate the quantitive aspects of the natural languages structures. Carrying out possibility analysis of language is a cardinal problem of quantitative descriptions

of language, that is very important from the point of view of creation of the future generations of computers, where communication of the consumer with a computer will occur

in a natural language.

The generalized information theory is constructed on the base of chromotheory of canonicaly conjugate fuzzy subsets. This allows us to process the objective and subjective information simultaneously and carry informational functions in the Hilbert space, where the colours will be presented as linear operator. Using this approaches we can establish the commutation conditions for operators that are corresponding to various colours, i. e. to receive the analogues of Heisenberg principle for knowledge presentation and construction of arithmetic of fuzzy numbers, which allows us development of fuzzy information processing methods and decision making algorithms.

On the base of canonically conjugate fuzzy subsets we construct the optimal arithmetic of canonically conjugate fuzzy real numbers, which allows consideration of the combined statistics (probabilistic and possibilistic) of natural language. Constructed formalism permits to investigate the quantitive aspects of the natural languages structures. Carrying out possibility analysis of language is a cardinal problem of quantitative descriptions

of language, that is very important from the point of view of creation of the future generations of computers, where communication of the consumer with a computer will occur

in a natural language.

The theory of canonically conjugate fuzzy subsets is presented. The Heizenberg’s principle’s analog principle is established. In Hilbert space the informational functions and joint membership functions are defined. In the work colour operators, Zadeh operators and corresponding commutativity relations are presented

The convergence of linearized difference scheme in Eulerian variables with non-linear regularizator to the smooth solutions for linear analog of two-dimensional Saint-Venant equations are considered for Cauchy problem with periodic (in spatial variables) solutions. The proof of convergence of difference scheme is performed by energetic method. In the class of sufficiently smooth solutions of the difference scheme is proved the convergence of solution of considered difference scheme in mesh norm L2 with speed O(h2).

In the present work the boundary-value problems for Poisson’s equations in the three-dimensional space on some two-dimensional structures with one-dimensional common part is given and investigated. This technique of investigation can be easily applied to the more complex initial data and equations. Such problems have practical sense and they can be used for mathematical modeling of specific problems of physics, engineering, ecology and so on. This problem is the generalization of boundary value problem for ordinary differential equations on graphs. This problem is investigated and correctness of the stated problem is proved in [1]. The special attention is given to construction and research of difference analogues. Estimation of precision is given. The formulas of double-sweep method type are suggested for finding the solution of obtained difference scheme.

The new chromo theory of canonically conjugate fuzzy subsets is presented. The Heisenberg’s principle’s analog principle is established. In Hilbert space the informational functions and joint membership functions are defined. In the work Zadeh operators, color operators and corresponding commutatively relations are presented.

In the class of sufficiently smooth solutions of the Kosh's problem for the two-dimensional Saint-Venan equations, written in Euler variables, the congruence of a linearised difference scheme in the L2 norm at velocity O(h^2) is proved.

The experton theory is presented in a such form that permits to use it directly for decision making. The new form of experton theory is applied to scientific themes presented on the concurs. Each them is characterized by some attributes in 10 point system. The algorithm of decision making is presented by some rules of matrix transformations. Shortly the expertons algebraic properties are considered.

The new chromotheory of canonicaly conjugate fuzzy subsets is presented. The Heizenberg’s principle’s analog principle is established. In Hilbert space the informational functions and joint membership functions are defined. In the work Zadeh operators, colour operators and corresponding commutativity relations are presented.  

For numerical modeling of difficult applied problems now is perspective to use the computing systems with parallel data processing. In the given work some parallel iterative methods for the solution of nonlinear systems of the equations for cluster systems are considered. 

The main task involved in this work is to investigate effectiveness of application of a method of generalized discrimination analysis [1] for automatic processing of digital seismological records in a problem of separating earthquakes and noise. The corresponding algorithm consists of two steps. Step I constructing of tabular-numerical database containing the information about known and well-identified events; step II-analysis of entry signal. For this purpose we use multifactor linear systems [2]. As data processing takes place simultaneously for several stations of operating seismic network the parallelization of above algorithm for whole network is suggested. Besides the algorithm includes modules where the calculations also can be parallelized.

The paper discusses the problems of constructing, researching and implementing synchronous parallel iterative methods for solving nonlinear equations. We also consider the problem of finding an self-similar solution of a mathematical model of movement of gas that arose under the action of a flat piston in the presence of volumetric flow points of mass, momentum and energy. The problems of constructing and implementing the iterative method of solving the obtained boundary problem are studied taking into account the peculiarities of parallel calculations. 

In the present talk the nonlocal initial-boundary problem for the linear parabolic equations with nonlinear boundary and initial conditions is considered.

The talk is devoted to the statement and investigation of one nonlocal problem for multidimensional elliptic equations with variable coefficients. The existance and uniqueness of the regular solution is proved.

Doctoral Thesis Referee

Master Theses Supervisor

Doctoral Thesis Supervisor/Co-supervisor

Scientific editor of monographs in foreign languages

Scientific editor of a monograph in Georgian

Editor-in-Chief of a peer-reviewed or professional journal / proceedings

Review of a scientific professional journal / proceedings

Member of the editorial board of a peer-reviewed scientific or professional journal / proceedings

Participation in a project / grant funded by an international organization

Participation in a project / grant funded from the state budget

Patent authorship

Membership of the Georgian National Academy of Science or Georgian Academy of Agricultural Sciences

Membership of an international professional organization

Membership of the Conference Organizing / Program Committee

National Award / Sectoral Award, Order, Medal, etc.

Honorary title

Monograph

Handbook

Research articles in high impact factor and local Scientific Journals

Publication in Scientific Conference Proceedings Indexed in Web of Science and Scopus