Mathematics

Subjects of research: the formation of stress-strain state of thin-walled structures.
Purpose of work: development of mathematical model and algorithms for the solution the static problem of plates bending with physical and geometrical nonlinearity, which would allow automating process of the problem solution, and giving a chance to spend multiple experimental researches.
Methods of research: In this paper for solving the problem was used the following approximate methods: the variation Ritz’s method, the method of clastic solutions by A. Ilyushin, the method of consecutive approximations.
The result obtained and their novelty: the mathematical model solutions for physically and geometrically nonlinear bending problem of plates based on which was made the algorithm of calculating the stress-strain state of plates using the Ritz’s method, the elastic solution by Ilyushin and the method of successive approximations were found. The novelty of the proposed work is as follows: was derived a mathematical model for solving the nonlinear problem on the basis of the Ritz’s method, an algorithm for solving the problem, and a set of software tools to automate the process of solving the problem have been created.
Practical value: the developed mathematical model, algorithm and software can be recommended for research and design institutes.
Degree of embed and economic affectivity: the results of research can be used in various industries that use different type of the metal plates for design. The significant economic can be achieved by reducing the time and complexity of design-development using the algorithm and software tools have been created.
Field of applications: mechanical engineering, shipbuilding, aircraft, power engineering, construction.

Objects of research: the formation of stress-strain state of thin-walled structures
Methods of research: In this paper for solving the problem was used the following approximate methods: the variation Ritz’s method, the method of clastic solutions by A. Ilyushin, the method of consecutive approximations.
The findings and their novelty: the mathematical model solutions for physically and geometrically nonlinear bending problem of plates based on which was made the algorithm of calculating the stress-strain state of plates using the Ritz’s method, the elastic solution by Ilyushin and the method of successive approximations were found.
The novelty of the proposed work is as follows: was derived a mathematical model for solving the nonlinear problem on the basis of the Ritz’s method, an algorithm for solving the problem, and a set of software tools to automate the process of solving the problem have been created.
Practical significance: the developed mathematical model, algorithm and software can be recommended for research and design institutes: the method of mathematical model constructing also as well as algorithm development and software can be used in designing organizations and specialized departments of the Universities of the Republic of Uzbekistan.
The degree of implementation and economic efficiency: the results of research can be used in various industries that use different type of the metal plates for design. The significant economic can be achieved by reducing the time and complexity of design-development using the algorithm and software tools have been created.
Range of applications: mechanical engineering, shipbuilding, aircraft, power engineering, construction

Objects of research: homogeneous and multilayered systems
Purpose of work: reception of effective approximately-analytical decisions for the quantitative and qualitative analysis at an estimation mass-transfer in multilayered systems.
Methods of research: widely used methods of mathematical physics, the theory of function complex variable, asymptotic methods are applied, integrated transformations Laplace, a sine and cosine transformations Fourier, method of fission and as a method of final elements.
The results obtained and their novelty: all of the main results of this work are new and consist of the following:
1. In the work the problem decision of a mass-transfer consisting of the equations in private derivatives of parabolic type of subordinates to certain initial and boundary conditions is received. The stream was from the outside carried out at a non-stationary mode from rectangular area with the account and without compressibility of environment in the bottom layers.
2. It is received problem decisions mass-transfer by a method of final elements when the equation contained the first derivative on a spatial variable.
3. Is received analytical decisions of a two-dimensional problem mass-transfer with method of the fission.
4. At use of method of final elements, the way of definition of an internal point for an element containing functions of forms of the second order at which for the fixed moment of time the approached decision has coincided with the exact is specified.
Practical value: the work has theoretical character.
Degree of embed and economic effectiveness: the received results can be used at reading of special courses for post-graduate students of faculties of a natural profile.
Field of application: it is considered in all problems of the mathematical physics, leading by the equation in private derivatives of parabolic type (thermal, gas diffusion, oil and gas extraction, and so forth).

The aim of the research is to increase the effectiveness of methods for mathematical modeling of gcofiltration processes of regional hydrogeological systems.
Scientific novelty of the research:
the concept of mathematical modeling of hydrogeological processes of a regional nature was developed, based on the principles of the theory of gcofiltration and gcomigration in complex hydrogeological conditions; methods for integrating the mathematical modeling of hydrogeological processes with information and communication technologies were improved;
numerical methods of mathematical modeling of hydrogeological processes of the regional plan were developed on the basis of modem G1S technologies, which allow to unite diverse models of gcofiltration within a single information and technological system;
a flexible system of geoinformation and mathematical modeling of gcofiltration processes of regional hydrogeological objects was proposed, based on the use of the principles of formation and joint application of models of different scale and spatial coverage;
the software, technologies and hardware-tools of the automated metering, registration and transfer of hydrogeological information, used for gcoinformation and mathematical modeling, as well as for monitoring the underground hydrosphere was devolcpcd;
the principles of organizing a database of geoinformation-and-mathematieal models of regional hydrogeological objects combining factographic and cartographic data with the possibilities of their subsequent integration into a single automated complex were developed;
principles and criteria for constructing a gcoinformation system in the integration of the mathematical model of salt transfer by interconnected flows of surface and groundwater for large-scale objects with complex hydrogeological conditions were developed.

The aim of the research work is mathematical modeling of applied hydrodynamic processes of compressible two-phase media on the basis of analytic functions "A" of complex variable in two-phase media.
Scientific novelty' of the research work is as follows:
Methods for generalizing the theory of an analytic function of a complex variable operator "A", algorithms and methods for solving some applied problems for two-phase media;
Cauchy’s and Montel’s theorems, Picard's large theorem are proved, the mechanisms for expanding Taylor and Laurent series for a classical generalization of analytic functions "A (z)" are developed;
Methods for generalizing the integral Poisson formula for a stationary system of poroelasticity;
An algorithm for obtaining the solution of the analogue of the Mindlin problem for the stationary system of the poroelasticity equation in a half-space and numerical study of the influence of various dynamic characteristics on the wave field is found;
Differential identities connecting velocities, pressures, and mass forces in the equations of two-velocity hydrodynamics with phase equilibrium from pressure are proved in a divergent form;
Methods for constructing a general solution for the stream function in the planar case of the Monge-Ampere system of equations are developed;
A numerical model to study the flow of incompressible viscous two-velocity fluids, taking into account the equilibrium of the phases with respect to pressure is developed.

