Equation In Polar Coordinates Research Articles

Dirac fermions in a spinning conical Gödel-type spacetime

Abstract In this paper, we determine the relativistic and nonrelativistic energy levels for Dirac fermions in a spinning conical Gödel-type spacetime in (2+1)-dimensions, where we work with the Dirac equation in polar coordinates and we use the tetrads formalism. Solving a second-order differential equation for the two components of the Dirac spinor, we obtain a generalized Laguerre equation, and the relativistic energy levels, where such levels are quantized in terms of the quantum numbers n and m j, and explicitly depends on the spin parameter s, spinorial parameter u, curvature and rotation parameters α and β, and on the vorticity parameter Ω. In particular, the quantization is a direct result of the existence of Ω (i.e., Ω acts as a kind of ``external field or potential''). We see that for m j>0, the energy levels do not depend on s and u; however, depend on n, m j, α, and β. In this case, α breaks the degeneracy of the energy levels and such levels can increase infinitely in the limit 4Ωβ/α→1. Already for m j<0, we see that the energy levels depend on s, u and n; however, it no longer depends on m j, α and β. In this case, it is as if the fermion ``lives only in a flat Gödel-type spacetime''. Besides, we also study the low-energy or nonrelativistic limit of the system. In both cases (relativistic and nonrelativistic), we graphically analyze the behavior of energy levels as a function of Ω, α, and β for three different values of n.

Normal form and Hopf bifurcation for the memory‐based reaction‐diffusion equation with nonlocal effect

In this paper, we provide the normal form for the Hopf bifurcation of a class of the reaction‐diffusion equation with memory‐based diffusion and nonlocal effect, where the delay is present in the differential term, similar to the chemotaxis model with time delay. The eigenvalue problems and the decomposition of the phase space are discussed in detail. Through a series of variable transformations, we obtain the third‐order truncated normal form of the model constrained on the central manifold and its equivalent equation in polar coordinates. Then, with the help of the dynamic analysis for the finite dimensional equations, the key parameters for determining the direction and stability of the Hopf bifurcation are given. These theoretical results are applied to the Bazykin's model, the stability, Turing bifurcation and Hopf bifurcation of the equilibrium are demonstrated through both theoretical and numerical methods.

A time splitting Chebyshev-Fourier spectral method for the time-dependent rotating nonlocal Schrödinger equation in polar coordinates

In this article, we propose a time splitting Chebyshev-Fourier spectral method to compute the dynamics of rotating nonlocal Schrödinger equation in polar coordinates. Since the rotation term is diagonalizable with Fourier spectral method in azimuthal direction, we split the Hamiltonian into a linear part, i.e., the Laplacian and rotation terms, and the nonlinear part. The linear part is discretized by Chebyshev-Fourier spectral method in space and integrated by Crank-Nicolson or exact matrix exponential. The nonlinear part is solved exactly in physical space, and the nonlocal potential, which is defined as convolution φ:=U⁎|ψ|2 with a singular kernel U, is computed via Kernel Truncation method with spectral accuracy. Then we construct a high order time splitting spectral method. Our scheme is spectrally accurate in space and of second/fourth order in the temporal direction. Extensive numerical results are presented to confirm the accuracy and efficiency for both the nonlocal potential and the wave function, together with one application to a two-component rotating dipolar Bose-Einstein condensates. In addition, we investigate the rotational symmetry preserving performance by a comprehensive comparison with existing scheme.

Measurements of electrical resistivity and Seebeck coefficient for disc-shaped samples

Herein we develop a methodology to measure the resistivity of discs by deriving a mathematic formula between resistivity and resistance from solving the electrostatic Laplace equation in polar coordinates. Then the resistivity and Seebeck coefficient of disc samples of p- and n-type bismuth telluride are measured experimentally either in nitrogen or helium atmosphere. The validity of Seebeck coefficient is demonstrated by the excellent linearity between Seebeck voltages and temperature differences. The bar samples are also measured for comparison. Finite element simulation is utilized to display the two-dimensional potentials and currents and have an error analysis. Furthermore, the resistivity error due to the probe distance error is discussed analytically based on the mathematic formula, and the probe distance can be optimized to minimize the resistivity error. The disclosed approach would extend the applicability of the present instruments to the disc-shaped samples and be useful in the emerging transverse thermoelectricity.

