Series In Small Parameter Research Articles

Three-Dimensional Magnetostatic Waves Caused by Electromagnetic Fields in Metallized Ferrite Plates

By using a well-known technique for finding the characteristics of magnetic fields based on the method of decomposition of unknown quantities in a series in small parameter (chosen as the ratio of amplitude of the tangential harmonic field to the amplitude of normal constant field) and restricting ourselves to the first two terms of decomposition, we establish the conditions of excitation and study the regularities of propagation of three-dimensional magnetostatic waves in metallized normally magnetized ferrite plates depending on the character of the external electromagnetic field and electrophysical characteristics of the material.

The plane piston problem with weak gravitational field in a dusty gas

In the present paper, the nature of a plane piston moving with constant velocity in a gas with dust particles in the presence of weak gravitational field is analyzed by using a combination of two methods, viz., perturbation method and similarity transformation. In perturbation method the physical variables are expanded as a series of small parameter. The zeroth order result represents the uniform flow which is affected by dust particles of the mixture without gravity. The first order result shows the consequence of applied gravity in a dusty gas. The effect of dust particles present in the gas and applied gravity on the distribution of flow variables has been discussed for the case of weak and strong shock wave. The structure of the shock wave front is also discussed.

Quantum model of a hysteresis in a single-domain magnetically soft ferromagnetic

A quantum model of a single-domain magnetically soft ferromagnetic is proposed. The α-Fe crystal in a state of the saturation magnetization and a variable magnetic field is considered as a sample. The method of an effective Hamiltonian, including the operators of the Zeeman energy, the spin-orbit interaction and the interaction with the crystal field, is used in the model. An expansion of trial single-electron wave function in a series in small parameter of the spin-orbit interaction is suggested to account for the magnetic anisotropy. Within the framework of the Heisenberg representation, the nonlinear equations of motion for the magnetization and the orbital moment of single domain are obtained. Parameters of the modelling Hamiltonian are found from a comparison with experimental data on the magnetic anisotropy of iron. A phenomenological term of the magnetic friction is introduced into equation of the magnetization motion. Nonlinear equations are solved numerically by the Runge-Kutta method. A dependence of the single domain magnetization on magnetic field intensity has a characteristic form of a hysteresis loop which parameters are quantitatively coordinated with experimental data of researches of magnetic properties of nanoparticles of iron and iron oxide. The method is extended for modelling the magnetization dynamics of multi-domain ferromagnetic in the approximation of a strong crystal field.

Modeling of Singularly Perturbed multi-criterion Optimal Control Workflows

One-criterion problem of control the complex metallurgical process presented by variable parameters in vector notation was considered. In the implementation vector notation presentation transformed into canonical form by means of piecewise continuous functions optimizing control variables on a given interval [0, T]. Pontryagin's maximum principle is used to determine the optimal conditions for solving one-criterion problem, which is formulated as regular multi-criteria (maximum performance, minimum loss of ferrous and nonferrous metals and energy, etc.) optimization problem. Assumed that a piecewise continuous equations in the zone u ⊂ U, and their existence in space U ⊂ E as the main way to solve multi-criteria optimization problems of metallurgical processes is acceptable convolution of trajectory one-criterion optimum problem. In the modeling of singularly perturbed multi-criteria optimal control problems of metallurgical processes was used the method of Whitham or the search of controlled variable and control action carried out in the form of an asymptotic series in small parameter with the highest differential equations derivatives. It is shown that the expansion of control actions equations system in an asymptotic series in the small parameter allows a distributed optimal control problem for considered process to be transformed to recurrent sequence of simple regular tasks. Finding the coefficients of the asymptotic expansion of the optimal control actions for multi-criteria metallurgical process is the desired solution with given initial conditions and singular perturbations.

New Methods for Solving Integral – Differential Transport Equations

The integral-differential equations describe many phenomena in different field of mechanical and nuclear engineering, chemistry, astronomy, biology and potential theory. The resolution of boundary problems for these equations are the subject of several papers in which the authors have approached in most cases numerical methods: Monte Carlo, truncated series of Chebyshev polynomials, the fictitious domain method, S N method and the finite element method (1-4). An exact solution was found only in the particular cases. Generally, these are obtained with the help of the methods of mathematical analysis, abstract functional analysis and the spectral methods (5). In recent years, the basic ideas of the homotopy, which is a concept of the topology and differential geometry, were used to obtain the approximate solutions for a wide class of differential, integral and integral-differential equations. We mention here the homotopy perturbation method (HPM) proposed by Ji-Huan He in 1998 and the homotopy analysis method (HAM) proposed by Liao in 1992. The perturbation methods approximate the solution of given problem by a series of small parameters. Unfortunately, the majority of non-linear problems have no small parameters and an unsuitable choice of these parameters can lead to bad effects. The new homotopy perturbation technique (HPM) embeds a parameter p that ranges from zero to one (6,7). When the embedding parameter is zero, we get a linear equation and if it is equal to one, we get the original transport equation. This embedding parameter that belongs to the interval (0, 1) can be considered as a small parameter. The homotopy analysis method (HAM) is one of the most effective methods to obtain the exact and approximate series solutions for the integral-differential equations (8). Being a great freedom in choosing of the initial guess functions and the auxiliary functions, are proposed for these such forms that correspond to the conditions imposed at the ends of the spatial interval and thus the algorithm becomes fast convergent. In the papers (9-11) is shown that the homotopy perturbation method, variational iteration method and Adomian decomposition method are only special cases of the homotopy analysis method solutions. Today, a new homotopy method is added to the above. With this we can solve the multipoint boundary problem for a stationary transport equation in the non-homogeneous media. Comparative studies between the solutions obtained with the help of mentioned algorithms for a numerical example with the appropriate exact solutions illustrate the accuracy and computational efficiency of these new methods.