The urgency and relevance of the theme of dissertation. There is a great interest in the study of nonlinear models of a variety of phenomena and processes occurring in mechanics, physics, technology, biophysics, biology, ecology, medicine and other fields which arc described by nonlinear differential equations widely in science. The basis of these models in particular constitute arc parabolic type partial differential equations. In research of properties of studies and numerical solutions of the Cauchy problems and boundary value problems, approximation methods were applied. Here, the main place get degenerate equations and systems of parabolic type, which arc simulate different nonlinear processes occurring in the natural sciences.
In the independence years of our Republic research and the practical application of nonlinear models of a variety of physical, biological, and chemical processing that arc relevant areas of applied mathematics. From this point, scientific works carrying out on a number of mathematical models, which express the heat conductivity processes, filtration, biological population that have a practical application in the fields of energetic, medicine, oil and gas.
Is currently widely spread in the world of mathematical models of processes described received degenerate quasilincar parabolic equations, it is because they arc derived from the fundamental conservation laws. Therefore, it is possible when two physical processes that in common do not have seemingly anything arc described by the same nonlinear diffusion equation, only with different numerical parameters. Currently, the implementation of scientific research and practical application of these equations is one of the important problems that arc carried out in the following areas: development of methods for the study of qualitative properties of nonlinear mathematical models; finding accurate estimates of solutions in different spaces; definition of nonlinear effects; development of efficient numerical schemes; creating a set of programs for the study of mathematical models of nonlinear processes and evolution dynamics of the process in time. Scientific studies, which are conducted in all of these areas, explain the relevance of the topic of this thesis.
This dissertation research in a certain extent is the implementation of the tasks provided in the Resolution of the President of the Republic of Uzbekistan PP-1730 «On measures for further implementation and development of modem information and communication technologies», dated March 21, 2012, PP-1442 «On the priorities of industrial development of Uzbekistan in 2011-2015» dated December 15, 2010 and the Cabinet of Ministers of the Republic of Uzbekistan №24 «On measures to create conditions for further development computerizing and information communication technologies in the field» of 1 February 2012, and also in other legal instruments adopted in this area.
The aim of research work arc the numerical and analytical investigation of qualitative properties of nonlinear mathematical models describing degenerate quasilincar parabolic equations of heat conduction processes (filtration, diffusion) in homogenous and in a medium with variable density to the source and the nonlinear boundary condition, development complex programs for the numerical investigation of nonlinear boundary value problems.
Scientific novelties of the dissertation research are as follows:
the conditions of global solvability and nosolvability of solutions for nonlinear heat conduction model in a inhomogeneous medium without power with nonlocal boundary condition arc determined; 
determined the effect of heterogeneity of the medium at the conditions of global solvability and nosolvability for the whole time of the solutions of nonlinear problems;
it is found the value of the type Fujita critical exponent for the model describing the Neumann problem in the case of slow and fast diffusion;
it was found the value of the critical exponent of the global existence of the solution for the model described by the second type of boundary value problem in the case of slow and fast diffusion;
the upper and lower bounds for the generalized solutions of the problem of slow-diffiision heat conduction in homogeneous and inhomogeneous medium arc constructed;
were obtained the principal terms of the asymptotic behavior of various selfsimilar solutions of double and triple nonlinear heat conduction problem by the method of standard equations;
computational schemes have been proposed for the study of qualitative properties of nonlinear mathematical models of thermal conductivity with variable density, developed algorithms, complex programs in Visual Studio 2012 (C #) and visualized solutions of nonlinear problems.
CONCLUSION
On the basis of studies on the doctoral thesis "Mathematical modeling of the heat conduction processes in a medium with double nonlinearity" arc presented the following conclusions:
1. For nonlinear mathematical model of heat propagation, non-Newtonian polytrophic filtration, diffusion, described by nonlinear parabolic equations with nonlocal boundary condition and with variable density studied conditions for global solvability and no solvability solutions in time is established.
2. The critical exponent type Fujita and a critical exponent of solvability for nonlocal problem of heat propagation in an inhomogeneous medium arc found.
3. The upper and lower bounds of global and unbonded generalized solutions for nonlinear mathematical models of thermal conductivity with variable density and nonlocal boundary condition.
4. Established properties of finite speed of propagation of disturbances and spatial localization of solutions for nonlinear mathematical model of polytrophic filtration with double non-linearity and with variable density in the case of slow diffusion.
5. The properties of the infinite speed of propagation of disturbances of the nonlinear mathematical model for the polytrophic filtration with double nonlinearity and with variable density in the case of fast diffusion.
6. We prove the asymptotic behavior of generalized solutions with compact support of the Cauchy problem for a degenerate heat equation in an inhomogeneous medium with the source and with variable density.
7. The condition of the global solvability and no solvability solutions in time and asymptotic representation of solutions of systems of nonlinear equations for the modeling of polytrophic filtration with a nonlocal boundary condition with variable density is proved.
8. Installed above the qualitative properties of solutions and estimates solution of nonlinear problems with nonlocal boundary conditions allowed to conduct numerical calculations, giving new nonlinear effects.
9. The computing schemes, algorithms and software systems in the environment of Visual Studio 2012 (C #) for the numerical simulation of nonlinear problems of filtration and visualization arc developed.

The aim of the research work is the development and improvement of mathematical models, computational algorithms and computer models for studying the regulatory mechanisms of the interaction of skin epidermal cells.
The scientific novelty of the research work is as follows:
the biological models that describe the interrelationships of the dividing, differentiating, fulfilling specific functions of skin epidermal cells were improved;
the system of equations for the regulatory mechanisms of skin epidermal cells based on biological models was developed taking into account the spatiotemporal organization;
the mathematical model of interaction of regulatory mechanisms of skin epidermal cells was created on the basis of a system of functional-differential equations with stumbling arguments;
the computational methods for mathematical models of regulatory mechanisms of skin epidermis were developed, taking into account the time of back action;
the software for computing experiments intended for solving medical problems in the field of construction of models of the interconnected processes of regulatory mechanisms of skin epidermal cells was created.