Some closed form series solutions for the time-fractional diffusion-wave equation in polar coordinates with a generalized Caputo fractional derivative

Some closed form series solutions for the time-fractional diffusion-wave equation in polar coordinates with a generalized Caputo fractional derivative

On the Equilibrium of Nonthin Cylindrical Shells with a Dent

On the basis of the linear three-dimensional theory of elasticity, we consider the problem of equilibrium of nonthin isotropic cylindrical shells with dents under certain boundary conditions imposed on the end faces. To describe the cross section of the reference surface, we use the Pascal snail equation in polar coordinates. By the method of separation of variables, with the help of the approximation of functions by discrete Fourier series, we reduce the three-dimensional boundary-value problem to a one-dimensional problem, which is solved by the stable numerical method of discrete orthogonalization. We estimate the accuracy of the obtained solutions. The results of solving the problems are presented in the form of plots and tables.

Numerical Solution of Electrostatic Potential Distribution on a Unit Circular Disc Using Possion Equation

In this paper, the two-dimensional electrostatic potential distribution problem has been solved numerically using the finite difference approach using Poisson’s equation in polar coordinates with Dirichlet’s boundary condition inside a unit circular disc. We also use Gauss elimination method to solve numbers of linear equations to obtain solutions of unknowns at each grid point. The numerical solution to the same problem is contrasted with the analytic solution. Finally, we examine the absolute errors throughout a range of iterations to evaluate the accuracy of the schemes.

Thermoelastic stress mitigation and weight reduction of functionally graded multilayer nonuniform thickness disc

In this paper, the particle swarm optimization method is used to reduce the weight of a multilayer rotating nonuniform thickness disc along with alleviation of the maximum tangential stress and the maximum tangential stress-jump at the interfaces. The proposed disc is made of functionally graded material and is subjected to both mechanical pressure and thermal loads. It is divided into several layers with each one having its unique volume fraction. These volume fractions are considered the design variables of the optimization problem along with two geometrical parameters related to the disc thickness. The equilibrium equation in polar coordinates are solved using the finite difference method. A punch of optimization results is calculated and discussed. It is concluded that the range of design variables’ variation widens by considering more layers. Finally, there is no potential disc configuration or geometry is found dominant to enhance the design parameters concurrently. Therefore, performing similar optimization analyses is compulsory to obtain an efficient and durable structure.

Telegraph equation in polar coordinates: Unbounded domain with moving time-harmonic source

The telegraph equation with moving time-harmonic source with angular frequency ω is considered in polar coordinates (r,φ). Two problems are studied: the source moving on a straight line with constant velocity v and the source traveling on a circumference of a circle of radius R with a constant orbital frequency Ω. The solutions are obtained using the integral transforms technique. Two limiting cases of the telegraph equation are also analysed: the Fourier heat conduction equation and the linear wave equation. The singularity of the solution to the wave equation at a point r=R+vt, φ=0 in the case of the source moving on a straight line is described. The relationship between the orbital frequency Ω and the polar coordinates at the wave front has been analyzed. The results of numerical simulation are presented graphically for a wide spectrum of nondimensional parameters.

The noncommutative quantum Hall effect with anomalous magnetic moment in three different relativistic scenarios

In the present paper, we investigate the bound-state solutions of the noncommutative quantum Hall effect with anomalous magnetic moment in three different relativistic scenarios, namely: the Minkowski spacetime (inertial flat case), the spinning cosmic string (CS) spacetime (inertial curved case), and the spinning CS spacetime with noninertial effects (noninertial curved case). In particular, in the first two scenarios, we have an inertial frame, while in the third, we have a rotating frame. With respect to bound-state solutions, we focus primarily on eigenfunctions (Dirac spinor and wave function) and on energy eigenvalues (Landau levels), where we use the flat and curved Dirac equation in polar coordinates to reach such solutions. However, unlike the literature, here we consider a CS with a non-null angular momentum and also the NC of the positions, and therefore, we seek a more general description for the QHE. Once the solutions are obtained, we discuss the influence of all parameters and physical quantities on relativistic energy levels. Finally, we analyze the nonrelativistic limit, and we also compared our problem with other works, where we verified that our results generalize some particular cases of the literature.

