ANALYSIS OF COMBINED DISKS WITH PIECEWISE THICKNESS

The combined constructions subjected to an action of expanding loads and consisting of separate sections are examined. Each of the mentioned sections has its own rigidity. These parts may be made from the same or from the various materials. The materials can be anisotropic or isotropic, homogeneous or inhomogeneous. The constructions under study have the round scheme and they are considered as circular disks with piecewise thickness. In the places of the separate parts conjugation the disks’ thickness can be discontinuous or continuous. The analytical approach is used. The solutions are obtained in closed form and expressed in terms of Legendre functions, Legendre, Gegenbauer and Laguerre polynomials.

NUMERICAL SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION USING RANDOMLY GENERATED FINITE GRIDS AND TWO-DIMENSIONAL FRACTIONAL-ORDER LEGENDRE FUNCTION

There are various methods to solve the physical life problem involving engineering, scientific and biological systems. It is found that numerical methods are approximate solutions. In this way, randomly generated finite difference grids achieve an approximation with fewer iterations. The idea of randomly generated grids in cartesian coordinates and polar form are compared with the exact, iterative method, uniform grids, and approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions. The most ideal and benchmarking method is the finite difference method over randomly generated grids on Cartesian coordinates, polar coordinates used for numerical solutions. This concept motivates the investigation of the effects of the randomly generated meshes. The two-dimensional equation is solved over randomly generated meshes to test randomly generated grids and the implementation. The feasibility of the numerical solution is analyzed by comparing simulation profiles.

Gauss Hypergeometric Representations of the Ferrers Function of the Second Kind

We derive all eighteen Gauss hypergeometric representations for the Ferrers function of the second kind, each with a different argument. They are obtained from the eighteen hypergeometric representations of the associated Legendre function of the second kind by using a limit representation. For the 18 hypergeometric arguments which correspond to these representations, we give geometrical descriptions of the corresponding convergence regions in the complex plane. In addition, we consider a corresponding single sum Fourier expansion for the Ferrers function of the second kind. In four of the eighteen cases, the determination of the Ferrers function of the second kind requires the evaluation of the hypergeometric function separately above and below the branch cut at [1,infty). In order to complete these derivations, we use well-known results to derive expressions for the hypergeometric function above and below its branch cut. Finally we give a detailed review of the 1888 paper by Richard Olbricht who was the first to study hypergeometric representations of Legendre functions.

Один подход к численному решению основного уравнения магнитостатики для конечного цилиндра в произвольном внешнем поле

The magnetostatics direct problem of calculating the resulting magnetic field strength from a homogeneous cylinder of finite dimensions placed in an external magnetic field of arbitrary configuration is considered. With the help of sufficiently voluminous analytical transformations using the basic properties of hypergeometric functions and Legendre functions, the solution of the basic three-dimensional magnetostatic equation for this configuration is reduced to solving of a certain number of systems of three one-dimensional linear integral equations. A simplified form of these systems for special cases of a constant external field and the resulting field on the cylinder axis is obtained.

Spherical symmetry breaking in electric, magnetic and toroidal multipole moment radiations in spherical toroidal resonant cavities and optimum-efficiency antennas

This Letter reports the breaking of the spherical symmetry in the complete electromagnetic multipole expansion when its sources are distributed on spherical toroidal surfaces, identifying the specic geometrical and physical changes fromthe familiar case of sources on a spherical surface. In fact, for spherical toroids dened by concentric spherical rings and symmetric conical rings, the boundary conditions at the latter are not compatible in general with integer values for the orbital angular momentum label of the multipole moments: the polar angle eigenfunctions become Legendre functions of order λ and associativity m represented as innite series with a denite parity, and their complementary associated radial functions are spherical Bessel functions of the same order λ. Consequently, the corresponding multipole sources for the electric, magnetic and toroidal moments and their connections are identied within the Debye formalism, and theappropriate outgoing wave Green functions are constructed in the new basis of eigenfunctions of the Helmholtz equation. Our familiarity with the exact solutions, for the cases of the complete sphere and of cylindrical toroids, allow us to give a preliminary account of the electromagnetic elds for the spherical toroids via the integration of their sources and the Green function for resonant cavities and optimum effciency antennas.

THE FLEXURE OF COMBINED PLATES WITH PIECEWISE VARIABLE THICKNESS. THE SOLUTIONS IN TERMS OF LEGENDRE FUNCTIONS

The work considers the problems of flexure of plates of circular form and consisting of separate parts with different laws of thickness variation. The each part of this plate has the form of a ring. The mentioned separate sections may be made from the same or from the different materials. The material of the piece-wise variable plate parts can be isotropic or orthotropic, homogeneous or nonhomogeneous. In the places of the separate parts conjugation the plate’s thickness can be continuous or discontinuous. The action of symmetric load on piece-wise variable plate is studied. The analytical approach is used, the solutions are obtained in the closed forms in terms of Legendre functions and Legendre polynomials. Rather wide set of plates’ profiles is determined. For the satisfaction of the conditions of the separate sections conjugation the special auxiliary functions are introduced. The computation of the plate, consisting of three sections and subjected to an action of discontinuous loads, is considered as an example.

Dirac Particles in Coulomb Like Field in FLRW–Space

Behaviour of the Dirac particle in Coulomb like field in FLRW space is investigated. Firstly, the Maxwell equations, in terms of the vector potentials are solved to identify the Lorentz and Coulomb like gauges. The radial Coulomb like potential is solved in terms of Legendre functions. Then the Dirac equation is generalized to include this potential and the angular part is separated and solved. The radial and temporal parts of the mass less case is also separated and solved. But the massive case remains coupled. This is still reduced to the case where the Dirac particle can be represented as being in a combined gravitational and electric potential. This effective potential is found to develop an attractive well, which may require a revisit to the recombination era.