The urgency and relevance of the theme of dissertation. Today in the world practice of the the natural sciences the development of methods of efficiency of the reaction-diffusion processes control system, study of nonlinear mathematical models is considered one of the most urgent tasks. According to the Elsevier information base the scientific works of scientists around the world devoted to the study of nonlinear reaction-diffusion equation, as the Cauchy problem and boundary-value problems to this equation and their practical applications.
In the Republic of Uzbekistan conducted extensive works on the effective organization of events dedicated to the development of automated systems for the computer visualization of diffusion processes, mathematical modeling of diffusion processes described by nonlinear equations with double nonlinearity in a heterogeneous environment. At the same time, carried out a series of research projects dedicated to the improvement of research methods and visualization of non-linear process, the creation of automated production systems, which play an important role in the study of mathematical models of nonlinear processes.
Currently, in the world a number of fundamental problems require mathematical modeling of nonlinear processes, the improvement of the methods and visualization tools, and applying to the practice of obtaining important results of the reaction-diffusion problems with double nonlinearity. At present, the study of equations with double non-linearity and practical application, conducting targeted research on the following areas is considered one of the most important tasks: the development of imaging methods in the study of nonlinear models; creating software systems that help the study of nonlinear processes; creating technology of computational experiment, monitoring the evolution over time of the process, the establishment of a computerized system of determining the properties of which depend on the parameters of the dynamics of change. Research carried out on the above areas of research, indicate the relevance of the topic of this thesis.
Research of this thesis, to a certain extent, serve to implement the objectives of all legal acts on this activity, the decree of the President on March 21, 2012 № PP-1730 «On measures for further implementation and development of modem information and communication technologies», dated December 15, 2010 № PP-1442 «On the priorities of industrial development of Uzbekistan in 2011-2015», Resolution of the Cabinet of Ministers dated February 1, 2012 № 24 «On measures to create conditions for further development of computerization and information and communication field of technology», as well as other legal documents adopted in this area.
The aim of research work is consists of mathematical modeling of reaction-diffusion, heat conductivity, luquids and gas distribution, filtration processes described degenerate parabolic equations and systems of equations with double nonlinearity with the influence of the source and absorption.
Scientific novelty of the research work is as follows:
the properties of finite speed of distribution of the disturbance and spatial localization, proof of the asymptotic behavior of blow-up solutions of reactiondiffusion systems with double nonlinearity and with variable density have been studied;
global solvability of the Cauchy problem for a model reaction-diffusion equation with double non-linearity to the source or absorption have been proven;
the estimates of the solution, and the front and for the class of systems of equations with double nonlinearity parabolic, asymptotic expressions of generalized solutions with compact support for degenerate nonlinear self-similar equations and systems have been produced;
the solution of type Zcldovich-Barcnblatt to the mutual diffusion system with double nonlinearity with a source and convective transfer have been found, the properties of finite speed of distribution of disturbations and spatial localization arc shown and an algorithm for determining the value of the critical exponent have been developed;
on based of the properties studied nonlinear mathematical models, have been built an iterative process that converges quickly;
nonlinear problem for a system or a degenerate parabolic equation with double non-linearity, take into account external factors and properties of the medium (variable density, medium conductivity, convective transfer, etc.) has numerically solved.
CONCLUSION
On the topic of the doctoral thesis «Mathematical modeling of reactiondiffusion systems with double nonlinearity» presented the following conclusions:
1. In solving problems of nonlinear models of reaction-diffusion, filtering, heat conductivity, as in homogeneous and in heterogeneous environments, based on the theoretical study by the self-analysis and the comparison principle, the analysis of the use of computational algorithms and software complexes isolated original properties and defined the further development of research.
2. The proposed methods arc used to study the properties of FSPD and localization solutions of nonlinear reaction-diffusion model with double nonlinearity for variable density environments by constraction solutions of type Zeldovich-Barcnblatt.
3. It is shown that the property FSPD and localization arc shown in moving nonlinear medium whose velocity depends on time.
4. For non-linear reaction-diffusion model in absorbing media or source found occurrence of localized wave structure.
5. Established property FSPD and spatial localization of a mathematical model of reaction-diffusion systems with double nonlinearity and with variable density.
6. It has been shown that there is a blow up property for the solutions of a system of self-rcaction-diffusion equations with double nonlinearity.
7. Built asymptotic behavior of generalized solutions with compact support and vanishing at infinity of solutions of self-similar equations or systems with double nonlinearity.
8. It is proved the global solvability of such problems for reaction-diffusion systems with double nonlinearity with the source or absorption.
9. At the critical exponent for the preparation of reaction-diffusion systems with double nonlinearity with the source or absorption and convective transfer use a universal algorithm.
10. A solution of type Zeldovich-Barcnblatt for nonlinear systems with cross property FSPD and spatial localization solutions.
11. The developed programs allows you to carry out computer simulations to study on the basis of the qualitative properties of nonlinear mathematical models of reaction-diffusion systems.
12. The developed computational schemes, algorithms and programm for solving a system of parabolic equations with double nonlinearity provide high performance in the study of the theory and process of numerical solution of such problems.