Метод численного решения уравнений квантовой теории рассеяния для системы нескольких частиц

A methods of numerical solution of the Faddeev equations for the bound and scattering state problems are presented. To solve the Faddeev’s equations with corresponding boudary conditions the mrthod is used wich is a generalization of the cooon scheme of solution of Schrodinger equations. In this method a finit-difference approximation of equations in polar coordinates is used. By means of the net with knots on the arc and knots on the beam the finit-difference approximation of Faddeev’s equations is performed. The increase the calculation accurace and to provide an effetive operation of the corresponding software support the net knots must be located nonuiformly are disscased. The net must be dens in the regions adjacent to an axes x = 0, while at → ∞ becomes less dense. Such an arrangment of the net rnots is necessary for sufficienty accurate description of the complex behaviour of wave function in the region, where x ≪ y. In the region → ∞ the wave function takes asymptotical depends smoothly on angular variable, hence in this region the net must be chosen to be less dense. At calculations on the inhomogeneous net, in very important to comply with the condition of self-consistence, which implies that the error due to introducing the cut off radius must be far less than the quantizing error in the vicinty of boundary. The stability and self-consistence of the calculations may be check by various methods determinations of the net, the cut off radius and the scattering amplitude. This approach to the numerical solution of the Faddeev’ equations combines the advantages of integral equations (the solution existens and uniqieness) with the computational scheme simplicity. Also it has the advantage that unknow two-particle quantities are specified not T-matrices but potentials and eigenfunctions corresponding to the bound states. The simplicity of the computational algorithm allows to consider few-body problems in real systems. The results of calculations binding energies and scattering state of are presented.

MATHEMATICAL APPARATUS FOR MODELING OF THE PROPAGATION THE MAGNETIC FIELD ELECTRIC MACHINES WITH A GIVEN ACCURACY

. The problem of modeling the propagation local magnetic fields and spatially dispersed sources is large errors compared to field measurements. An important aspect of adequate modeling is the use of the correct mathematical apparatus. It is shown that in order to obtain reliable models of the propagation magnetic fields around electrical machines (generators, electric motors of different power, geometric dimensions and poles), it is advisable to apply the Gauss equation for a scalar potential. The solution of the equation in polar coordinates makes it possible to take into account not only the fundamental, but also other harmonics of the magnetic field (dipole, quadrupole, octupole). This allows, depending on the number of spatial harmonics taken into account, to obtain a model with the required accuracy (error) for predicting the magnetic field strength at any point around the machine. It is considered in the paper that an electronic machine is an object of base radius R0. The presented approach makes it possible to unambiguously determine the location of zero field points at a distance from the source (for a quadrupole source and zero field lines, for an octupole source). The results of modeling and their verification by full-scale measurements for the most common four-pole machines (quadrupole source) are presented. The main task of modeling the propagation the magnetic field of such sources is to ensure the required accuracy based on the goals of modeling. It is shown that the modeling accuracy and the presence of zero field points are due to different field levels near the electrical machine housing for different harmonics. The dipole harmonic at the cabinet is 20% of its own harmonic. But it falls more slowly with distance. This necessitates taking into account a different number of harmonics depending on the value of the ratio R0/R, R is the distance to the point of determining the field strength from the source. Therefore, with the ratio R0/R=2/3, the eighth harmonic is essential. At R0/R=1/5, already the fourth spatial harmonic can be neglected. Such data allow you to choose a rational number of harmonics. This reduces the amount of calculations and simplifies the process of modeling the propagation of the magnetic field around the source.

