EXACT SOLUTIONS OF A MANY-ANYON PROBLEM

1992 ◽  
Vol 07 (32) ◽  
pp. 7931-7942 ◽  
Author(s):  
STEFAN V. MASHKEVICH
Keyword(s):  
Exact Solutions ◽  
Stationary State ◽  
Arbitrary Number ◽  
Statistical Parameter ◽  
Analytic Solutions ◽  
Wave Functions ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Exact Analytic Solutions ◽  
State Functions

A scheme for obtaining exact analytic solutions of the problem of an arbitrary number of anyons in a harmonic well is developed. Its essence consists in establishing a set of wave functions with the demanded interchange properties followed by finding stationary state functions within this set by the method of successive approximations. The energy of the corresponding states depends linearly on the statistical parameter. The discussion of the obtained states is carried out.

scholarly journals New Soliton Solutions for the Higher-Dimensional Non-Local Ito Equation

2021 ◽  
Vol 10 (1) ◽  
pp. 374-384
Author(s):  
Mustafa Inc ◽  
E. A. Az-Zo’bi ◽  
Adil Jhangeer ◽  
Hadi Rezazadeh ◽  
Muhammad Nasir Ali ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Solution Process ◽  
Soliton Solutions ◽  
Analytic Solutions ◽  
Bright Solitons ◽  
Exact Analytic Solutions ◽  
Higher Dimensional ◽  
Non Local ◽  
Free Parameters ◽  
Ito Equation

Abstract In this article, (2+1)-dimensional Ito equation that models waves motion on shallow water surfaces is analyzed for exact analytic solutions. Two reliable techniques involving the simplest equation and modified simplest equation algorithms are utilized to find exact solutions of the considered equation involving bright solitons, singular periodic solitons, and singular bright solitons. These solutions are also described graphically while taking suitable values of free parameters. The applied algorithms are effective and convenient in handling the solution process for Ito equation that appears in many phenomena.

FINDING NON-STATIONARY STATE PROBABILITIES OF G-NETWORK WITH SIGNALS AND CUSTOMERS BATCH REMOVAL

2017 ◽  
Vol 31 (4) ◽  
pp. 396-412 ◽  
Author(s):  
Mikhail Matalytski
Keyword(s):  
Stationary State ◽  
Recurrence Relations ◽  
Queueing Network ◽  
Stationary Regime ◽  
Time Dependent ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Kolmogorov Equations ◽  
Modified Method ◽  
Dependent State

This paper is devoted to the research of an open Markov queueing network with positive customers and signals, and positive customers batch removal. A way of finding in a non-stationary regime time-dependent state probabilities has been proposed. The Kolmogorov system of difference-differential equations for state probabilities of such network was derived. The technique of its building, based on the use of the modified method of successive approximations combined with a series method, has been proposed. It is proved that the successive approximations converge over time to the stationary state probabilities, and the sequence of approximations converges to the unique solution of the Kolmogorov equations. Any successive approximation can be represented as a convergent power series with infinite radius of convergence, the coefficients of which satisfy the recurrence relations; that is useful for estimations. Model example illustrating the finding of time-dependent state probabilities of the network has been provided.

ANALYSIS OF THE NETWORK WITH MULTIPLE CLASSES OF POSITIVE CUSTOMERS AND SIGNALS AT A NON-STATIONARY REGIME

2018 ◽  
Vol 33 (3) ◽  
pp. 404-416 ◽  
Author(s):  
M. Matalytski ◽  
D. Kopats
Keyword(s):  
Stationary State ◽  
Recurrence Relations ◽  
Stationary Regime ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Network Systems ◽  
Modified Method ◽  
Computer Calculations ◽  
Modeling Behavior ◽  
Selection Of

The object of research is G-network with positive customers and signals of multiple classes. The present paper describes an analysis of this network at a non-stationary regime, also provided a description of method for finding non-stationary state probabilities.At the beginning of the article, a description of the network with positive customers and signals is given. A signal when entering the system destroys a positive customer of its type or moves the customer of its type to another system. Streams of positive customers and signals arriving to each of the network systems are independent. Selection of positive customers of all classes for service – randomly. For non-stationary state probabilities of the network, the system of Kolmogorov difference-differential equations (DDE) has been derived. It is solved by a modified method of successive approximations, combined with the method of series. The convergence of successive approximations with time has been proved to the stationary distribution of probabilities, the form of which is indicated in the article, and the sequence of approximations converges to the unique solution of the DDE system. Any successive approximation is representable in the form of a convergent power series with an infinite radius of convergence, the coefficients of which satisfy recurrence relations, which is convenient for computer calculations.The obtained results can be applied for modeling behavior of computer viruses and attack in computer systems and networks, for example, as model impact of some file viruses on server resources.

scholarly journals The occurrence and properties of molecular vibrations with V ( x ) = ax 4

1945 ◽  
Vol 183 (994) ◽  
pp. 328-337 ◽  
Keyword(s):  
Harmonic Oscillator ◽  
Energy Levels ◽  
Wave Functions ◽  
Successive Approximations ◽  
Fourth Power ◽  
Method Of Successive Approximations ◽  
Group Iii ◽  
Modes Of Vibration ◽  
The Difference ◽  
Four Levels