The topicality and significance of the subject of dissertation. Recently science has seen a rapid development of theory related fields, i.e. the mutual influence of two or more physical fields, in particular, a typical example of this direction of research is magneto-elasticity. Electromagnetic sensors arc in high demand at the present time in the world, according to forecasts only for the automotive market the proceeds of their sales in 2012 amounted to 812,2 million US dollars, next 2013 year they increased by 9,5%, and in subsequent two years this indicator grew by 6-7%, and the end of 2016 it is expected that the amount of revenue will reach 1,1 billion U.S. dollars1.
In Uzbekistan, held large-scale activities on the use of magneto-thin bodies in the technical designs and identify the influence of electromagnetic fields on the on deformation state of thin electro-conductive bodies. In this area, it is important to develop methods for determining the effects of electromagnetic fields on the deformation state of thin electro-conductive bodies of complex configuration, development of methods and algorithms for solving systems of differential equation in partial derivatives with initial-boundary conditions defining the magnctoelastic thin plates and shells of complex structural shapes, aimed at study the principles of creating magnctocumulativc generators for plasma confinement devices in fusion devices, magneto-hydrodynamic accelerators of contactless magnetic poles moving systems, high-quality and long-term use of measuring equipment, operating in the area of influence of electromagnetic fields.
In world practice, focuses on process modeling effects of electromagnetic fields on the deformation state of thin clcctro-conductivc bodies, development of mathematical models and numerical-analytical methods for solving partial differential equations derived from the initial and boundary conditions governing magnctoelastisity thin plates and shells of complex structural forms, using the method of R -functions formation of systems and structures of solutions satisfying the boundary conditions for the magnctoelastic plates and shells of complex configuration that is of particular interest from the scientific community. In this area, the implementation of targeted research arc priority tasks, including scientific research in the following areas: development of numerical and analytical methods and algorithms for solving systems of differential equations in partial derivatives with initial-boundary conditions, describing the influence of electromagnetic fields on the thin electro-conductive bodies (plates, shells) of complex configuration; development of complex software tools using the method of R-functions, magnctoelasticity thin bodies of complex shape, calculation algorithms of the class of problems of magnctoelasticity thin plates and shells of complex shape; conducting computational experiments to determine the degree of influence of electromagnetic fields on thin plates and shells with complex structural form, the development of algorithms for solving problems of statics and dynamics magnctoelasticity thin bodies.
In Uzbekistan, the modeling of the effects of electromagnetic fields on the state of deformation of thin clcctro-conductivc bodies, theory of magnetic elasticity for the interaction of the deformation field and the electromagnetic field in a solid clastic body is aimed at the study of the principles of creating magneto-cumulative generators, devices for plasma confinement in thermonuclear facility, magneto-hydrodynamic accelerators, contactless magnetic bearing of the moving systems, measuring equipment, working in the field of action of electromagnetic fields. In various industries technical and economic reliability from the practical application of magneto-elastic sensors is characterized by the error of their component errors of 2-3%.
The thesis is directly serve the implementation of the tasks set out in the following provisions of the President of the Republic of Uzbekistan: PP-1730 of 21 March 2012 «On measures for further implementation and development of modern information and communication technologies», the PP-1442 of 15 December 2010 «On the priorities of industrial development of Uzbekistan in 2011-2015 years», and in the decree of the Cabinet of Ministers of the Republic of Uzbekistan №64 of 7 March 2012 «On additional measures on decreasing production expenses and reduction of production cost in industry» and also in other standard legal documents accepted to the sphere.
The use of thin electro-conductive bodies in the elements of constructions of devices and machines under the influence of electromagnetic fields in modem electronic, medical and other measuring systems, as well as in communication devices, radio engineering and computer science establishes topicality of the research problems of mutual influence of electromagnetic fields and electro-conductive thin bodies having a complex configuration by R-function method (RFM).
The purpose of research is to develop algorithms and software tools of mathematical modeling of electromagnetic fields influence on the deformation state of the thin electro-conductive bodies of complex configuration using the R-functions and numerical-analytical methods.
Scientific novelty of dissertational research consists in the following:
a mathematical model describing the processes of influence of electromagnetic fields on the deformation state of thin electro-conductive bodies is built on the basis of the generalized variational Hamilton-Ostrogradsky principle with the terms of the linear theory of elasticity and Lorentz electromagnetic forces, the mathematical model of magnetic elasticity of thin plates and shells is constructed;
qualitative numerical-analytical methods and algorithms for solving systems of differential equations with initial-boundary conditions describing the effect of electromagnetic fields on clcctro-conductivc thin bodies (plates and shells) with a complex structural shape of the joint application of the variational Bubnov-Galcrkin method and the structural R-functions method is developed;
solutions structures and systems to practical boundary conditions at rigid-clamped, hinged-simply supported edge magnetoelastic plates and shells with complex configuration (with cuts) is formed;
complex software for calculation of magneto-elasticity of thin plates and shells is developed on conducted algorithms for solving problem classes of thin plates and shells magneto-elasticity with complex structural form;
the convergence of the computational algorithm concerning the number of coordinate functions of the solutions structure is shown, the practical applicability of the method and the reliability of the obtained numerical calculation results of magnetic elasticity of thin bodies by comparing with the exact solutions is proved;
the algorithms for carrying out computational experiments to study the static and dynamic effects of electromagnetic fields on the deformation state of the thin perfectly conducting bodies with complex structural form is developed.
CONCLUSION
On the basis of studies on the doctoral thesis « Mathematical modeling of processes of the electromagnetic fields’ effects on deformational condition of thin conductive bodies by the method of R-function» presented the following conclusions:
1. The fundamental geometric and physical relationships of the linear elasticity theory and linear electrodynamics arc defined taking into account properties of the structure and mechanical characteristics of the material for electro-conductive thin bodies under the influence of electromagnetic forces;
2. new mathematical models arc developed and a two-dimensional mathematical model of magnetic elasticity of thin shells and plates is built on the basis of generalized principle of Hamilton-Ostrogradsky using the Kirchhoff-Lyav hypothesis for thin bodies taking into account the linear Cauchy relations and Hooke's law of elasticity and relations of the linear theory of electrodynamics, in particular, Maxwell's equations, the influence of the electromagnetic field is determined by the volume of Lorentz pondcromotive forces but the surface and contour forces arc defined by Maxwell's electromagnetic tensor.
3. analytical and numerical methods and algorithms for solving systems of differential equations with initial-boundary conditions describing the effect of electromagnetic fields on the deformation state of the conductive thin bodies (plates and shells) complex shape with a joint application of the variational method of Bubnov-Galcrkin method and the structural R-functions method arc developed and the resolving equations (discrete model) arc obtained.
4. solution structure (sequence of coordinate functions) to the basic boundary value problem of magneto-clastic plates and shells with complex configuration area (a circle with two and four circular cutouts, polygon, rectangle with rounded comers, etc.) by the method of R-functions is formed and normalized equations for complex fields of the thin bodies, using card operations of algebraic the R-functions theory is constructed;
5. Vector-matrix equations for discrete models of magnetic elasticity of the subtle bodies, formed by the corresponding block of the matrix of damping, etc. when modeling thin-walled structures defined by systems of linear algebraic and ordinary differential equations with initial conditions and numerical methods for solving these systems of equations based on the use of quadrature sums, methods of Newmark and Gaussian elimination is developed;
6. software in the form of a complex of programs for calculation of magnetic elasticity of thin bodies by the method of R-functions on the computer, consisting of ten core modules is developed on the base of modular analysis of algorithms for solving problem classes of magnetic elasticity of thin plates and shells with complex shape;
7. numcrically-analytical methods arc developed and the validity of the obtained numerical calculation results of magnetic elasticity for thin plates of areas a classic shape (square, circle) by comparing exact and approximate solutions by the R-functions method is substantiated, moreover the plates having rigidly-clamped and hingcd-simply supported boundary conditions arc considered. The convergence of the computational algorithm of calculating the magnetic elasticity of thin shells and plates with complex structural form with regard to the number of coordinate functions of the structure of the solutions built by R-functions method and on the number of nodes (points) when calculating double integrals is studied. As the basic polynomial is selected by power polynomial, and good convergence is observed when the degree of the polynomial 3-4 (which corresponds to 10-15 coordinate functions).
8. computational experiments on the solution of problems of statics of magnetic elasticity for thin plates of complex configuration (with two circle and four circular cutouts, complex polygon shape, a ring) arc described on the basis of the developed algorithmic and software Toolkit (software package). The effect of static electromagnetic field on the deformation state of the plate with rigidly-clamped and hinged boundary conditions at a given magnetic field with different values and directions of the magnetic field is shown;
9. The dynamic effect of the electromagnetic field on the deformation state of the plate with rigidly-clamped and hinged boundary conditions on the basis of the developed algorithmic and software complex and computational experiments on problems of dynamics of magnetic elasticity of thin bodies for areas with a complex configuration by the R-functions method is studied. Plates of constant thickness, made of a material with finite electrical conductivity in an external electromagnetic field arc considered. This problem is solved in two stages: the first is the problem of electrostatics and determine the values of the magnetic field, the second solves the problem of magnetic elasticity using the values of the magnetic field. The dynamic effect of the electromagnetic field on the deformation state of the thin bodies of complex structural forms is defined.
10. The results obtained in the form of algorithmic and software tools arc implemented in the solution of specific problems of magnetic elasticity of thin shells and plates with complex configuration in the framework of the contract and economic efficiency in the amount of 127.8 million soums is obtained as a result of the implementation.

