Hasil untuk "Analytic mechanics"

Fred Cooper, A. Khare, U. Sukhatme

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multisoliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large N expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order p.

A.D. Akhmetshin, M.T. Omarov, R.Z. Toleukhanova

This article deals with the fundamental problems in the mathematical theory of fractional differential equations, specifically focusing on the analytical solvability of boundary value problems in time-dependent domains. The relevance of the study implies the necessity of developing methods for equations with nonlocal operators modeling anomalous diffusion. A one-dimensional diffusion equation containing a RiemannLiouville fractional derivative with respect to time is examined. The characteristic features of the problem, posed in a non-cylindrical domain bounded by a moving linear boundary and a fixed spatial coordinate, are analyzed. The need to handle inhomogeneous boundary data is identified, and the problem is initially reduced to one with homogeneous conditions. On the basis of the study, the author constructs the fundamental solution in a quarter-plane by means of the bilateral Laplace transform and obtains the Green function for the Dirichlet problem. It is shown that the solution can be expressed through an integral representation in terms of a specific boundary density. This density satisfies a Volterra-type integral equation with a weakly singular kernel. Using the contraction mapping principle, it is proved that this equation has a solution. Consequently, the existence of a regular solution to the original boundary value problem is established.

N.T. Orumbayeva, M.T. Kosmakova, T.D. Tokmagambetova et al.

This paper investigates a class of second-order partial differential equations describing wave processes with nonlocal effects, including cases involving fractional derivatives. Such equations often arise in the theory of elasticity, aerodynamics, acoustics, and electrodynamics. The presented equations include both integral and differential terms, evaluated either at a fixed point x = x0 or x = α(t). An equation with a fractional derivative of order 0 ≤ β < 1 is considered, making it possible to model memory effects and other nonlocal properties. For each equation, supplemented by initial conditions, either a closed-form analytical solution is obtained or the main steps of its derivation are outlined. The article employs the Laplace transform to solve the resulting integral equation, enabling the solution to be presented in an explicit form.

Q. Ma, Huanyu Cheng, Kyung-In Jang et al.

Е. Толеугазы, К.Е. Кервенев

The embedding theory of spaces of differentiable functions of many variables studies important connections and relationships between differential (smoothness) and metric properties of functions and has wide application in various branches of pure mathematics and its applications. Earlier, we obtained the embedding theorems of different metrics for Nikol’skii-Besov spaces with a dominant mixed smoothness and mixed metric, and anisotropic Lorentz spaces. In this work, we showed that the conditions for the parameters of spaces in the above theorems are unimprovable. To do this, we built the extreme functions included in the spaces from the left sides of the embeddings and not included in the “slightly narrowed” spaces from the spaces in the right parts of the embeddings.

M.B. Ismoilov, R.A. Sharipov

The theory of m-convex (m − cv) functions is a new direction in the real geometry. In this work, by using the connection m − cv functions with strongly m-subharmonic (shm) functions and using well-known and rich properties of shm functions, we show a number of important properties of the class of m−cv functions, in particular, we study Hessians Hk(u), k = 1, 2, ..., n − m +1, in the class of bounded m − cv functions.

A.R. Yeshkeyev, O.I. Ulbrikht, M.T. Omarova

First of all, we have to note that in this article, we introduced the new concepts of relations between Jonsson theories in the class of cosemanticness for some considered Jonsson spectrum. All consideration of this new approach was done under sufficiently important class of Jonsson theories, which we called as normal Jonsson theories class. The main result, that we obtained, describes the model-theoretical properties of syntactical and semantical similarities inside the fixed cosemanticness class. For all new concepts in the article, we provided classical samples. The main result of this paper is considering normal Jonsson theories class by similarity to some fixed class of polygons (S-acts).

