Дэлгэрэнгүй мэдээлэл

In frame of De Launay model, the expression for elements of dynamical matrix of graphene have been deduced with accounting interatomic forces on the first six neighbor atoms. The calculation of phonon dispersion of graphene in ГK and ГM directions is done via known values of radial and tangential force constants on the first five neighbor atoms. Calculated phonon dispersion is in satisfactory agreement with experimental phonon spectra of graphite measured by another authors. In long wave approximation, the equations for estimation of elastic constants of graphene have been obtained from the expressions of elements of dynamical matrix.

In frame of De Launay model we calculated phonon dispersion of graphene in ГM direction using radial and tangential force constants for first four neighbor atoms. Calculated phonon dispersion is in satisfactory agreement with experimental phonon spectra of graphite. In long wave approximation we estimated elastic modulus of graphene in comparison with experimental results for graphite and carbon nanotube.

We study a class of time fractional diffusion-wave equations with variable coefficients using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of equations. Group invariant solutions are given explicitly corresponding to every element in an optimal system of Lie algebras generated by infinitesimal symmetries of equations in the class. We express the solutions in terms of Mittag-Leffler functions, generalized Wright functions, and Fox H-functions. These solutions contain previously known solutions as particular cases.

In this study, we present discretization schemes based on Generalized Integral Representation Method (GIRM) for numerical simulation of the Boussinesq wave. The schemes numerically evaluate the coupled Boussinesq equation for different solitary wave phenomena, namely, propagation of a single soliton, head-on collision of two solitons and reflection of a soliton at a fixed wall boundary. In these three soliton interactions, we utilize different Generalized Fundamental Solutions (GFS) along with piecewise constant approximations for the unknown functions. For the case of soliton reflection at a wall, time evolution in GIRM is coupled with the Green’s function in order to cope with the complicated boundary conditions that arise from the GIRM derivation. We conduct numerical experiments and obtain satisfactory approximate result for each case of the soliton interactions.

Here we present a simple numerical scheme for simulation of the coupled Boussinesq equation. The scheme is based on Generalized Integral Representation Method and numerically simulate the Boussinesq wave for reflection of a solitary wave at a vertical wall fixed on the boundary. In this particular wave interaction, we utilize two different Generalized Fundamental Solutions (GFS) in our scheme, namely, common Gaussian GFS and harmonic GFS, for comparison purpose. The unknown functions in the Boussinesq equation are expressed via piecewise constant approximations. We emphasize that time evolution in the scheme, is coupled with the Green’s function in order to cope with the complicated boundary conditions that arise during derivation of the numerical scheme.

In this work8 the diffuse scattering on polycrystalline Mg-10at.%In was measured using X-ray diffactometer. From the intensity of diffuse scattering we defined short-range order parameters on the first eight shells in Mg-10at.%In with accounting static displacements.

There is a recurrence relation for calculating an integral of associated Legendre function Pml. Here we try to find a direct formula for the integral by using generating function and we successfully developed a formula for m = l case.

The Boussinesq wave equation is a simpli ed model used to describe the propagation of long waves in shallow water. This equation combines the effects of dispersion and nonlinear wave interactions. Solving the Boussinesq equation can be challenging, and analytical solutions are often not feasible. Various numerical methods, such as  nite di erence,  nite element, or spectral methods, are commonly employed to approximate the solution. On the other hand, Generalized Integral Representation Method (GIRM) is a mathematical technique used for solving differential equations, integral equations, and partial di erential equations. The GIRM technique involves expressing the solution of a given problem as a weighted integral of a known function. This integral representation can then be used to obtain the solution for any point within the domain of the problem. In this study, we discuss our discretization schemes based on GIRM for numerical study of the 1D Boussinesq wave in case of soliton reflection at a fixed boundary.