Subjects of the inquiry: the study heat and mechanical processes in electronic charge with provision for anisotropies mechanical and heat features of the electronic charges.
Aim of the inquiry: the development algorithm decisions initial-marginal problems to anisotropic theories to bounce and the numerical realization of the marginal problems with reference to calculation element radio electronics equipments (REE).
Method of inquiry: it is used new recurrence-operators method of the decision of the linear differential equations and their systems with provision for anisotropies, dissipation and heat conductivity.
The results achieved and their novelty: Constructed the new decisions of the modified equations of heat conductivity will built in two variants, equations unbound theories heat elasticity, equations to Lame, complemented member, taking into account viscosity and dissipation energy, as well as is for the first time solved problems of the fluctuation pivotal and flat design REE in new wave production.
Practical value: the got results of the analytical decision of the boundary problems allows on the one hand to value accuracy of the numerical methods of the decision of these problems, but will on the other hand get the more reliable results, for their account when designing REE.
Degree of embed and economic effectivity: the results of the work arc introduced in the training processes and can be enclosed in automated system of the provision to stability and quality of the equipment that allows in total to reduce the expenses under automatic designing REE.
Sphere of usage: the got results can be using not only when designing REE, but also in the other branch of the technology, where is researched heat and mechanical processes in anisotropic building design, in shipbuilding, machine building and others.

Subjects of research: low-frequency digital signals, and architecture of digital signal processors.
Purpose of work: To develop high-speed signal processing techniques to be submitted in the form of an algebraic polynomial on the basis of the signal spectrum, and their software implementation on modern signal processors.
Methods of research: a theory of functional analysis, spectral analysis in the Fourier bases, methods of calculating polynomials and elementary functions, theory of numbers and matrices.
The results obtained and their novelty: A method for transforming the signal in the area of polynomial representation, and finding the polynomial coefficients using the spectral approach, the algorithms and software for polynomial signal processing using digital signal processors; the qualitative characteristics of the developed algorithms was examined, A polynomial approach for calculating the biosignal parameters and to solve the problems of compression and audio signals smoothing.
Practical value: The algorithm for calculating the coefficients of algebraic polynomials, a set of digital signal processing was established applications, developed applications are protected by the Patent Office evidence of the Republic of Uzbekistan.
Degree of embed and economic effectivity: the basic theoretical and practical results of the thesis inculcated in the Institute of Physiology and Biophysics, Academy of Sciences of Uzbekistan and Institute of Microelectronics, as well as embedded in the learning process at the Department of «Computer Systems» Tashkent University of Information Technology. The total economic effect is 10 million sum in a year.
Field of application: the thesis methods, algorithms and software designed in dissertation work can be used in medicine, biology, geophysics, ecology, seismology, speech processing and audio signals.