Flow pattern induced by the plane piston moving in a non-ideal gas with weak gravitational field

A systematic perturbation scheme is used to analyze the flow pattern induced by the motion of a plane piston moving with constant velocity in a non-ideal gas with weak gravitational field. The flow variables are expanded as a series of small parameter ε, which is the ratio of the typical escape velocity to the plasma velocity. The zeroth order solution gives the uniform flow which is influenced by the parameter of non-idealness without gravity. The first order relationships show the influence of non-idealness of the gas as well as the applied gravity on the flow field. Distribution of the flow variables has been presented graphically for various values of parameter of non-idealness. The structure of the shock front in case of small perturbation theory is also discussed.

Convergence of Liapunov series for Huygens–Roche Figures

According to the classical theory of equilibrium figures, surfaces of equal density, potential and pressure concur (let us call them isobars). Isobars can be represented by means of Liapunov power series in small parameter q, up to the first approximation coinciding with the centrifugal to gravitational force ratio at the equator. Liapunov has proved the existence of the universal convergence domain: the above mentioned series converge for all bodies (satisfying a natural condition that the density ρ decreases from the center to the surface) if |q| < q*. Using Liapunov’s algorithm and symbolic manipulation tools, we have found q*= 0.000370916. Evidently, the convergence radius q* may be much greater in common situations. To comfirm it it is reasonable to consider two limiting and one or two intermediate cases for the density behaviour: ρ is a constant, the Dirac’s δ-function, linear function of the distance from the center, etc. And indeed, in the previous paper we find a three orders of magnitude greater value for homogeneous figures. In the present paper we find that in the opposite case of Huygens-Roche figures (a point-mass surrounded by a weightless atmosphere) the convergence radius is unexpectedly large and coincides with the well-known biggest possible value q*= 0.541115598 for such a class of figures. To ascertain it we ought to use numerical calculations, so our main result is demonstrated by means of a computer assisted proof.

Linearization and correction method for nonlinear problems

A new perturbation-like technique called linearization and correction method is proposed. Contrary to the traditional perturbation techniques, the present theory does not assume that the solution is expressed in the form of a power series of small parameter. To obtain an asymptotic solution of nonlinear system, the technique first searched for a solution for the linearized system, then a correction was added to the linearized solution. So the obtained results are uniformly valid for both weakly and strongly nonlinear equations.

Convergence of Liapunov Series for Maclaurin Ellipsoids

According to the classical theory of equilibrium figures surfaces of equal density, potential and pressure concur (let call them isobars). Isobars may be represented by means of Liapunov power series in small parameter q, up to the first approximation coincident with centrifugal to gravitational force ratio on the equator. A. M. Liapunov has proved the existence of the universal convergence radius q ⋆: above mentioned series converge for all bodies if ∣q∣ < q ⋆. Using Liapunov's algorithm and symbolic calculus tools we have calculated q ⋆ = 0.000370916. Evidently, convergence radius q 0 may be much greater in non-pathological situations. We plan to examine several simplest cases. In the present paper, we find q 0 for homogeneous liquid. The convergence radius turns out to be unexpectedly large coinciding with the upper boundary value q 0 = 0.337 for Maclaurin ellipsoids.

Thermoelasticity problem for a multilayer coating-substrate assembly irradiated by a laser beam

An analytical solution of a two-dimensional thermoelasticity problem is determined for a thin coating attached to a substrate. The coating is treated as a stratified multilayer structure composed of layers with varying thermophysical and optical characteristics. The developed mathematical procedure of solution is based one expansion of Laplace and Hankel transforms of temperature, displacement and strain distributions in series of small parameter which is equal to the ratio of a coating thickness to a laser beam radius. Expressions for determination of the two-dimensional nonstationary temperature, strain and stress distributions in a multilayer assembly are derived in a closed analytical form which can be used for development of an algorithm of laser photothermal method for detection of failures in the multilayer coating-substrate systems.

Optimal design of a thick-walled pipeline cross-section against creep rupture

Cylinder under combined loadings (pressure, bending, axial force) is subject to non-linear creep described by Norton-Odqvist creep law. In view of bending a circularly-symmetric cross-section is no longer optimal in this case. Hence we optimize the shape of the cross-section; minimal area being the design objective under the constraint of creep rupture. Kachanov-Sdobyrev hypothesis of brittle creep rupture is applied. The solution is based on the perturbation method (expansions into double series of small parameters), adjusted to optimization problems.