It is shown that in certain modes of vibration of plane rings the potential energy for small displacements is proportional to the fourth power of the displacement, provided that there is free rotation about the bonds of the ring. This type of vibration is termed a ‘fourth-power vibration’. It is likely to occur in cyclobutane and its derivatives, in a number of halides having the formula X 2 Y 6 , and in the hydrides of group III elements. The energies and wave functions of the first four levels of a one-dimensional oscillator with V ( x ) = ax 4 have been derived by a method of successive approximations, and asymptotic formulae are given for the higher levels. The wave functions are qualitatively similar to those of a harmonic oscillator, but the energy levels differ considerably. A comparison is made between energy levels for oscillators with V ( x ) = a q | x q | and different values of q . The selection rule for dipole radiation from a fourth-pow er vibration is discussed. Overtones will be more numerous than in the spectrum of a harmonic oscillator. Estimates are made of the spectrum frequencies of fourth-power vibrations in actual molecules, with special reference to cyclobutane and diborane. For these two molecules there are observed infra-red frequencies of approximately the expected value. The isotope effect should provide a means of discriminating experimentally between harmonic and fourth-power vibrations. The contribution of a fourth-power vibration to any thermodynamic function will differ from that of a harmonic vibration with the same fundamental spectrum frequency. Figures are given for the specific heat, where the difference should be detectable experimentally. In the general case V ( x ) = a q | x q | the energy levels derived from the quantum theory lead to expressions for the thermodynamic functions which agree with the predictions of classical theory at high temperatures.

Exact Solutions for Two-Dimensional Time-Dependent Flow and Deformation Within a Poroelastic Medium

1999 ◽  
Vol 66 (2) ◽  
pp. 536-540 ◽  
Author(s):  
S. I. Barry ◽  
G. N. Mercer
Keyword(s):  
Exact Solutions ◽  
Applied Pressure ◽  
Arbitrary Point ◽  
Analytic Solutions ◽  
Time Dependent ◽  
Finite Domain ◽  
Two Dimensional ◽  
Poroelastic Medium ◽  
Exact Analytic Solutions ◽  
Dependent Flow

Exact analytic solutions are derived for the time-dependent deformation of a poroelastic medium within a two-dimensional finite domain. Solutions are given with a specific set of boundary conditions for the case of a source of fluid at an arbitrary point and for an applied pressure on the boundary. These solutions are ideal for testing numerical schemes for poroelastic flow and deformations due to their relative simplicity.

scholarly journals Lie Symmetry Reductions and Exact Solutions to the Rosenau Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Ben Gao ◽  
Hongxia Tian
Keyword(s):  
Exact Solutions ◽  
Analytic Solutions ◽  
Lie Symmetry ◽  
Series Method ◽  
Lie Symmetry Analysis ◽  
Power Series Method ◽  
Exact Analytic Solutions ◽  
Rosenau Equation ◽  
Physical Phenomena ◽  
Similarity Reductions

The Lie symmetry analysis is performed on the Rosenau equation which arises in modeling many physical phenomena. The similarity reductions and exact solutions are presented. Then the exact analytic solutions are considered by the power series method.

Exact solutions for steady bubbles in a Hele-Shaw cell with rectangular geometry

2001 ◽  
Vol 444 ◽  
pp. 175-198 ◽  
Author(s):  
GIOVANI L. VASCONCELOS
Keyword(s):  
Surface Tension ◽  
Exact Solutions ◽  
Arbitrary Number ◽  
Integral Form ◽  
Analytic Solutions ◽  
Elliptic Integrals ◽  
General Solutions ◽  
Special Cases ◽  
Hele Shaw Cell

Exact solutions are presented for an arbitrary number of steadily moving bubbles in a Hele-Shaw channel when surface tension is neglected. According to the symmetry displayed by the bubbles, the solutions are classified into two groups: (i) solutions where the bubbles are symmetrical about the channel centreline and (ii) solutions in which the bubbles have fore-and-aft symmetry. The general solutions are expressed in integral form but in some special cases analytic solutions in terms of elliptic integrals are found. The possible relevance of these exact solutions to experiments is also briefly discussed.

Numerical Methods for Solving Physically Nonlinear Problems for Inhomogeneous Thick-Walled Shells

2017 ◽  
Vol 865 ◽  
pp. 325-330 ◽  
Author(s):  
Vladimir I. Andreev ◽  
Lyudmila S. Polyakova
Keyword(s):  
Numerical Method ◽  
Nonlinear Problems ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Deformation Diagram ◽  
Method Of Solution ◽  
Properties Of Materials ◽  
Physical Fields ◽  
Cylinders And Spheres ◽  
Numerical Method Of Solution

The paper proposes the numerical method of solution the problems of calculation the stress state in thick-walled cylinders and spheres from physically nonlinear inhomogeneous material. The urgency of solved problem due to the change of mechanical properties of materials under the influence of different physical fields (temperature, humidity, radiation, etc.). The deformation diagram describes the three-parameter formula. The numerical method used the method of successive approximations. The results of numerical calculation are compared with the test analytical solutions obtaining the authors with some restrictions on diagram parameters. The obtained results can be considered quite satisfactory.

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