Subject of research: multi-phase filtration of fluids in porous media.
Purpose of work: The development of computational algorithms and their basis the creation of mathematical software an automated system of calculating oil and gas fields.
Methods of research: methods of computational mathematics, mathematical modeling, development and testing of software systems.
The results achieved and their novelty: investigated mathematical models and developed computational algorithms for solving the problems of filtration of multi-phase fluids in porous media; developed software an automated system for solving the problems of filtration of multi -phase fluids in porous media; conducted computational experiments to calculate main indicators of oil and gas fields.
Practical value: proposed mathematical and special software that allows us to development and quickly carry out serial calculations to predict the main indicators of oil and gas fields.
Degree of embed and economic affectivity: The developed software is protected by certificate of evidence of the Agency for Intellectual Property of the Republic of Uzbekistan, № DGU 02001. The software tool implemented in the "Mubarak neftgaz" and showed its cost-effectiveness.
Field of application: The developed method of calculation and software tools allow us to investigate the development of the oil and gas fields, theoretical positions and research findings can be used to delivery special courses for undergraduate and master students on specialty "Mathematical and software of computers, computer systems and networks."

Geometrik masalalami yechishda vektor metodi so’ngi paytlarda yetakchi metodlardan biri hisoblanadi. Ushbu maqolada vektor metodning amaliy tadbiqlariga katta e'tibor qaratilgan.

Subject of the inquiry: semigroup operators in Banach - Kantorovich spaces and Markov processes in Banach - Kantorovich spaces E[LP].
Aim of the inquiry: The aim of the thesis is generalization of the theory of semigroup operators for Banach - Kantorovich spaces.
Methods of inquiry: In the work methods of measurable Banach bundles, of functional analysis, of the theory of Banach - Kantorovich spaces, of Markov processes are used.
The results obtained and their novelty: All obtained results of the thesis arc new and consist of the following:
- representation of the Lo -bounded semigroup of Lo -bounded LQ -linear operators in Banach - Kantorovich spaces in the form of a measurable bundle of semigroups of bounded operators;
- representation of strongly continuous semigroups of operators in Banach -Kantorovich spaces with of strongly continuous semigroups of operators of bundles is discribcd;
- representation of infinitczimial operators of the semigroup of Lo -bounded Lo -linear operators with the help of measurable bundle of semigroups of operator is given;
- description of the semigroup operators appeared in the result of Markov processes in Banach - Kantorovich spaces E[Lp] and the variants of the static and individual ergodic theorem for all.
Practical value: The work has a theoretical character.
Degree of embed and economic effectivity: The results and methods introduced in the work can be used in special courses on functional analysis, of the theory of Banach - Kantorovich spaces and the of ergodic theory.
Field of application: The theory of Banach - Kantorovich spaces, the ergodic theory.

Ishda parametrga bog‘liq chiziqli tengsizliklar sistemasining parametming parallelepipeddagi barcha qiymatlarida yechimi mavjud yoki mavjud emasligini aniqlash masalasi maksimin masalasining qiymati yoki optimal qiymati yordamida hal qilingan.

Subjects of inquiry: Algebra of measurable operators, non commutative Arens algebras, local derivations.
Aim of the inquire: Description of local derivations on algebras of measurable operators.
Methods of the inquire: In the work general methods of functional analysis, of theory operator algebras are used.
The results achieved and their novelty: a description of local derivations on the non commutative Arens algebras associated with von Neumann algebra and faithful normal semi-finite trace is obtained; it is proved that every tT-continuous linear operator Д on the algebra 5(Л/,т) satisfying the identity A(p) = A(p)/? + /?A(p) is a derivation, where AY be a von Neumann algebra with a faithful normal semi-finite trace r; it is proved that every linear operator D: 4(X) —> B(X) satisfying the identity Z)(x") = У^хА 'D(x)x"~*, хбЛ(Т) *=i
is a spatial derivation, where n > 3 - some fix number; necessary and sufficient conditions for the existence of local derivations which are not derivations on algebras S(M) and S(M,r) affiliated with a commutative von Neumann algebra are obtained; a description of local derivations of the algebras LS(M), S(M) and S(M,r) concerning type I von Neumann algebras without abelian direct summands is obtained.
Practical value: The results of the dissertation have a theoretical character.
Degree of embed and economic effectivity: The results, presented in the work can be used in special courses on functional analysis and theory of operator algebras for masters and post-graduate students.
Field of application: Functional analysis, theory of operator algebras, mathematical physics and its applications.

Subjects of research: local and non-local boundary-value problems for parabolic-hyperbolic equations with three lines of type changing.
Purpose of work: formulation of local and non-local boundary-value problems for parabolic-hyperbolic equation with three lines of type changing and investigation for the existence and uniqueness of solution of formulated problems.
Methods of research: methods of integral equations and energy integrals are used.
The results obtained and their novelty: local and non-local boundary problems for parabolic-hyperbolic equations with three lines of type changing are formulated and the existence, the uniqueness of solution for formulated problems is proved.
Practical value: the results of the dissertation work have a theoretical character.
Degree of embed and economic effectiveness: on the base of achieved results, the special course for the master- students can be taught and may be used in the subsequent theoretical development of this field.
Field of application: results of the dissertation work can de used at future development of the theory of partial differential equations and also at studying mathematical questions of problems of physics, mechanics and biology.

