Hasil untuk "Analytic mechanics"

K. Prasad, Indubala, M. Kumari

In this article, we study a novel extension of the classic balancing numbers, referred to as the higher-order balancing numbers and denoted by. This sequence is analogous to the higher-order Fibonacci numbers and follows the same recurrence relation as the balancing sequence itself. The case k=1 gives the classic balancing numbers (A001109) and for k=2 gives the sequence A029547, thus establishing a direct link to existing number sequences. Here, we first establish the Binet-like formula and then, with its help, present various algebraic properties of this newly introduced sequence, such as recurrence relations, generating functions (both ordinary and exponential), partial sums, binomial sums, combined identities, and more. We also obtain the limiting ratio and establish several well-known identities, including Catalan’s identity, d’Ocagane’s identity, Vajda’s identity, Honsberger’s identity, using the Binet-like formula. Finally, we give some mixed identity and series sum formulae. In this study, the obtained identities and algebraic properties are expressed in terms of the existing balancing and Lucas-balancing numbers.

K.T. Karimov, M.R. Murodova

The paper investigates a Dirichlet-type boundary value problem for a three-dimensional elliptic equation with three singular coefficients in the first octant. The uniqueness of the solution within the class of regular solutions is established using the energy integral method. To prove the existence of a solution, the Hankel integral transform method is employed. The use of the Hankel transform is particularly appropriate when the variables in the equation range from zero to infinity. This transform is an effective method for obtaining solutions to such problems. In three-dimensional space, to derive the image equation, the Hankel integral transform is applied to the original equation with respect to the variables x and y. As a result, a boundary value problem for an ordinary differential equation in the variable z arises. By solving this problem, a solution to the original boundary value problem is constructed in the form of a double improper integral involving Bessel functions of the first kind and Macdonald functions. To justify the uniform convergence of the improper integrals, asymptotic estimates of the Bessel functions of the first kind and Macdonald functions are utilized. Based on these estimates, bounds for the integrands are obtained, which ensure the convergence of the resulting double improper integral, that is, the solution to the original boundary value problem and its derivatives up to second order, inclusively, as well as the theorem of existence within the class of regular solutions.

G.F. Azhmoldaev, K.A. Bekmaganbetov, G.A. Chechkin et al.

We consider the reaction–diffusion system of equations with rapidly oscillating terms in the equation and in boundary conditions in a domain with locally periodic oscillating boundary. In the subcritical case (the Fourier boundary condition is changed to the Neumann boundary condition in the limit) we proved that the trajectory attractors of this system converge in a weak sense to the trajectory attractors of the limit (homogenized) reaction–diffusion systems in domain independent of the small parameter, characterizing the oscillation rate. To obtain the results we use the approach of homogenization theory, asymptotic analysis and methods of the theory concerning trajectory attractors of evolution equations. Defining the appropriate functional and topological spaces with weak topology, we prove the existence of trajectory attractors and global attractors for these systems. Then we formulate the main Theorem and prove it with the help of auxiliary Lemmata. Applying the homogenization method and asymptotic analysis we derive the homogenized (limit) system of equations, prove the existence of trajectory attractors and global attractors and show the convergence of trajectory and global attractors.

M. Ashyraliyev, M.A. Ashyralyyeva

This paper presents a numerical study of source identification problems for one-dimensional telegraphparabolic equations subject to Dirichlet and Neumann boundary conditions. In these inverse problems, the unknown source terms are assumed to be space-dependent, which introduces both analytical and computational challenges. The study begins by discretizing the considered problems using the finite difference method – first in space and subsequently in time – resulting in a system of discrete equations. Stability results for the solutions of the resulting finite difference schemes are established to ensure the reliability of the numerical approach. A numerical algorithm is proposed for solving the discrete inverse problems. The algorithm begins by eliminating the unknown source terms, which transforms the original discretized problem into a new nonlocal problem with unknown initial data. To approximate this initial data, an iterative procedure based on fixed-point iterations is constructed. Once the transformed nonlocal problem is solved, the solution of the main finite difference scheme and approximations of the unknown source term are recovered. Numerical results for two test problems are presented to illustrate the proposed method in practice. The findings confirm the accuracy of the approach in solving space-dependent inverse source problems.

