Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математика. Механика. Физика
Все статьи: 733
Non-classical equations of mathematical physics. Linear Sobolev type equations of higher order
Статья научная
The article presents the review of authors' results in the field of non-classical equations of mathematical physics. The theory of Sobolev-type equations of higher order is introduced. The idea is based on generalization of degenerate operator semigroups theory in case of the following equations: decomposition of spaces, splitting of operators' actions, the construction of propagators and phase spaces for a homogeneous equation, as well as the set of valid initial values for the inhomogeneous equation. The author uses a proven phase space technology for solving Sobolev type equations consisting of reduction of a singular equation to a regular one defined on some subspace of initial space. However, unlike the first order equations, there is an extra condition that guarantees the existence of the phase space. There are some examples where the initial conditions should match together if the extra condition can't be fulfilled to solve the Cauchy problem. The reduction of nonclassical equations of mathematical physics to the initial problems for abstract Sobolev type equations of high order is conducted and justified.
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On a Q-boundary value problem with discontinuity conditions
Статья научная
In this paper, we studied q-analogue of Sturm-Liouville boundary value problem on a finite interval having a discontinuity in an interior point. We proved that the q-Sturm-Liouville problem is self-adjoint in a modified Hilbert space. We investigated spectral properties of the eigenvalues and the eigenfunctions of q-Sturm-Liouville boundary value problem. We shown that eigenfunctions of q-Sturm-Liouville boundary value problem are in the form of a complete system. Finally, we proved a sampling theorem for integral transforms whose kernels are basic functions and the integral is of Jackson’s type.
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On basis property of root functions for a class of the second order differential operators
Статья научная
It is well known that the Sturmian theory is an important tool in solving numerous problems of mathematical physics. Usually, eigenvalue parameter appears linearly only in the differential equation of the classic Sturm-Liouville problems. However, in mathematical physics there are also problems, which contain eigenvalue parameter not only in differential equation, but also in the boundary conditions. In this paper, we consider a Sturm-Liouville equation with the eigenparameter dependent boundary condition and with transmission conditions at two points of discontinuity. The aim of this paper is to investigate the completeness, minimality and basis properties of rootfunctions for the considered boundary value problem.
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On determination of minor coefficient in a parabolic equation of the second order
Статья научная
An inverse problem of recovering the minor time-dependent coefficient in a parabolic equation of the second order is considered. The unknown coefficient is the controlling parameter. The inverse problem lies in finding the solution of an initial-boundary value problem for this parabolic equation and this time-dependent coefficient using data of the initial-boundary value problem and point conditions of overdetermination. Cases of the Dirichlet boundary conditions and oblique derivative conditions are considered. Conditions under which the theorem of existence and solution uniqueness is applicable for the given inverse problem is described; the numerical solution method is described, and its justification is given. All the considerations are carried out in Sobolev spaces. Solution of the direct problem is based on the finite element method and the finite difference method. The proposed algorithm for the numerical solution consists of three stages: initialization of the massive that describes geometry of the area and the boundary vector; implementation of integrative calculation of the desired coefficient using the finite element method; implementation of the finite difference method. Results of numerical experiments are presented, and numerical solution of the model inverse problem is constructed in the case of Neumann boundary conditions; dependency of an error in calculation of the controlling parameter on the variation of the equation coefficients and the noise level of the overdetermination data for domains with different number of nodes that depend on an observation point is described. Results of the calculations show a good convergence of the method. In the case when introduced noise level is 10 %, the error between the desired and the obtained solution increases from 8 to 35 times, though the graph of recovered coefficient remains close to the solution graph and repeats its outlines.
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On one equation of internal waves
Статья научная
The Cauchy-Dirichlet problem is considered for the equation of internal waves. This equation has various applications in hydrodynamics, for example, in the study of waves in the ocean. The article provides an analytical study of one equation of internal waves. This equation characterizes propagation of waves in a homogeneous incompressible stratified fluid. The equation of internal waves is reduced to an abstract semilinear Sobolev type equation of the second order. The study of the equation is carried out within the framework of the theory of polynomially bounded operator pencils. In this work, we construct propagators for the equation of internal waves. Also, we present two model examples, where the domain D is represented in the form of a cylinder and a parallelepiped. The result of the work is an analytical solution to the considered cases for the equation of internal waves.
