Michael Shatalov - Academia.edu (original) (raw)

Papers by Michael Shatalov

International Journal of Applied Mathematics - IAENG, 2020

The Morse potential, which is the most widely used potential in evaluating the vibrational energi... more The Morse potential, which is the most widely used potential in evaluating the vibrational energies of diatomic molecules, is studied and its unknown parameters were estimated in this paper. The integral-differential and integral approaches which are relatively more accurate approaches of the objective least squares function method were discussed and applied in this paper. The approaches were used to estimate the Classical and Generalized Morse potential parameters. The estimates obtained for the Classical Morse potential was used to obtain the Morse potential parameters. The approaches were used to identify new unknown parameters of the Classical and Generalized Morse potential. They were also used to approximate parameters which were fundamental to graphically identifying the potentials using potential energy curves. The approach consists of recognizing the functional form of the hybrid forms of the Morse potential as the solution of some second order ordinary differential equatio...

Nanoscience & Nanotechnology - Asia, 2020

This paper proposes an approach for parameter estimation of the Classical and Generalized Morse p... more This paper proposes an approach for parameter estimation of the Classical and Generalized Morse potential functions. A new potential, Modified Generalized Morse potential which is a modification of the Generalized Morse potential was proposed as parameter estimates yielded complex conjugate roots using gold atom for simulation. Existing methods of parameter estimation requires the provision of initial guess values of which convergence to the optimal solution is not always guaranteed. This makes provision of initial guess values that guarantees convergence to the optimum solution more of an art than a science. The proposed objective least squares function method does not require provision of initial guess values and it involves the minimization of two formulated objective functions using the differential numerical approach and least squares method. The built-in “Minimize” function of Mathematica is also used to minimize the formulated objective function. Potential energy curves were ...

Журнал вычислительной математики и математической физики

The Extended-Rydberg potential has wide applicability in determining the properties of diatomic m... more The Extended-Rydberg potential has wide applicability in determining the properties of diatomic molecules. In this paper, we estimate the Extended-Rydberg potential using a novel approach based on the objective least square functions of differential, integral-differential and integral approaches for the estimation of the potential. Interesting research results are obtained as the numerical differentiation (differential approach), integration (integral-differential and integral approach) are in agreement with the experimental data sets of gold atoms. It is a well-known fact that the more parameters a semiempirical interatomic potential has, the more flexible and accurate it is for experimental curve fitting but it takes longer computational time. We establish via CPU time the efficiency and novelty of our approach for the five-parameter Extended-Rydberg potential.

Background: The knowledge of parameter estimation for interatomic potentials is useful in the com... more Background: The knowledge of parameter estimation for interatomic potentials is useful in the computation of the vibrational structure of van der Waals molecules. Methods: On the estimation of the Generalized Morse and Classical Lennard-Jones potential energy functions, complex conjugates eigenvalues may be obtained. Different approaches can be used to solve this resulting problem. A method that uses the objective least squares function method to estimate parameters of the interatomic potentials is employed. Results: Numerical simulation of the systems using metal atoms yields complex conjugates eigenvalues at some initial point. Conclusion: Other approaches of solving the complex conjugates eigenvalues problem are discussed comprehensively.

A simplified diffusive predator-prey model of the Lotka-Volterra type is considered for annular ... more A simplified diffusive predator-prey model of the Lotka-Volterra type is considered for annular habitat which is used for description of predator and prey coexistence at habitats surrounding lakes, mountains at particular heights,etc. The model is formulated as a system of two partial differential equations in which unknown populations of the predator and prey are described by functions depending on time and polar angle. A mixed problem is formulated so that the boundary conditions are 2pi2\pi2pi-periodic and in the initial conditions is assumed that populations of the predator and prey are completely separated on the annular habitat. The problem is solved by the method of lines by means of which the original system of partial differential equation is converted to the system of several hundred nonlinear ordinary differential equations. It's shown that predator and prey start slowly propagating through the annular habitat and their interaction commences after a time interval in the course of which the population of the predator is decreasing and the population of the prey is increasing. An intensive interaction of the predator and prey occurs after their meeting. The dynamics of the transient populations of predator and prey and the tendency of their steady state is analyzed.

BIOMATH

In this paper we undertake to consider the inverse problem of parameter identification of nonline... more In this paper we undertake to consider the inverse problem of parameter identification of nonlinear system of ordinary differential equations for a specific case of complete information about solution of the Holling-Tanner model for finite number of points for the finite time interval. In this model the equations are nonlinearly dependent on the unknown parameters. By means of the proposed transformation the obtained equations become linearly dependent on new parameters functionally dependent on the original ones. This simplification is achieved by the fact that the new set of parameters becomes dependent and the corresponding constraint between the parameters is nonlinear. If the conventional approach based on introduction of the Lagrange multiplier is used this circumstance will result in a nonlinear system of equations. A novel algorithm of the problem solution is proposed in which only one nonlinear equation instead of the system of six nonlinear equations has to be solved. Diff...