Jun Zhong, Shuang Liu, Chao Sun

We perform a two-dimensional numerical study on the thermal effect of porous media on global heat transport and flow structure in Rayleigh–Bénard (RB) convection, focusing on the role of thermal conductivity $\lambda$ of porous media, which ranges from $0.1$ to $50$ relative to the fluid. The simulation is carried out in a square RB cell with the Rayleigh number $Ra$ ranging from $10^7$ to $10^9$ and the Prandtl number $Pr$ fixed at $4.3$. The porosity of the system is fixed at $\phi =0.812$, with the porous media modelled by a set of randomly displayed circular obstacles. For a fixed $Ra$, the increase of conductivity shows a small effect on the total heat transfer, slightly depressing the Nusselt number. The limited influence comes from the small number of obstacles contacting with thermal plumes in the system as well as the counteraction of the increased plume area and the depressed plume strength. The study shows that the global heat transfer is insensitive to the conduction effect of separated porous media in the bulk region, which may have implications for industrial designs.

T.B. Akhazhanov, N.А. Bokayev, D.T. Matin et al.

In this paper, we introduce the concept of a variational modulus of continuity for functions of several variables, give an estimate for the sum of the coefficients of a multiple Fourier-Haar series in terms of the variational modulus of continuity, and prove theorems of absolute convergence of series composed of the coefficients of multiple Fourier-Haar series. In this paper, we study the issue of the absolute convergence for multiple series composed of the Fourier-Haar coefficients of functions of several variables of bounded p-variation. We estimate the coefficients of a multiple Fourier-Haar series in terms of the variational modulus of continuity and prove the sufficiency theorem for the condition for the absolute convergence of series composed of the Fourier-Haar coefficients of the considered function class. This paper researches the question: under what conditions, imposed on the variational modulus of continuity of the fractional order of several variables functions, there is the absolute convergence for series composed of the coefficients of multiple Fourier-Haar series.

M. Y¨ucel, F.S. Muhtarov, O.Sh. Mukhtarov

The main goal of this study is to adapt the classical differential transformation method to solve new types of boundary value problems. The advantage of this method lies in its simplicity, since there is no need for discretization, perturbation or linearization of the differential equation being solved. It is an efficient technique for obtaining series solution for both linear and nonlinear differential equations and differs from the classical Taylor’s series method, which requires the calculation of the values of higher derivatives of given function. It is known that the differential transformation method is designed for solving single interval problems and it is not clear how to apply it to many-interval problems. In this paper we have adapted the classical differential transformation method for solving boundary value problems for two-interval differential equations. To substantiate the proposed new technique, a boundary value problem was solved for the Weber equation given on two non-intersecting segments with a common end, on which the left and right solutions were connected by two additional transmission conditions.

E.F. Abdyldaeva, A.K. Kerimbekov, M.T. Zhaparov

In the paper, the solvability of the nonlinear boundary optimization problem has been investigated for the oscillation processes, described by the integro-differential equation in partial derivatives with Fredholm integral operator. It has been established that the components of the boundary vector control are defined as a solution to a system of nonlinear integral equations of a specific form, and the equations of this system have the property of equal relations. An algorithm for constructing a solution to the problem of nonlinear optimization has been developed.

E.T. Karimov, A. Hasanov

We aim to study a unique solvability of a boundary-value problem for a time-fractional diffusion equation involving the Prabhakar fractional derivative in a Caputo sense in a bounded domain. We use the method of separation of variables and in time-variable, we obtain the Cauchy problem for a fractional differential equation with the Prabhakar derivative. Solution of this Cauchy problem we represent via Mittag-Leffler type function of two variables. Using the new integral representation of this two-variable Mittag-Leffler type function, we obtained the required estimate, which allows us to prove uniform convergence of the infinite series form of the solution for the considered problem.

M.T. Kassymetova, N.M. Mussina

In this article, strongly minimal geometries of fragment hybrids are considered. In this article, a new concept was introduced as a family of Jonsson definable subsets of the semantic model of the Jonsson theory T, denoted by JDef(CT). The classes of the Robinson spectrum and the geometry of hybrids of central types of a fixed RSp(A) are considered. Using the construction of a central type for theories from the Robinson spectrum, we formulate and prove results for hybrids of Jonsson theories. A criterion for the uncountable categoricity of a hereditary hybrid of Jonsson theories is proved in the language of central types. The results obtained can be useful for continuing research on various Jonsson theories, in particular, for hybrids of Jonsson theories.