Энд бид тухайн уламжлалт дифференциал тэгшитгэлийн шийдийн онолын талаар өөрсдийн хүрээнд судалсан судалгааныхаа тоймыг танилцуулж байна. Tухайн уламжлалт дифференциал тэгшитгэлийн судалгаанд аналитик шийд, түүнийг олох аргуудаас (хувьсагчийг ялгах арга, интеграл хувиргалтын аргууд, Гриний функцийн арга, конформ буулгалтын арга гэх мэт) гадна сул,“mild” болон тоон шийд, харгалзах онол, арга (вариац томъёолол, Semigroup, операторын онол, төгсгөлөг ялгаврын арга, төгсгөлөг элементийн арга, төгсгөлөг эзэлхүүний арга) техниктэй холбоотой судалгаанууд ихээхэн байр суурь эзэлдэг. Тухайлбал, тухайн уламжлалт дифференциал тэгшитгэлийн аналитик (тасралтгүй дифференциалчлагддаг) шийд оршин байх нь мужийн хүрээ нь хангалттай гөлгөр эсэх, бодлогын өгөгдлүүдийн гөлгөр чанар, захын нөхцөлүүдийн төрөл өөрчлөгдөх цэгүүдээс хамаардаг. Эдгээр шаадлагуудаас биелэгдэхгүй үед тасралтгүй биш функц тухайн уламжлалт дифференциал тэгшитгэлийн шийд болох ба түүнийг өргөтгөсөн эсвэл сул шийд гэж тайлбарладаг. Иймд шийдийн тухай ойлголтуудыг тодорхойлох, аналитик болон сул шийдийн ялгааг тодруулах зайлшгүй шаардлагатай болсон байдаг.

МУИС-д 1960-аад оны сүүлчээр Тооцон бодох математикийн анхны лекцүүдийг Киевийн их сургуулийн профессорууд уншиж байсан тэр цаг үеэс эхэлсэн математикийн энэ салбар Moнгол улсын ууган шавь сургуулиудын нэг болон эрчимтэй хөгжиж иржээ. Ѳнөөдрийг хүртэлх тавь гаруй жилийн хугацаанд МУИС-иас Тооцон бодох математикийн чиглэлээр үндэсний эрдэмтэн, судлаачид олноор төрөн гарч, тэдний бүтээл дотоод, гадаадын мэргэжлийн өндөр түвшний сэтгүүл, монографад хэвлэгдэж, бусад орны судлаачдын бүтээлүүдэд эшлэгдэж байна. Манай эрдэмтэн, багш нар сурах бичиг эх хэл дээрээ бичиж, бакалаврын оюутнуудаа мэргэшүүлэн сургаж, ахисан түвшний судлаачдаа удирдан ажиллаж, төрөл бүрийн судалгааны төсөл хөтөлбөрүүдийг хэрэгжүүлж ирсний үр дүнд Тооцон бодох математик нь математикийн бусад салбаруудтай төдийгүй бусад шинжлэх ухаан, технологи, үйлдвэрлэлтэй нягт холбоотой хөгжиж байгаа билээ. Үүний нэг тод жишээ гэвэл сүүлийн үед манай судлаачдын эрдэм шинжилгээ, судалгааны үр дүн олон салбар, чиглэлийн дундын бүтээл болон гарч байна. Энэхүү өгүүллээр зөвхөн МУИС-ийн Тооцон бодох математикийн онолын судалгааг илүү тодотгохын зорьж, судалгааны голлох үр дүнгүүдийн тухай товч танилцуулсан болно.

In this work, we use Newton-like methods to solve systems of nonlinear equations numerically. This work will be divided in 2 parts: Theoretical and Application part. The theoretical part covers the study of convergence of two and three-step Newton-like methods. Let us find the root x of F(x) = 0 here F is a system of nonlinear equations.

We discuss of computer implementation of Generalized Integral Representation Method (GIRM) for one-dimensional diffusion problem on regular meshes. Although GIRM requires the initial matrix inversion of the given problem, the solution is stable and the accuracy is satisfactory. Moreover, it can be applied to an irregular mesh. In order to confirm the theory, we apply GIRM to the one-dimensional Initial and Boundary Value Problem for advective diffusion equation. The numerical experiments are conducted and the approximate solutions coincide with the exact ones in both cases. The computer code is implemented in Matlab is given and discussed in detail.