N.A. Abiev

In the present paper sets related to invariant Riemannian metrics of positive sectional and (or) Ricci curvature on generalized Wallach spaces are considered. The problem arises in studying of the evolution of such metrics under the influence of the normalized Ricci flow. For invariant Riemannian metrics of the Wallach spaces which admit positive sectional curvature and belong to a given invariant surface of the normalized Ricci flow equation we establish that they form a set bounded by three connected and pairwise disjoint regular space curves such that each of them approaches two others asymptotically at infinity. Analogously, for all generalized Wallach spaces with coincided parameters the set of Riemannian metrics which belong to the invariant surface of the normalized Ricci flow and admit positive Ricci curvature is bounded by three space curves each consisting of exactly two connected components as regular curves. Mutual intersections and asymptotical behaviors of these components are studied as well. We also establish that curves corresponding to Ka¨hler metrics of spaces under consideration form separatrices of saddles of a three-dimensional system of nonlinear autonomous ordinary differential equations obtained from the normalized Ricci flow equation.

Durvudkhan Suragan, Bolys Sabitbek

Nazarbayev University in Astana, Kazakhstan, will host the 15th International ISAAC Congress from July 21–25, 2025. The International Society for Analysis, its Applications, and Computation (ISAAC) Congress is a prestigious event that continues a successful series of meetings previously held across the globe.

M.T. Kosmakova, K.A. Izhanova, L.Zh. Kasymova

In the paper, the boundary value problem for the loaded heat equation is solved, and the loaded term is represented as the Riemann-Liouville derivative with respect to the time variable. The domain of the unknown function is the cone. The order of the derivative in the loaded term is less than 1, and the load moves along the lateral surface of the cone, that is in the domain of the desired function. The boundary value problem is studied in the case of the isotropy property in an angular coordinate (case of axial symmetry). The problem is reduced to the Volterra integral equation, which is solved by the method of the Laplace integral transformation. It is also shown by direct verification that the resulting function satisfies the boundary value problem.

Yu.P. Apakov, D.M. Meliquzieva

In this paper, we consider a boundary value problem in a rectangular domain for a fourth-order homogeneous partial differential equation containing the third derivative with respect to time. The uniqueness of the solution of the stated problem is proved by the method of energy integrals. Using the method of separation of variables, the solution of the considered problem is sought as a multiplication of two functions X (x) and Y (y). To determine X (x),we obtain a fourth-order ordinary differential equation with four boundary conditions at the segment boundary [0,p], and for a Y (y) – third-order ordinary differential equation with three boundary conditions at the boundary of the segment [0,q]. Imposing conditions on the given functions, we prove the existence theorem for a regular solution of the problem. The solution of the problem is constructed in the form of an infinite series, and the possibility of term-by-term differentiation of the series with respect to all variables is substantiated. When substantiating the uniform convergence, it is shown that the “small denominator” is different from zero.

D. Dauitbek, Y. Nessipbayev, K. Tulenov

This paper provides a number of examples of relatively weakly compact sets in Orlicz spaces. We show some results arising from these examples. Particularly, we provide a criterion which ensures that some Orlicz function is increasing more rapidly than another (in a sense of T. Ando). In addition, we point out that if a bounded subset K of the Orlicz space LΦ is not bounded by the modular Φ, then it is possible for a set K to remain unbounded under any modular Ψ increasing more rapidly than Φ.

S. Shaimardan, N.S. Tokmagambetov

In this paper, we obtain exact solutions of a new modification of the Schrödinger equation related to the Bessel q -operator. The theorem is proved on the existence of this solution in the Sobolev-type space Wq2(R+q ) in the q -calculus. The results on correctness in the corresponding spaces of the Sobolev-type are obtained. For simplicity, we give results involving fractional q -difference equations of real order a> 0 and given real numbers in q -calculus. Numerical treatment of fractional q -difference equations is also investigated. The obtained results can be used in this field and be supplement for studies in this field.