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On some classes of inverse parabolic problems of recovering the thermophysical parameters
Статья научная
In the article we examine the question of regular solvability in Sobolev spaces of parabolic inverse coefficient problems. A solution is sought in the class of regular solutions that has all derivatives occurring in the equation summable to some power. The overdetermination conditions are the values of a solution at some collection of points lying inside the domain. The proof is based on a priori estimates and the fixed point theorem.
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Статья научная
In this paper, we consider conformable equal width wave (EW) equation in order to construct its exact solutions. This equation plays an important role in physics and gives an interesting model to define change waves with weak nonlinearity. The aim of this paper is to present new exact solutions to conformable EW equation. For this purpose, we use an effective method called Improved Bernoulli Sub-Equation Function Method (IBSEFM). Based on the values of the solutions, the 2D and 3D graphs and contour surfaces are plotted with the aid of mathematics software. The obtained results confirm that IBSEFM is a powerful mathematical tool to solve nonlinear conformable partial equations arising in mathematical physics.
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Статья научная
We consider the problem of determining point sources for mathematical models of heat and mass transfer. The values of a solution (concentrations) at some points lying inside the domain are taken as overdetermination conditions. A second-order parabolic equation is considered, on the right side of which there is a linear combination of the Dirac delta functions δ(x-xi) with coefficients that depend on time and characterize the intensities of sources. Several different problems are considered, including the problem of determining the intensities of sources if their locations are given. In this case, we present the theorem of uniqueness of solutions, the proof of which is based on the Phragmén-Lindelöf theorem. Next, in the model case, we consider the problem of simultaneous determining the intensities of sources and their locations. The conditions on the number of measurements (the ovedetermination conditions) are described which ensure that a solution is uniquely determined. Examples are given to show the accuracy of the results. This problem arises when solving environmental problems, first of all, the problems of determining the sources of pollution in a water basin or atmosphere. The results are important when developing numerical algorithms for solving the problem. In the literature, such problems are solved numerically by reducing the problem to an optimal control problem and minimizing the corresponding objective functional. The examples show that this method is not always correct since the objective functional can have a significant number of minima.
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Optimal control of solutions to the initial-final problem for the model of linear waves in a plasma
Статья научная
The optimal control problem for a Sobolev type equation of higher order with a relatively polynomially bounded operator pencil is investigated in the paper. The results are applied to the study of the optimal control of solutions to the initial-final problem for the model of linear waves in plasma. The first results on the investigation of equation that describes the linear ion-acoustic waves in an unmagnetized plasma and on the study of some properties of these waves were obtained by Yu.D. Pletner. The initial-final conditions posed for the fourth-order Sobolev type equation are the generalization of the conditions in the Cauchy problem that is unsolvable at the arbitrary initial values. The research is based on the phase space method developed by G.A. Sviridiuk and the theory of relatively polynomially bounded operator pencil developed by A.A. Zamyshlyaeva. The article considers an equation that describes ion-acoustic waves in a plasma in an external magnetic field.
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Optimal control over solutions of a multicomponent model of reaction-diffusion in a tubular reactor
Статья научная
This article studies a mathematical model of reaction-diffusion in a tubular reactor based on degenerate equations of reaction-diffusion type defined on a geometric graph. It is precisely the degenerate case that is studied, since when building the mathematical model it is taken into account that the speed of one sought function is significantly higher than the speed of the other. This model belongs to a wide class of semilinear Sobolev-type equations. We give sufficient conditions for the simplicity of the phase manifold of the abstract Sobolev-type equation in the case of s-monotone and p-coercive operator; we prove the existence and uniqueness of a solution to the Showalter-Sidorov problem in the weak generalized sense, and the existence of optimal control over weak generalized solutions to this problem. On the basis of the abstract theory, we find sufficient conditions for the existence of optimal control for a mathematical model of neural signal transmission.