Darboux transformations for Dirac equations in polar coordinates with vector potential and position-dependent mass

Darboux transformations for Dirac equations in polar coordinates with vector potential and position-dependent mass

A Method for Directly Extracting the Jacobian Matrix of the Power Flow Equations of a Power System in Polar Coordinates

This paper presents a method to directly extract the Jacobian matrix of a power system’s power flow (PF) equations in polar coordinates (termed as DEJMP method). This method is designed to reduce the computational complexity of the extraction process and improve the computational efficiency of the relevant PF algorithm. Direct extraction of the Jacobian matrix from the complex power equation in polar coordinates precludes an increase in both the number of procedural steps and the computational expense, which is a problem associated with the conventional PF (C-PF) algorithm due to its requirement that the derivatives of the active and reactive bus power equations be taken in separate steps. The DEJMP method avoids the trigonometric calculations used in the conventional method for computing the Jacobian matrix, resulting in a significant increase in computational efficiency. In addition, simulation results obtained for IEEE-300, S-1047, S-2383, and S-9241 bus systems validate the efficiency of the DEJMP-based PF algorithm. The DEJMP-based PF algorithm requires more than 20% less time to calculate the PF than the C-PF algorithm. This decrease in computing time is more pronounced for large systems.

Online MTPA Operation of IPMSM Based on Dual-Loop Control in Polar Coordinates

In this article, a maximum online torque per ampere (MTPA) control is proposed for torque control of interior permanent magnet synchronous motors. First, torque and MTPA equations are derived based on flux variables, where the influence of magnetic saturation and cross-coupling effects is carefully considered. These equations can be described in polar coordinates, in which the output torque and MTPA operation are mainly determined by the current magnitude and current angle, respectively. Consequently, a dual-loop controller is proposed to solve the torque and MTPA equations in polar coordinates with a lower computational burden. It consists of a current magnitude and a current angle controller, where a torque reference limiter is embedded to satisfy the current limit. Moreover, the current angle is set within a stable range to avoid the stuck of the controller. The effectiveness of the proposed method is verified through simulation and experimental results. Applying the proposed algorithms, both MTPA operation and torque accuracy can be achieved with less computational burden.

Representation of Weierstrass integral via Poisson integrals

In our research, we have presented a second-order linear partial differential equation in polar coordinates. Considering this differential equation on the unit disk, we have obtained a one-dimensional heat equation. It is well-known that the heat equation can be solved taking into account the boundary condition for the general solution on the unit circle. In our paper, the boundary-value problem is solved using the well-known method called the separation of variables. As a result, the general solution to the boundary-value problem is presented in terms of the Fourier series. Then the expressions for the Fourier coefficients are used to transform the Fourier series expansion for the general solution to the boundary-value problem into the so-called Weierstrass integral, which is represented via the so-called Weierstrass kernel. A representation of the Weierstrass kernel via the infinite geometric series is derived by a way allowing a complicated function to be parameterized via a simplified function. The derivation of the corresponding parametrization is based on two well-known integrals. As a result, a complicated function of the natural argument is represented in the form of a double integral that contains a simplified function of the same natural argument. So, the double-integral representation of the Weierstrass kernel has been derived. To obtain this result, the integral representation of the so-called Dirac delta function is taken into account. The expression found for the Weierstrass kernel is substituted into the expression for the Weierstrass integral. As a result, it was found that the Weierstrass integral can be considered a double-integral that contains the Poisson and conjugate Poisson integrals.

Representation of Weierstrass integral via Poisson integrals

In our research, we have presented a second-order linear partial differential equation in polar coordinates. Considering this differential equation on the unit disk, we have obtained a one-dimensional heat equation. It is well-known that the heat equation can be solved taking into account the boundary condition for the general solution on the unit circle. In our paper, the boundary-value problem is solved using the well-known method called the separation of variables. As a result, the general solution to the boundary-value problem is presented in terms of the Fourier series. Then the expressions for the Fourier coefficients are used to transform the Fourier series expansion for the general solution to the boundary-value problem into the so-called Weierstrass integral, which is represented via the so-called Weierstrass kernel. A representation of the Weierstrass kernel via the infinite geometric series is derived by a way allowing a complicated function to be parameterized via a simplified function. The derivation of the corresponding parametrization is based on two well-known integrals. As a result, a complicated function of the natural argument is represented in the form of a double integral that contains a simplified function of the same natural argument. So, the double-integral representation of the Weierstrass kernel has been derived. To obtain this result, the integral representation of the so-called Dirac delta function is taken into account. The expression found for the Weierstrass kernel is substituted into the expression for the Weierstrass integral. As a result, it was found that the Weierstrass integral can be considered a double-integral that contains the Poisson and conjugate Poisson integrals.