A.Kh. Attaev, M.I. Ramazanov, M.T. Omarov

The paper studies the problems of the correctness of setting boundary value problems for a loaded parabolic equation. The feature of the problems is that the order of the derivative in the loaded term is less than or equal to the order of the differential part of the equation, and the load point moves according to a nonlinear law. At the same time, the distinctive characteristic is that the line, on which the loaded term is set is at the zero point. On the basis of the study the authors proved the theorems about correctness of the studied boundary value problems.

A. Ashyralyev, M. Sadybekov

This issue is a collection of 15 selected papers of foreign and national scientists. All these have been accepted after peer-reviewing and contain numerous new results in the fields of construction and investigation of solutions of well-posed and ill-posed boundary value problems for partial differential equations and their related applications.

B.R. Bayraktar, A.Kh. Attaev

In this paper, we obtained several new integral inequalities using fractional Riemann-Liouville integrals for convex s-Godunova-Levin functions in the second sense and for quasi-convex functions. The results were gained by applying the double Hermite-Hadamard inequality, the classical Holder inequalities, the power mean, and weighted Holder inequalities. In particular, the application of the results for several special computing facilities is given. Some applications to special means for arbitrary real numbers: arithmetic mean, logarithmic mean, and generalized log-mean, are provided.

M.A. Bobodzhanova, B.T. Kalimbetov, G.M. Bekmakhanbet

In this paper, the regularization method of S.A.Lomov is generalized to the singularly perturbed integrodifferential fractional-order derivative equation with rapidly oscillating coefficients. The main goal of the work is to reveal the influence of the oscillating components on the structure of the asymptotics of the solution to this problem. The case of the absence of resonance is considered, i.e. the case when an integer linear combination of a rapidly oscillating inhomogeneity does not coincide with a point in the spectrum of the limiting operator at all points of the considered time interval. The case of coincidence of the frequency of a rapidly oscillating inhomogeneity with a point in the spectrum of the limiting operator is called the resonance case. This case is supposed to be studied in our subsequent works. More complex cases of resonance (for example, point resonance) require more careful analysis and are not considered in this work.

C. Anderson

T. Kling, Daniel Vogler, L. Pastewka et al.

Mohamed El-Zeadani, M. Saifulnaz, Y. H. M. Amran et al.

Abstract Partial-interaction due to sliding between the steel bars, adhesively attached FRP plates and their bordering concrete surface, accompanied with the detachment of the FRP plates due to intermediate crack (IC) debonding make the deflection of strengthened RC beams difficult to anticipate. Previous research and design rules on determining the deflection of strengthened RC beams using FRP plates have opted for a full-interaction moment-curvature design technique where the deflection was measured by either deriving average effective moment of inertia and using elastic deflection equations or integrating the curvature along the beam’s length. Therefore, IC deboning of the plate and the slip resulting from the formation and broadening of new cracks were not directly considered. In this study, a partial-interaction moment-rotation analysis of an adhesively plated beam segment was used to derive analytical equations for the rotation of individual crack faces. The analytical expressions were used to compute the rotation at a crack for a given moment; subsequently, the influence of each crack to the midspan deflection of the RC beams was calculated. As for the uncracked region of the beam, the deflection contribution was measured by integrating the curvature over the uncracked span. The deflection results from the mechanics solution seem to compare well with experimental results. The analytical mechanics solution accounts for the partial-interaction between the steel bars, externally bonded FRP plate and their bordering concrete surface, and also the detachment of the external plate through IC debonding. Further, due to its generic nature and non-reliance on empirical data, the mechanics solution can be adopted to forecast the deflection of strengthened RC beams with novel types of reinforcement materials.