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Population kinetics of localized states in nanocomposite materials under exposure to radiation
Статья научная
We simulated the population of localized states in nanocomposite materials using Rouse-Fowler model. The following radiation effects were considered: prolonged irradiation (over 3 s) with the low absorbed dose rate (0,002 W/kg) and pulsed irradiation (100 ns) with the high absorbed dose rate (over 105 W/kg) of ionizing radiation. We investigated the role of localized states in electrical conductive properties of nanocomposite materials on the example of nanocomposite materials with hole conductivity (polymathimethacrilate (PMMA) + CdS) and electron conductivity (α-Al2O3+SrO), as well as in pure PMMA and α-Al2O3. Our results indicate that the small traps influence the speed of relaxation to the equilibrium radiation induced electrical conductivity, while the deep traps, the depth of which is much greater than kT, have an impact on the sensitivity to an absorbed dose of ionizing radiation. Moreover, pure PMMA and nanocomposite materials based on it are unsuitable for dosimetry due to a large share of the small traps in the spectrum of intrinsic localized states. On the contrary, aluminum oxide is an almost perfect material for the accumulation of the information about the ionizing radiation, since its spectrum of localized states includes only deep traps. On the whole, the most interesting materials from the dosimetry viewpoint are nanocomposites based on aluminum oxide, where the concentration of impurity centers does not exceed the concentration of intrinsic states, and the nanoparticle radius is no more than 2 nm in case of small share of the small traps in impurity spectrum.
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Promising quantum engineering of optical even/odd Schrodinger cat states
Статья научная
We propose an efficient way to implement new family of continuous variable (CV) states of definite parity. Measurement induced CV states of definite parity states are realized after subtraction of an arbitrary number of photons from the initial single-mode squeezed vacuum (SMSV) state using a photon number resolving (PNR) detector. Optical design requires irreducible number of optical elements for implementation of the CV states of definite parity. The potential of using the CV states in optical quantum information processing can be high. As an example, we show the possibility of using a family of the CV states of definite parity for quantum engineering of optical even/odd Schrödinger cat states (SCSs). In particular, we report the possibility of implementing the CV states of definite parity that approximate even/odd SCSs of amplitude slightly greater than 4 with fidelity prevailing 0,99 after subtraction of 50,51 photons from original SMSV. The success probability being the third key parameter of the optical design, decreases with an increase in the number of photons, but generally remains at an acceptable level for further use in quantum information processing in the case of a small number of subtracted photons.
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Recovering of lower order coefficients in forward-backward parabolic equations
Статья научная
We study the issue of recovering a lower order coefficient depending on spatial variables in a forward-backward parabolic equation of the second order. The overdetermination condition is an analog of the final overdetermination condition. A solution at the initial and final moments of time is given. Equations of this type often appear in mathematical physics, for example, in fluid dynamics, in transport theory, geometry, population dynamics, and some other fields. Conditions on the data are reduced to smoothness assumptions and some inequalities for the norms of the data. So it is possible to say that the obtained results are local in a certain way. Under some condition on the data, we prove that the problem is solvable. Uniqueness of the theorem is also described. The arguments rely on the generalized maximum principle and the solvability of theorems of the periodic direct problem. The results generalize the previous knowledge about the multidimensional case. The used function spaces are the Sobolev spaces.
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Self-stabilizing mobile medical robots scattering algorithm
Статья научная
The paper is devoted to the possibility of determinate and probabilistic scattering under various assumptions of the state of the locations of medical robots in both fault-tolerance and vulnerable environments. The topicality of the work is due to the need to place medical robots in the coordinate space having disjoint polygons (robot bodies) which is absolutely unacceptable in the case of medical applications. As a limitation it is assumed that the medical robot sees its nearest neighbors and local monitor of multiplicity is functioning, which can determine the situation when robots occupy intersecting spaces. We propose a probabilistic scattering algorithm which describes the initial states of medical robots and the proper transient state algorithm which can predict the movement of robots to a location where they can intersect. It is shown, that when using the algorithm the states and motion algorithms can be estimated in a fault-tolerance (robots do not fail and the medium is stationary) and vulnerable (the robot may fail and the byzantine problem is not solved, the environment changes faster than the robot can react) environments. The estimates for the computational complexity of the algorithm working without the mission planner are given.