In this paper a methodology for the recognition of multiple Gaussian patterns by estimating suffi... more In this paper a methodology for the recognition of multiple Gaussian patterns by estimating sufficient parameters of a finite mixture model (FMM) is proposed. Regular methods of FMM identification require initial guess values (IGVs) that may result in high computation time, slow convergence and or even fail to converge if the provided IGVs are far from the optimal solution. The FMM is firstly decomposed into it's even and odd parts, which are linearised through differential techniques. Secondly the ordinary least squares (OLS) method is employed to estimate the unknown parameters in the linearised models. A Monte Carlo simulation is done to evaluate the performance of the proposed method (PM). It is shown that numerical results of the PM compare well with the simulated values. The study indicates that (i) the PM can be used symbiotically with the regular methods to compute IGVs; and (ii) can be used to estimate a general n-component Gaussian model.

Applied Mathematics and Computation, 2014

ABSTRACT In 1890 G.H. Bryan observed that when a vibrating structure is rotated with respect to i... more ABSTRACT In 1890 G.H. Bryan observed that when a vibrating structure is rotated with respect to inertial space, the vibrating pattern rotates at a rate proportional to the inertial rate of rotation. This effect, called “Bryan’s effect”, as well as the proportionality constant, called “Bryan’s factor”, have numerous navigational applications. Using a computer algebra system, we present a numerically accurate method for determining fundamental eigenvalues (and some of the overtone eigenvalues) as well as the corresponding eigenfunctions for a linear ordinary differential equation (ODE) boundary value problem (BVP) associated with a slowly rotating vibrating disc. The method provides easy and accurate calculation of Bryan’s factor, which is used to calibrate the resonator gyroscopes used for navigation in deep space missions, stratojets and submarines. Bryan used “thin shell theory” to calculate Bryan’s factor for fundamental vibrations. Apart from the high accuracy achieved, the numerical routine used here is more robust than “thin shell theory” because it determines (at least for low frequencies) the fundamental as well as the first three overtone frequencies and each Bryan’s factor associated with these vibration modes. The theory involved and the calculation of results with this numerical method are quick, easy and accurate and might be applied in other disciplines that need to solve suitable eigenvalue problems. Indeed, results are obtained directly using commercial software to numerically solve a system of linear ODE BVPs without having to formulate the extremely technical solution that is traditionally used (viz: solve the governing system of partial differential equations via Helmholtz potential functions and the necessary numerical calculation of Bessel and Neumann functions).

Theoretical Foundations of Chemical Engineering, 2006

Journal of Sound and Vibration, 2009

When a vibrating structure is rotated with respect to inertial space, the vibrating pattern rotat... more When a vibrating structure is rotated with respect to inertial space, the vibrating pattern rotates at a rate proportional to the inertial rate of rotation. Bryan first observed this effect in 1890. The effect, called Bryan&amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;amp;#39;s effect in the sequel, has numerous navigational applications and could be useful in understanding the dynamics of pulsating stars and earthquake series in astrophysics

Days on Diffraction, St. Petersburg, Russia 2006 It was found by G. Bryan in 1890 that vibrating ... more Days on Diffraction, St. Petersburg, Russia 2006 It was found by G. Bryan in 1890 that vibrating pattern of a rotating ring follows to a direction of the inertial rotation of this ring with an angular rate of the vibrating pattern smaller than the inertial rate. In 1979 E. Loper and D. Lynch proposed a hemispherical vibrating bell gyroscope utilising the Bryan’s effect, which can measure an inertial angular rate and angle of rotation about the symmetry axis of the hemispherical shell. All these works exploited the precession properties of thin vibrating shells subjected to an inertial rotation around their axes of symmetry. In 1985 V. Zhuravlev generalized the abovementioned results and shown that the Bryan’s effect has a three dimensional nature, i.e. that a vibrating pattern of an isotropic spherically symmetric body, arbitrary rotating in 3-D space, follows the inertial rotation of the solid body with a proportionality factor depending on the vibrating mode. This result had a qua...

