The predecessors are the Department of Membrane Theory, which was established when the Institute was founded in 1968, the Department of Continuous Body Mechanics Problems, and later the Department of Elongation Theory, established in 1973. The aim of the section is to investigate the new boundary and initial-boundary problems posed in the theory of shells and the theory of elongation for ordinary and choreographically differential equations and to develop methods for their solution, for which: i. Vekua dimension reduction method; Methods of theory of functions of complex variables; Potential method; The method of fundamental solutions; The method of singular integral equations; Finite element method; Boundary element method and other known methods. In addition, the aim of the department is to develop new research methods and obtain numerical results for specific practical tasks.
The department was established in 1968 when the institute was founded. The existing Department of Numerical Methods and Programming and the later established Department of Numerical Methods (1970) and the Department of Projection Methods (1973) were engaged in individual research at the Institute from June 2009 to September 2018 under short-term contracts. They have experience in creating software packages using spatial building calculations. The section continues, expands and generalizes the traditional themes related to the modeling, foundation, creation of analytical-discrete schemes of their calculation and numerical realization of the continuous environment using programming languages using programming languages.
The Department of Differential Equations and Optimal Management is the legal successor to the two pre-existing divisions at the institute - the Division of Differential Equations and Management Theory and the Division of Conventional Differential Equations.
The condition of continuous and discrete systems (biological, economic, physical, etc.) at a given point in time usually depends on the behavior of the system at different points in time. In such a case it is said that the evolution of the system is influenced by the factor of delay, the factor of heredity and skewed arguments, and so on. For example, in the differential model of the immune response, the delay factor is the time delay of the body's immune response to viruses that enter the body, while in the economic growth model, the delay factor arises if the company takes into account past earnings when investing. Adequate mathematical models of continuous and discrete systems containing the above factors are functionally-differential and functionally-differential equations, respectively.
The aim of the department is to develop the qualitative theory of functional-differential and functional-differential equations and the theory of optimal management:
1) Determine the properties of A or B for functional-differential and differential equations; Proof of comparison theorems with respect to property A or B; Determine the volatility and non-volatility conditions of the solution; Obtain sufficient (necessary and sufficient) conditions for the determination of properties A or B for the generalized equation of Emdenfowler;
2) To investigate the correctness of the Cauchy problem for the neutral equation by summing and distributing the delay for the functional-differential equations, to determine the formulas for the variation of the solution and to identify the various effects in them, to investigate the optimal control problems.
From a theoretical and practical point of view, each new result obtained in the directions described in points 1) and 2) is relevant because it allows to determine the nature of the process evolution and the effects of process concerns, as well as the criteria for optimal process data selection.
The study and approximate solution of Partial Differential Equations was one of the main scientific directions in the Problem Laboratory of Applied Mathematics at Tbilisi State University since 1966, and then in the Institute of Applied Mathematics established in 1968 on the basis of this laboratory. In 1979, on the initiative of A. Bitsadze, the institute seminar "Partial Differential Equations" was launched, and in 1980 a department of the same name was established. The department is currently investigating approximate differential and integro-differential models related to the solution of many theoretical and practical problems. The thematic section studies the problems initiated by models reflecting the state of elastic equilibrium of solids, including membranes, Maxwell systems of electromagnetic field propagation in the environment, Boltzmann models of radiation transfer, and mathematical modeling of other processes.
The department is the legal successor of the department of the same name established in 1982 at the Institute. Its aim is to develop traditional themes for the predecessor department, namely the study of function spaces and their application to Fourier series congruence issues, and to give it an applied character in line with the objectives of the other departments of the Institute.
The Division of Complex Analysis and its Applications was established in 1976. The initiator of its creation and the first scientific leader was the founder of the institute I. Vekua. By his invitation, G.Manjavidze became the head of the department of complex analysis and its applications in 1977. From the very beginning, the scientific topics of the section were presented in the fundamental directions of classical complex analysis, such as boundary tasks for analytical and generalized analytic functions, quasi-conformal reflections, systems of elliptical equations with complex points on complex planes with complex planes, as well as various models.
Currently, research is underway in the following areas: to study the solution space of a system of generalized Beltram equations and to establish its relation to complex structures of diversity; Study of the spatial structure of systems of elliptic equations with properties on planes and Riemannian surfaces, solvability of systems of singular integral equations; Boundary problems for generalized analytical vector functions and related soluble models related to theoretical physics.
The Department of Probability Theory and Mathematical Statistics has been functioning in the Problem Laboratory of Applied Mathematics at Tbilisi State University since 1966. In 1968, the Institute of Applied Mathematics was established on the basis of this laboratory, in which the above-mentioned department continued to function until 1973, When the Sector of Economic-Mathematical Research and Mathematical Statistics started functioning in the Institute of Economics and Law of the Georgian Academy of Sciences, where the whole staff of Probability Theory and Mathematical Statistics of Ilia Vekua Scientific-Research Institute was transferred. In 2016, after the restoration of the status of an independent scientific-research structural unit of the Ivane Javakhishvili Tbilisi State University for the Ilia Vekua Institute of Applied Mathematics, the Department of Probability Theory and Mathematical Statistics was also restored in this institute.
The main directions of scientific development of the Department of Probability Theory and Mathematical Statistics:
1) The theory of nonparametric estimation of the functional characteristics of the law of observational distribution, which implies the theory of modern nonparametric estimation of probability distribution density and the theory of nonparametric estimation of the regression function (this theory was founded in parallel in 1964 in Georgia and the USA);
2) Methods for statistical estimation of unknown parameters of probabilistic distribution;
3) Statistical hypothesis testing methods;
4) Theory of Stochastic Differential Equations (Problems of Evaluation of Solutions of Differential Equations with Random Coefficients, Based on Absolute Continuity of Measures and Properties of Logarithmic Derivatives).
5) Realization of the obtained results and practical application.
The Department of Discrete Mathematics was established in 1983 at the Institute. This section deals with current topics in modern discrete mathematics:
1) Problems of discrete, combinatorial and convex geometry (point systems with certain properties, generalized Euler-Venn diagrams, multidimensional multivariate structure and problems related to their equivalence, etc.);
2) Problems of infinite combinatorics and their applications in set theory, mathematical analysis and size theory (Ramsay theory, large cardinal numbers, almost disjunctive sets of sets);
3) Models associated with different types of set-theoretical axioms (internal models, Frankel-Mostowski models, etc.);
4) Structural properties of finite and infinite graphs and their specific realizations.
The pre-unit was the Systems Programming Department, which operated at the Institute from 1973-2006.
The purpose of the department is to conduct scientific research in computer science and computational logic, with the aim of bringing the results to a level that can be used in practice (including obtaining results in the form of a computer software product, compiling instructions for their usage, etc.).
The laboratory is the successor of the Mathematical Logic and Algorithm Theory department existing at the institute from 1969 to 2006 and the Applied Logic and Programming laboratory established in 2007 at the Faculty of Exact and Natural Sciences of I.Javakhishvili Tbilisi State University. The later was moved to the institute by the decision of the university administration.
The aim of the laboratory is to conduct scientific research in the fields of applied logic and logic programming. The term "applied logic" is considered in a broad sense and includes subfields of mathematical logic and artificial intelligence such as computer and computational logic, proof theory, automated reasoning, knowledge representation, semantic web and the like.
One of the important functions of the laboratory is to ensure that students of different levels of education at TSU perform laboratory work and undergo various types of practices (for example, creating different software modules).