The urgency and relevance of the dissertation topic. Many scientific and applied studies conducted at the world level, in many cases, reduce to the study of ill-posed boundary-value problems for partial differential equations. The basis of the theory of ill-posed problems laid in the middle of the last century and they are associated with problems of great practical importance. The main object of the theory of inverse and ill-posed problems is the model of applied research in the field of geophysical observation, gas dynamics, the propagation of acoustic waves, etc. Since the study of ill-posed problems for mixed-composite equations on conditional correctness and the construction of an approximate solution, it is insufficient to develop a study of ill-posed problems for such equations is an actual problem.
The aim of the research work is to establish conditionally correctness and finding approximate solutions on the set of correctness of ill-posed problems for high order partial differential equations of mixed, composite, mixed-composite types.
The tasks of research work:
- investigation of ill-posed boundary value problems for high orders partial differential equations of composite and mixed-composite types;
-studies of ill-posed boundary value problems for high orders partial differential equations of mixed type;
-finding regular solutions, determining estimates of the norms of the difference between exact and approximate solutions in the corresponding function spaces;
-finding formulas for calculating regularization parameters, implementing numerical solutions and graphical results.
The object of the research work is the high order partial differential equation of mixed, composite and mixed-composite types.
Scientific novelty of the research work is as follows:
- a priori estimates of the solution of ill-posed boundary-value problems for partial differential equations of mixed, composite and mixed-composite types of high orders are obtained;
- sets of correctness defined for ill-posed boundary-value problems for partial differential equations of mixed, composite and mixed-composite types of high orders, and uniqueness and conditional stability theorems are proved;
-approximate solutions are constructed, estimates of the norms of the difference between the exact and approximate solutions in the corresponding spaces are determined, formulas for calculating the regularization parameters are derived;
- implemented programs in a visual c # environment that outputs numerical and graphical results of an exact and approximate solution based on computational algorithms.

In the given article author describes the ordinary differentiated nonlinear super singular point where presented integrals through two arbitrary constants and
researched the problem Koshi types

Subjects of the inquiry: nonlinear evolution equations with self-consistent source.
Aim of the inquiry: To deduce the scattering data of spectral problem connected with nonlinear evolution equations with self-consistent source.
Method of the inquiry: used research methods include the methods of mathematical physics, theory differential equations, the theory of functions of complex variables, spectral theory of differential and difference operators.
I hc results achieved and their novelty: The main results of this work are new and consist of the following:
1) the law of changing on t of spectral data of Stunn-Liouville operator with potential which is the solution of general Korteweg - de Vries equation with source in the class of ‘rapidly decreasing’ functions is deduced;
2) the evolutions of scattering data of Sturm-Liouville operator with potential which is the solution of general Korteweg - de Vries equation in the class of “steplike” functions is defined;
3) the integration of general Korteweg - de Vries equation with source from “steplike” initial data is studied;
4) the inverse scattering method is used to solve various nonlinear evolution equations with self -consistent source in the case the simple eigenvalues of corresponding spectral problems;
5) it is shown that the inverse scattering method may be used for integration of sin-Gordon equation with self-consistent source in the case the multiple eigenvalues of Dirac’s operator;
6) the solution of Toda lattice with self-consistent source is expressed in terms of the inverse scattering method for the discrete Sturm-Liouville operator.
Practical value: the work has a theoretical character.
Degree of embed and economic cffcctivity: a special course will be read for post-graduate students on the basis of the received results.
Sphere of usage: the obtained results may be used in mathematical physics for integration of nonlinear evolution equations.

Maqolada tabiatdagi jarayonlar: issiqlik tarqalishi, sterjen tebranishi, tor tebranishi, magnit maydon impulsi, mayatnik tebranishi va hakazolar. Tabiiy protsess o’lchangani uchun uni modellashtirish natijasida hosil bo’lgan masalalar taqribiy yechiladi. Bu erda yechim protsessni kuzatish (bir nuqtada o’lchash) bilan hisoblashga harakat qildik. Bunda masala uchun qo’shma operator tuzilib, nokorrekt masala (boshlang’ich qiymati yo’q holat) shartli korrekt xolatga o'tkaziladi.

Subject of inquiry: Gibbs measures for q-eomponenl model and Potts model with competing interactions on a Cayley tree.
Aim of the inquiry: We study Gibbs measures and periodic ground states of the Potts and q-componcnt models with competing interactions on a Cayley tree.
Methods of the inquiry: Methods of contours on a Cayley tree, methods of Pirogov-Sinay theory, measure theory and contractive maps.
The results achieved and their novelty: The obtained results arc new. They consist of the following:
• For q-componcnt models on a Cayley tree contours and ground states arc constructed.
• For q-componcnt models, at sufficiently low temperatures, by a contour method on a Cayley tree existence of at least q different Gibbs measures is proved.
• For a Potts model with competing interactions on a Cayley tree the set of periodic ground states is constructed.
• It is shown that the Pcicrls’s condition is satisfied for the Hamiltonian of the Potts model.
• At sufficiently low temperatures, for the Potts model with competing interactions and three spins existence of at least three Gibbs measures is proved.
• On parameters of a model with the interaction radius two a sufficient conditions arc found under which the periodic configurations arc the ground states of this model.
Practical value: the results of the dissertation work have theoretical character. They can be applied in problems of statistical physics.
Sphere of usage: results of the work can be used in measure theory, theory of phase transitions, theory of probability, theoretical and mathematical physcs.

The aim of the research work is study location of essential spectrum and a number of eigenvalues out of essential spectrum of certain generalized Friedrichs model, corresponding to a system of no more than two particles on lattice.
Scientific novelty of the research work is as follows:
It is found location of essential spectrum of certain generalized Friedrichs model, corresponding to a system of no more than two particles on lattice, interacting via creation and annihilation operators and two particle Schrocdinger operator, interacting via contact potentials;
It is shown existence at least one eigenvalue out of essential spectrum of certain generalized Friedrichs model, interacting via two particle Schrodinger operator, corresponding to a system of no more than two particles on one and two dimensional lattice and interacting via contact potentials;
It is shown the existence of eigenvalue out of essential spectrum of certain generalized Friedrichs model, corresponding to a system of no more than two particles on a lattice, represented by two particle Schrocdinger operator, interacting via contact potentials on lattice with dimension no less than three, or its absence depending on parameters of the operator;
It is proved the existence at least one eigenvalue lying below the essential spectrum of certain generalized Friedrichs model, corresponding to a system of no more than two particles on one dimensional lattice, interacting via creation and annihilation operators and two particle Schrodinger operator, interacting via contact potentials, and established that existence or absence second eigenvalue depends on parameters of the operator.