G. Agarwal, E. Wolf

N. Kausar, M. Munir, M. Gulzar et al.

In this article, we have presented some important charcterizations of the ordered non-associative semigroups in relation to their ideals. We have initially characterized the ordered AG-groupoid through the properties of the their ideals, then we characterized the two important classes of these AG-groupoids, namely the regular and intragregular non-associative AG-groupoids. Our aim is also to encourage the research and the maturity of the associative algebraic structures by studying a class of non-associative and non-commutative algebraic structures called the ordered AG-groupoid.

Yao Wang, Sergei Alexandrov, Elena Lyamina

The boundary conditions significantly affect solution behavior near rough interfaces. This paper presents general asymptotic analysis of solutions for the rigid plastic double slip and rotation model in the vicinity of an envelope of characteristics under plane strain and axially symmetric conditions. This model is used in the mechanics of granular materials. The analysis has important implications for solving boundary value problems because the envelope of characteristics is a natural boundary of the analytic solution. Moreover, an envelope of characteristics often coincides with frictional interfaces. In this case, the regime of sticking is not possible independently of the friction law chosen. It is shown that the solution is singular in the vicinity of envelopes. In particular, the profile of the velocity component tangential to the envelope is described by the sum of the constant and square root functions of the normal distance to the envelope in its vicinity. As a result, some components of the strain rate tensor approach infinity. This finding might help to develop an efficient numerical method for solving boundary value problems and provide the basis for the interpretation of some experimental results.

Zh.N. Tasmambetov, A.A. Issenova

In this paper, the systems with solutions in the form of degenerate hypergeometric Humbert functions of two variables reduced to Bessel functions of two variables are established and studied. The connections between the Humbert and Bessel functions of two variables are revealed, their differential properties are investigated. The addition and multiplication theorems are proved. In future, these proven properties allow us to establish recurrent relations between degenerate hypergeometric functions of two variables, similarly to extend these properties to the case of many variables. The connection between type systems of Bessel and Whittaker is shown. Using the Frobenius - Latysheva method, the singularities of constructing normalregular solutions of the newly established Bessel - type system are studied.

G.A. Yessenbayeva, D.N. Yesbayeva, N.K. Syzdykova et al.

The article is devoted to the application of the collocation method to solving differential equations, which are the basis for calculating many problems of mechanics. In this article the structure of this method is presented, its main components are highlighted; its types are characterized, as well as its classical approaches. The research of the problem of rectangular plates bending is carried out by the method of collocations in this article. The collocation method, like all numerical - analytical approximate methods, has a number of advantages and disadvantages, which are also noted in this article. The article is focused mainly on mechanics, engineers and technical specialists.

M. Wertheim

A. Kalybay, R. Oinarov, S. Shalginbayeva

В статье мы находим необходимые и достаточные условия для выполнения дискретного неравенства Харди с весами, записанного в разностной форме. Задача исследуется на множестве финитных последовательностей. Ключевой результат данной статьи - это нахождение оценок для точной константы исследуемого неравенства. Данные оценки в дальнейшем будут нами применены для установления качественных характеристик, таких как условия осцилляторности и неосцилляторности, некоторых разностных уравнений. Более того, как следствие основных результатов, мы находим критерии вложения некоторых пространств и компактности этого вложения.

A.V. Zenkov

M-Группа есть пара (G;*); где G решеточно упорядоченная группа ( l -группа) и * есть убывающий автоморфизм 2 - го порядка G: В статье получено описание m-транзитивных представлений произвольной m-группы. Найдены необходимые и достаточные условия того, что m-группа допускает точное m-транзитивное представление. Изучено строение решетки конгруэнций произвольного m-транзитивного представления, введены понятия m-2-транзитивного и m-примитивного представлений. Получено описание m-транзитивных примитивных представлений в терминах стабилизаторов. Указаны необходимые и достаточные условия m-2-транзитивности и изучены некоторые свойства таких представлений. Кроме того, введено понятие сплетения m-групп подстановок и доказано, что m-транзитивная группа подстановок вложима в сплетение подходящих m-транзитивных m-групп подстановок. Как следствие, установлено, что произвольная m-транзитивная группа из произведения двух многообразий m-групп вложима в сплетение подходящих m-транзитивных групп из этих многообразий.

J. Marko, E. Siggia