scholarly journals Stability problems and numerical integration on the Lie group SO(3) × R3 × R3

2018 ◽  
Vol 16 (1) ◽  
pp. 219-234
Author(s):  
Camelia Pop ◽  
Remus-Daniel Ene
Keyword(s):  
Nonlinear System ◽  
Numerical Integration ◽  
Lie Group ◽  
Asymptotic Method ◽  
Analytic Solutions ◽  
Stability Problems ◽  
Optimal Homotopy Asymptotic Method

AbstractThe paper is dealing with stability problems for a nonlinear system on the Lie group SO(3) × R3 × R3. The approximate analytic solutions of the considered system via Optimal Homotopy Asymptotic Method are presented, too.

scholarly journals Stability and Analytic Solutions of an Optimal Control Problem on the Schrödinger Lie Group

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 549-558
Author(s):  
Remus-Daniel Ene ◽  
Camelia Pop ◽  
Camelia Petrişor
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Numerical Simulations ◽  
Periodic Solutions ◽  
Nonlinear Stability ◽  
Lie Group ◽  
Asymptotic Method ◽  
Analytic Solutions ◽  
Optimal Homotopy Asymptotic Method

AbstractThe nonlinear stability and the existence of the periodic solutions for an optimal control problem on the Schrödinger Lie group are discussed. The analytic solutions via optimal homotopy asymptotic method of the dynamics and numerical simulations are presented, too.

scholarly journals Stability Problems and Analytical Integration for the Clebsch’s System

2019 ◽  
Vol 17 (1) ◽  
pp. 242-259
Author(s):  
Camelia Pop ◽  
Remus-Daniel Ene
Keyword(s):  
Numerical Integration ◽  
Periodic Orbits ◽  
Nonlinear Stability ◽  
Asymptotic Method ◽  
Approximate Solutions ◽  
Equilibrium States ◽  
Stability Problems ◽  
Analytical Integration ◽  
Optimal Homotopy Asymptotic Method

Abstract The nonlinear stability and the existence of periodic orbits of the equilibrium states of the Clebsch’s system are discussed.. Numerical integration using the Lie-Trotter integrator and the analytic approximate solutions using Multistage Optimal Homotopy Asymptotic Method are presented, too.

scholarly journals Approximate Solution of Nonlinear System of BVP Arising in Fluid Flow Problem

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
A. K. Alomari ◽  
N. Ratib Anakira ◽  
A. Sami Bataineh ◽  
I. Hashim
Keyword(s):  
Fluid Flow ◽  
Approximate Solution ◽  
Nonlinear System ◽  
Boundary Value Problems ◽  
Asymptotic Method ◽  
Series Solution ◽  
Flow Problem ◽  
Point Boundary ◽  
Optimal Homotopy Asymptotic Method ◽  
First Time

We extend for the first time the applicability of the Optimal Homotopy Asymptotic Method (OHAM) to find approximate solution of a system of two-point boundary-value problems (BVPs). The OHAM provides us with a very simple way to control and adjust the convergence of the series solution using the auxiliary constants which are optimally determined. Comparisons made show the effectiveness and reliability of the method.

scholarly journals The Optimal Homotopy Asymptotic Method for Solving Two Strongly Fractional-Order Nonlinear Benchmark Oscillatory Problems

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2218
Author(s):  
Mohd Taib Shatnawi ◽  
Adel Ouannas ◽  
Ghenaiet Bahia ◽  
Iqbal M. Batiha ◽  
Giuseppe Grassi
Keyword(s):  
Fractional Order ◽  
Asymptotic Method ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Auxiliary Function ◽  
Approximate Methods ◽  
Homotopy Analysis ◽  
Optimal Homotopy Asymptotic Method ◽  
Relativistic Oscillator ◽  
Convergence Of Approximate Solutions

This paper proceeds from the perspective that most strongly nonlinear oscillators of fractional-order do not enjoy exact analytical solutions. Undoubtedly, this is a good enough reason to employ one of the major recent approximate methods, namely an Optimal Homotopy Asymptotic Method (OHAM), to offer approximate analytic solutions for two strongly fractional-order nonlinear benchmark oscillatory problems, namely: the fractional-order Duffing-relativistic oscillator and the fractional-order stretched elastic wire oscillator (with a mass attached to its midpoint). In this work, a further modification has been proposed for such method and then carried out through establishing an optimal auxiliary linear operator, an auxiliary function, and an auxiliary control parameter. In view of the two aforesaid applications, it has been demonstrated that the OHAM is a reliable approach for controlling the convergence of approximate solutions and, hence, it is an effective tool for dealing with such problems. This assertion is completely confirmed by performing several graphical comparisons between the OHAM and the Homotopy Analysis Method (HAM).

An Optimal Homotopy Asymptotic Method Solution of Injection of a Newtonian Fluid Through One Side of a Long Vertical Channel

2011 ◽  
Vol 42 (3) ◽  
pp. 267-283
Author(s):  
Rehan Ali Shah ◽  
Saeed Islam ◽  
A. M. Siddiqui ◽  
Ishtiaq Ali ◽  
Manzoor Ellahi
Keyword(s):  
Newtonian Fluid ◽  
Asymptotic Method ◽  
Vertical Channel ◽  
Optimal Homotopy Asymptotic Method

scholarly journals A Drift-Free Left Invariant Control System on the Lie Group SO(3)×R3×R3

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Camelia Pop
Keyword(s):  
Control System ◽  
Numerical Integration ◽  
Lie Algebra ◽  
Lie Group ◽  
Poisson Structure ◽  
Dual Space ◽  
Free System ◽  
Geometrical Properties ◽  
Free Left ◽  
Invariant Control

A controllable drift-free system on the Lie group G=SO(3)×R3×R3 is considered. The dynamics and geometrical properties of the corresponding reduced Hamilton’s equations on g∗,·,·- are studied, where ·,·- is the minus Lie-Poisson structure on the dual space g∗ of the Lie algebra g=so(3)×R3×R3 of G. The numerical integration of this system is also discussed.

scholarly journals Optimal Homotopy Asymptotic Method for a thin film flow of a pseudo plastic fluid draining down or lifting up on a cylindrical surface

2013 ◽  
Vol 19 (4) ◽  
pp. 513-527
Author(s):  
Kamran Alam ◽  
M.T. Rahim ◽  
S. Islam ◽  
A.M. Sidiqqui
Keyword(s):  
Thin Film ◽  
Asymptotic Method ◽  
Cylindrical Surface ◽  
Film Flow ◽  
Vertical Cylinder ◽  
Stokes Number ◽  
Velocity Shear ◽  
Thin Film Flow ◽  
Fluid Velocities ◽  
Optimal Homotopy Asymptotic Method

In this study, the pseudo plastic model is used to obtain the solution for the steady thin film flow on the outer surface of long vertical cylinder for lifting and drainage problems. The non-linear governing equations subject to appropriate boundary conditions are solved analytically for velocity profiles by a modified homotopy perturbation method called the Optimal Homotopy Asymptotic method. Expressions for the velocity profile, volume flux, average velocity, shear stress on the cylinder, normal stress differences, force to hold the vertical cylindrical surface in position, have been derived for both the problems. For the non-Newtonian parameter ?=0, we retrieve Newtonian cases for both the problems. We also plotted and discussed the affect of the Stokes number St, the non-Newtonian parameter ? and the thickness ? of the fluid film on the fluid velocities.

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