The aim of the research work is to study the existence of eigenvalues and also to obtain a convergent expansion for the eigenvalue lying outside the essential spectrum of the Schrodinger operator corresponding to system of two identical particles (fermions or bosons) interacting via pairwise short-range potentials on one and two dimensional lattices.
Scientific novelty of the research work is as follows:
the existence of eigenvalue and explicit form of the corresponding eigenfunction of the two-particle Schrodinger operator associated to a system of two identical particles(bosons) interacting via contact attractive or repulsive potential on one and two dimensional lattices is established;
a convergent expansion for eigenvalue at the coupling constant threshold of the two-particle Schrodinger operator associated to a system of two identical particles (bosons) interacting via contact attractive or repulsive potential on one and two dimensional lattices;
it is shown that the left threshold of the essential spectrum may be a virtual level (resonance) or an eigenvalue for the two-particle Schrodinger operator associated to a system of two particles (fermions) interacting at neighboring sites on one-dimensional lattice and the existence or absence of eigenvalue lying to the left of the essential spectrum is proved;
a convergent expansion at the coupling constant threshold and quasi-momentum threshold for the eigenvalue of the two-particle Schrodinger operator associated to the two-particle system (fermions) with pair interactions at neighboring sites on a one-dimensional lattice is found.

Actuality and demand of the theme of dissertation. Numerous scientific and applied research conducted on a global level show that everywhere in physics stable complex objects arc usually formed as a result of action of attractive forces that allow the component parts to reduce the energy in their binding. However, recent years scientists have proved that in the ordered medium stable complex objects can exist even in the case of repulsive interactions. Bose-Hubbard model is used to describe the repulsive pairs, i.e. Schrodinger operator on a lattice is the theoretical basis of experimental observations and theoretical basis for the application. Therefore, the development of research of Schrodinger operators corresponding Hamiltonians of the systems of particles on a lattice, which arc found in models of solid state physics and lattice field theory is one of the priorities.
In our country in the years of independence much attention has been paid to directions of applied importance, in particular, special attention was paid to the study of Schrodinger operators corresponding to the system of particles on an integer lattice. For the Schrodinger operators significant results were achieved in determining the conditions for the existence of bound states which is located outside of the essential spectrum and for their number.
Since the spectrum of the Schrodinger operators associated to the systems of two quantum particles on lattice is quite sensitive to changes in the quasi-momentum of the system, solving problems related to studies of the spectrum of the operator and to show the existence of bound states as well as determine their number plays an important role . In this regard, the implementation of targeted investigation in the following areas is one of the most important problems: investigate the discrete spectrum of the Schrodinger operator corresponding to a system of two arbitrary particles with short-range pair potentials on lattice, to establish the threshold phenomenon below the bottom or above the top essential spectrum for the operator. Research carried out in the aforementioned areas confirms the actuality of the dissertation topic.
This dissertation, to some extent, serves the tasks specified in the Decrees of the President of the Republic of Uzbekistan № DP-436 dated August 7, 2006 "On Measures for Improving the Coordination and Management of the Development of Science and Technology" and № DP-916 dated July 15, 2008 "Encouraging the introduction of innovative projects and technologies in production ", № DP -2789 dated February 17, 2017 "On measures to further improve the organization, management and financing of research activities and activities of the Academy of Sciences " and № DP -4947 dated February 8, 2017 "On strategy actions for the further development of the Republic of Uzbekistan ", as well as in other normative-legal acts on this activity.
The aim of the research is to study location of essential and discrete spectra as well as the number of eigenvalues of the Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact potential on lattice.
The scientific novelty of the research is as follows:
it is proven the existence of eigenvalue located to the right of essential spectrum of discrete Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact repulsive (//>0) potential on d>3 dimensional lattice and it is proven regularity of corresponding eigenstate finding its exact form;
it is proven that the discrete Schrodinger operator has virtual level at the right edge of essential spectrum if the dimension is J = 3,4 and it is established that the virtual state is integrable;
it is defined for the fixed value of quasi-momentum the values of coupling constant which the operator has virtual level and for the fixed value of coupling constant it is separated the set of quasimomentum which the operator has eigenvalue or has not eigenvalue or has a virtual level;
it is shown that the right edge of essential spectrum is eigenvalue of the Schrodinger operator if the dimension is d > 5;
it is proven the existence of eigenvalue located to the left of essential spectrum of discrete Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact attractive potential on d>3 dimensional lattice and it is shown regularity of corresponding eigenstate finding its exact form
it is proven that the discrete Schrodinger operator has virtual level at the left edge of essential spectrum if the dimension is d = 3,4 and it is established that the virtual state is integrable;
it is shown that the left edge of essential spectrum is eigenvalue of the Schrodinger operator if the dimension is d > 5.
Conclusion
The dissertation is devoted to study essential and discrete spectra of the two particle Schrodinger operator corresponding to system of two arbitrary particles on lattice interacting via a pair contact potential.
The main results of the research arc as follows:
1. It is proven the existence of eigenvalue above the top of essential spectrum of discrete Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact repulsive (/z > 0) potential on d > 3 dimensional lattice and it is shown the corresponding eigenstate in coordinate representation exponentially decreases at infinity;
2. It is proven that the discrete Schrodinger operator has virtual level at the right edge of essential spectrum if the dimension is J = 3,4 and it is established that the virtual state approaches to zero at infinity;
3. It is defined for the fixed value of quasi-momentum the set of the values of coupling constant which the operator has virtual level and for the fixed value of coupling constant the set of quasimomentum;
4. It is shown that the right edge of essential spectrum is eigenvalue of the Schrodinger operator if the dimension is d > 5;
5. It is proven the existence of eigenvalue below the bottom of essential spectrum of discrete Schrodinger operator associated to a system of two arbitrary particles interacting via a pair contact attractive (// > 0) potential on 4>3 dimensional lattice and it is shown that the corresponding eigenstate in coordinate representation exponentially decreases at infinity;
6. It is proven that the discrete Schrodinger operator has virtual level at the left edge of essential spectrum if the dimension is <7 = 3,4 and it is established that the virtual state approaches to zero at infinity;
7. It is shown that the left edge of essential spectrum is eigenvalue of the Schrodinger operator if the dimension is d > 5;