Methods like the maximum likelihood and Newton's iterative techniques are usually employed to est... more Methods like the maximum likelihood and Newton's iterative techniques are usually employed to estimate the parameters of the classical Rydberg interatomic potential function. However, such approaches require initial guess values (IGV) to compute the optimal solutions of the unknowns. In this work, a multiple objective function approach that is used to compute the parameter estimates via the least-squares method is presented. The new approach does not necessitate the uniqueness and existence of the initial guess values to compute the required estimates. The classical Rydberg interactomic potential is firstly considered as a general solution of a second order linear differential equation with constant coefficients. Secondly, the characteristic equation of the general solution of the potential function is formulated and it's constraints derived. Two objective functions are constructed from the assumed ordinary differential equation. The first objective function is constrained and selected parameters are estimated using the least-squares method and the second objective function is directly minimised by formulating normal equations that can be solved for exact solutions of the unknowns. The new method and the computer algebraic system (CAS) are tested using experimental datasets of copper, silver ions and silver-copper alloy. Estimates from both approaches are compared and used to construct potential energy curves of the atoms considered. Three potential surfaces, that is, for experimental data, new method and the CAS for each atom are plotted on the same axis for better comparison. It is observed that estimates from the new method are greater in absolute terms than those from the CAS. However, all the estimates have the same direction. The potential curves from the experimental data, the new method and the CAS all have the same shape and close to each other, especially at the minimum of the potential well and at that point they are all indistinguishable. Estimates from the new method can be trusted as they produce a potential curve that is close to that of the experimental data and the CAS. As the present iterative methods usually converge to the required solutions given " good " IGVs, it is proposed that the new method be used symbiotically with the present methods to systematically compute the necessary IGVs. Hence the new method can be used in both theoretical and practical applications to estimate parameters of the Rydberg potential function since it does not require provision of IGVs to the unknown parameters.

British Journal of Mathematics & Computer Science, 2016

Two approaches to parameter estimation for a mixture of two univariate Gaussian distributions are... more Two approaches to parameter estimation for a mixture of two univariate Gaussian distributions are numerically compared. The proposed method (PM) is based on decomposing a Continuous function into its odd and even components and estimating them as polynomials, the other is the usual maximum likelihood (ML) method via the expected maximisation (EM) algorithm. An overlapped mixture of two univariate Gaussian distributions is simulated. The PM and ML are used to re-estimate the known mixture model parameters and the measure of performance is the absolute percentage error. The PM produces comparable results to those of to the ML approach. Given that the PM produces good estimates, and knowing that the ML always converges given good initial guess values (IGVs), it is thus recommended that the PM be used symbiotically with the ML to provide IGVs for the EM algorithm.

Abstract Two new approaches (method I and II) for estimating parameters of a univariate normal pr... more Abstract Two new approaches (method I and II) for estimating parameters of a univariate normal probability density function are proposed. We evaluate their performance using two simulated normally distributed univariate datasets and their results compared with those obtained from the maximum likelihood (ML) and the method of moments (MM) approaches on the same samples, small n = 24 and large n = 1200 datasets. The proposed methods, I and II have shown to give significantly good results that are comparable to those from the standard methods in a real practical setting. The proposed methods have performed equally well as the ML method on large samples. The major advantage of the proposed methods over the ML method is that they do not require initial approximations for the unknown parameters. We therefore propose that in the practical setting, the proposed methods be used symbiotically with the standard methods to estimate initial approximations at the appropriate step of their algorit...

Transcendental models are often solved by using a different approach, which can be a derivative f... more Transcendental models are often solved by using a different approach, which can be a derivative free, direct optimisation or iterative linearization method. All these approaches require guess values for the unknown parameters to start the iteration procedure. However, if the transcendental model involves
several parameters, some of these methods become very cumbersome and computationally expensive. A new method for computing parameter estimates which are then used as initial values for the unknown model parameters to start the iteration process was proposed. Confidence intervals for the estimated parameters were constructed using the bootstrap method. We generated two randomised datasets that
simulated the decay and growth processes. A three parameterized single exponential model f (x)  exp(x)  was identified using the simulated datasets in each case. The absolute percentage errors were used as a measure of comparison between the proposed method and the
current Levenberg-Marquardt (L-M) method. Tables and figures were used to present results from both methods. The proposed method appeared to produce better results than the current L-M method. The superiority of the proposed method over the current methods is that it does not require initial guess
values and it guarantees convergences. Thus the proposed method could be adopted to solve real life problems.

The equations of motion for a dynamically adjustable gyroscope are derived in terms of the Lagran... more The equations of motion for a dynamically adjustable gyroscope are derived in terms of the Lagrange function, of generalized coordinates and velocities and of time, this function assumed to be analytic. The derivation is based on applying an Euler operator to the corresponding Lagrange equation of the second kind and analyzing the Lagrange function as the difference between kinetic energy and potential energy. Linear differential equations of motion are derived in this way, first for a gyroscope with single wheel and then for one with n wheels in tandem. The method is also applicable to derivation of linearized nonlinear equations or their higher-order approximations.