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Sobolev type mathematical models with relatively positive operators in the sequence spaces
Статья научная
In the sequence spaces which are analogues of Sobolev function spaces we consider mathematical model whose prototypes are Barenblatt - Zheltov - Kochina equation and Hoff equation. One should mention that these equations are degenerate equations or Sobolev type equations. Nonexistence and nonuniqueness of the solutions is the peculiar feature of such equations. Therefore, to find the conditions for positive solution of the equations is a topical research direction. The paper highlights the conditions sufficient for positive solutions in the given mathematical model. The foundation of our research is the theory of the positive semigroups of operators and the theory of degenerate holomorphic groups of operators. As a result of merging of these theories a new theory of degenerate positive holomorphic groups of operators has been obtained. The authors believe that the results of a new theory will find their application in economic and engineering problems.
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Some Sobolev spaces as Pontryagin spaces
Статья научная
We show that well known Sobolev spaces can quite naturally be treated as Pontryagin spaces. This point of view gives a possibility to obtain new properties for some traditional objects such as simplest differential operators.
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Structural and elastic properties of Fe-Ge alloys: ab initio studies
Статья научная
In this paper, with the help of the density functional theory, the structural and elastic properties of A2, B2, D03, and L12 phases of Fe100-xGex alloys (12,5 ≤ x ≤ 28,125 at. %) have been studied. The electronic and full ionic relaxations were used for the investigation of crystal structures. The concentration dependencies of the atomic volumes, structural phase transition temperatures, tetragonal and rhombohedral shear moduli have been calculated. We show that the atomic volume curves correlate with the sequence of phase transitions observed experimentally: A2→B2→D03 (x ≤ 22 at. % of Ge content). The structural phase transition temperatures increase with the Ge concentration. The calculated tetragonal moduli for the D03, A2, and L12 structures decrease with the increasing of the Ge content, what agrees with the experimental results. The dependence of rhombohedral shear moduli as a function of Ge concentration does not change significantly with increasing Ge atoms. The C44 is increased for the D03 phase, while for A2, B2, and L12, it decreases.
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Статья научная
In this article, we present the result of the first step of tabulation of prime links in the thickened surface of genus 2 that admit diagrams with no more than 4 crossings. Namely, we describe all three steps of tabulation of prime link projections in the surface of genus 2 with no more than 4 crossings. First, we define primality of a link projection in the surface of genus 2. Second, we tabulate prime link projections in the surface of genus 2 with no more than 4 crossings. For this purpose, it is sufficient to consider graphs having special type and enumerate all possible embeddings of the graphs into the surface of genus 2 giving prime link projections. At this step, we prove some auxiliary statements to simplify enumeration of the embeddings. Finally, we show that all obtained projections are nonequivalent in the sense of homeomorphism of the surface of genus 2 onto itself. Our main result states that there exist exactly 15 pairwise nonequivalent prime link projections in the surface of genus 2 with no more than 4 crossings. Several new and known tricks allow rigorously theoretically prove the completeness of the obtained tabulation, as well as to keep the process within reasonable limits. Further, we intend to use the obtained table to classify prime diagrams, i.e. to obtain table of prime links.
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Статья научная
The article is devoted to the study of the unique solvability of the Barenblatt-Zheltov-Kochina equation, equipped with the Neumann boundary condition and a multipoint initial-final value condition. This equation is degenerate or, in other words, it belongs to the Sobolev type equations. To study this equation, the authors used the methods of the theory of degenerate operator semigroups, created by Prof. G.A. Sviridyuk, and further developed by him and his students. We would also like to note that the equation under study is supplied with a multipoint initial-final value condition, which is not just a generalization of the Cauchy problem for the Sobolev type equations. This condition makes it possible to avoid checking the consistency of the initial data when finding a solution.
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The analysis and processing of information for one stochastic system of the Sobolev type
Статья научная
This article is devoted to the analysis and processing of information for a stochastic model based on the equation of potential distribution in a crystalline semiconductor with the Nelson-Glicklich derivative and the Showalter-Sidorov initial condition. By semiconductors, we mean substances that have a finite electrical conductivity that rapidly increases with increasing temperature. It is assumed that the initial experimental data may be affected by random noise, which leads to the study of the stochastic model. An analysis of the stochastic model of the potential distribution in a crystalline semiconductor is given. Conditions under which there are step-by-step solutions of the model under study with the Showalter-Sidorov initial condition are found. Further, on the basis of the theoretical results, an algorithm for the numerical analysis of the system is given. Its implementation is presented in the form of a computational experiment, which is necessary for the further processing of information.
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