Sobolev orthogonal Legendre rational spectral methods for exterior problems

The purpose of this paper is to develop the diagonalized Legendre rational spectral method for exterior problems. We first consider the exterior problems of two-dimensional elliptic and parabolic equations in polar coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. Then we consider the exterior problems of three-dimensional elliptic and parabolic equations in spherical coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. The main advantages of the suggested approaches are that the discrete systems are diagonal and the numerical solutions can be represented as truncated Fourier series. The numerical results show their effectiveness and accuracy.

ЗАДАЧА АНИЗОТРОПИИ УПРУГОСТИ И ПРОЧНОСТИ ВОЛОКНИСТЫХ АНИЗОТРОПНЫХ МАТЕРИАЛОВ

Introduction: The paper presents new results of studies on the anisotropy of fiber materials with cylindrical anisotropy, which include filament-wound composite materials reinforced with various fibers. Methods: We suggest a mathematical solution to a fourth-order partial differential equation in polar coordinates with two variables for an orthotropic anisotropic body. To solve this equation, we converted it into Cartesian coordinates and presented the stress function as a sum of polynomials. Results and Discussion: As a result of the solution, we obtained two relationships between the elastic constants in the principal directions of anisotropy (so-called elasticity parameters). One of them was obtained for the first time, and the other results from the solution of the anisotropy problem for an orthotropic curved body, suggested by S. G. Lekhnitsky. The obtained solution does not contradict Lekhnitsky’s solution. Thus, in our opinion, orthotropic materials can be divided into two groups. In one group, when shifting from the radial to the tangential direction, the elastic constants take on extreme values when the layers are at angles of 0, 60, and 90°. In the other group, there is no intermediate extreme value and the elastic constants take on extreme values when the layers are at angles of 0 and 90°. The obtained results can be applied in the development of new high-strength composite materials and new technologies for the design and manufacture of building structures, as well as in the design of high-strength structures from synthetic composite materials.

РАСЧЕТ ТОЛЩИНЫ ТЕПЛОИЗОЛЯЦИИ И ВЕЛИЧИНЫ ЗАГЛУБЛЕНИЯ ПОДЗЕМНЫХ ПОЛИЭТИЛЕНОВЫХ ТРУБОПРОВОДОВ ТЕПЛОСНАБЖЕНИЯ В РЕГИОНАХ С МНОГОЛЕТНЕМЕРЗЛЫМИ ГРУНТАМИ

The process of thawing and freezing of the host soil of an underground polyethylene heat supply pipeline with thermal insulation, operated in areas of permafrost distribution, is considered. The task is to determine the thickness of the thermal insulation of the pipeline and the size of its depth based on the analysis of the temperature field in the ground during long-term cycle. The temperature field in the "insulated pipe-ground" system, which changes over time, is proposed to be determined by solving the heat conduction equations in polar coordinates. In this case, a bundle of pipes is considered as one pipe of equivalent radius. This approach allows to simplify the interface conditions of the pipe, thermal insulation, polyethylene shell and the host soil, as well as to use a simple numerical method for solving the equation – the finite difference method. The results of calculations of the maximum depth of soil thawing from the depth of the heat supply pipe at different thicknesses of thermal insulation are presented. It is proposed to determine the optimal depth of the underground heat supply pipeline from the condition of annual establishment of the zero isotherm of the host soil at the level of the boundary of the active soil layer by the beginning of the heating season. The possibility of finding a zero isotherm at this level is illustrated by calculations of isotherms by the beginning of the heating season (September) at various depths and thicknesses of thermal insulation. Based on the results of calculations the method of estimation the insulation thickness and depth of the pipeline from the condition of minimizing the functional that characterizes the measure of deviation of the ordinates of the zero isotherm on the left-hand boundary of the region from the depth of the active soil layer at the beginning of the heating season.