In 2009, Prof. R. Picard showed that a number of initial boundary-value problems of classical mathematical physics is generally represented in the linear operator equation and established its well-posedness and causality in a Hilbert space setting. We say that a problem is well-posed if the problem has a unique solution and the solution continuously depends on given data. The independence of the future behavior of a solution until a certain time indicates the causality of the solution. We shall establish the well-posedness and causality of the solution of the evolutionary problems with a perturbation, which is defined by a quadratic form. As an example, we will consider Fisher’s problem.

We present discretization schemes based on Generalized Integral Representation Method (GIRM) for numerical study of the Boussinesq wave. The schemes numerically evaluate the coupled Boussinesq equation for three different solitary wave phenomena, namely, propagation of a single soliton, head-on collision of two solitons and reflection of soliton at a fixed boundary. In each case of the soliton interactions, we utilize different Generalized Fundamental Solutions (GFS) along with piecewise constant approximations for the unknown functions.

Firstly, Lloyd D. Fosdick applied the Monte Carlo Method in calculation of short range order parameters in Cu3Au alloy using pairwise effective potentials on the first shell. In this work we we calculated short range order parameters in first two shells in Cu3Au alloy using pairwise effective potentials on the first eleven shell calculated in [2]. Table 1 shows result of our calculation in comparison with experimental results [3]. Results of calculation are in satisfactory agreement with experimental data.

In this paper, we are proposing a novel method to estimate static displacements of atoms caused by size effects in fcc substitutional binary polycrystalline solid solutions. Fourier transforms of static displacements of the atoms on every considered shell were calculated using the equations that include dynamical matrix and Fourier transform of interatomic forces. Short-range order parameters on the first seven shells of Ni-14 at. % Ir alloy have been identified from X-ray diffuse scattering intensity by accounting microscopic static displacements of atoms on a particular shell. Pairwise interatomic potentials on the considered shells and critical temperature of disorder-order phase transition were calculated using values of short-range order parameters

We study a class of time fractional diffusion-wave equations with variable coefficients using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of equations. Group invariant solutions are given explicitly corresponding to every element in an optimal system of Lie algebras generated by infinitesimal symmetries of equations in the class. We express the solutions in terms of Mittag-Leffler functions, generalized Wright functions, and Fox H-functions. These solutions contain previously known solutions for particular cases.

In this study, we derive a discretization scheme based on Generalized Integral Representation Method (GIRM) for numerical evaluation of a diffusion problem in 2D. In particular, we solve a Dirichlet problem for the unsteady diffusion equation in a circular domain. By its nature, the scheme does not require continuity of the approximate solution across the computational elements. Therefore the numerical validation, we provide examples with different initial conditions such that there is a discontinuity between the initial and boundary conditions.

Рассчитаны значения радиальных и тангенциальных силовых постоянных межатомного взаимодействия на первых шести координационных сферах -Fe из фононного спектра, сопоставлены с ранее полученными данными. Расчеты проводились методом наименьших квадратов путем описания экспериментальных фононных ветвей с помощью динамической матрицы, которая построена в модели Де Лане. Для проверки достоверности полученных значений силовых постоянных с их применением построены фононные спектры и рассчитаны упругие постоянные -Fe, которые удовлетворительно согласуются с соответствующими экспериментальными данными.

Short-range order parameters on the first three shells of Cu-10, 17 and 25 at %Au alloys have been identified from X-ray diffuse scattering intensity by accounting microscopic static displacements of atoms on a particular shell.

We calculated the elastic constants of gamma-Fe from the phonon spectra, which was previously measured by other authors. The calculations were carried out using the least squares method by fitting of the experimental phonon branches using a dynamical matrix, which is described in frame of DeLaunay model.

By using X-ray diffuse scattering method were determined short range order parameters on first eight shells .