scholarly journals APPROXIMATE SOLUTIONS OF DAMPED NON LINEAR SYSTEM WITH VARYING PARAMETER AND DAMPING FORCE

2019 ◽  
pp. 13-18
Author(s):  
REZAUL KARIM ◽  
PINAKEE DEY ◽  
SOMI AKTER ◽  
MOHAMMAD ASIF AREFIN ◽  
SAIKH SHAHJAHAN MIAH
Keyword(s):  
Approximate Solution ◽  
Dynamical Systems ◽  
Linear System ◽  
Harmonic Balance ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Damping Force ◽  
Kbm Method ◽  
Non Linear System

The study of second-order damped nonlinear differential equations is important in the development of the theory of dynamical systems and the behavior of the solutions of the over-damped process depends on the behavior of damping forces. We aim to develop and represent a new approximate solution of a nonlinear differential system with damping force and an approximate solution of the damped nonlinear vibrating system with a varying parameter which is based on Krylov–Bogoliubov and Mitropolskii (KBM) Method and Harmonic Balance (HB) Method. By applying these methods we solve and also analyze the finding result of an example. Moreover, the solutions are obtained for different initial conditions, and figures are plotted accordingly where MATHEMATICA and C++ are used as a programming language.

scholarly journals Approximate Solution of a Fourth Order Weakly Non-Linear Differential System with Strong Damping and Slowly Varying Coefficients by Unified KBM Method

1970 ◽  
Vol 34 (1) ◽  
pp. 71-82
Author(s):  
M Alhaz Uddin ◽  
MA Sattar
Keyword(s):  
Approximate Solution ◽  
Differential System ◽  
Fourth Order ◽  
Initial Conditions ◽  
Numerical Procedure ◽  
The Other ◽  
Oscillatory Process ◽  
Strong Damping ◽  
Varying Coefficients ◽  
Kbm Method

The unified Krylov-Bogoliubov-Mitropolskii (KBM) method is used for determining theanalytical approximate solution of a fourth order weakly nonlinear differential system with strongdamping and slowly varying coefficients when a pair of eigen-values of the unperturbed equationis a multiple (approximately or perfectly) of the other pair or pairs. In a damped case, one of thenatural frequencies of the linearized equation may be a multiple of the other. The analytical firstorder approximate solution for different initial conditions shows a good coincidence with thoseobtained by the numerical procedure. The method is illustrated by an example.Key words: Perturbation method; Weak nonlinearity; Oscillatory process; Strong damping; Varying coefficientsDOI: 10.3329/jbas.v34i1.5493Journal of Bangladesh Academy of Sciences, Vol.34, No.1, 71-82, 2010

Comparison of the approximate solutions of a noisy non-linear system based on Monte Carlo simulation

1976 ◽  
Vol 24 (5) ◽  
pp. 719-731
Author(s):  
Y. SAWARAGI ◽  
T. SOEDA ◽  
T. NAKAMIZO ◽  
S. OMATU ◽  
Y. TOMITAS
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Linear System ◽  
Approximate Solutions ◽  
Non Linear ◽  
Non Linear System

scholarly journals The Laplace-Adomian-Pade Technique for the ENSO Model

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Yi Zeng
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Decomposition Methods ◽  
High Accuracy ◽  
Padé Method

The Laplace-Adomian-Pade method is used to find approximate solutions of differential equations with initial conditions. The oscillation model of the ENSO is an important nonlinear differential equation which is solved analytically in this study. Compared with the exact solution from other decomposition methods, the approximate solution shows the method’s high accuracy with symbolic computation.

scholarly journals Fractional Multi-Step Differential Transformed Method for Approximating a Fractional Stochastic SIS Epidemic Model with Imperfect Vaccination

2019 ◽  
Vol 16 (6) ◽  
pp. 973 ◽  
Author(s):  
Salah Abuasad ◽  
Ahmet Yildirim ◽  
Ishak Hashim ◽  
Samsul Abdul Karim ◽  
J.F. Gómez-Aguilar
Keyword(s):  
Dynamical Systems ◽  
Epidemic Model ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
Graphical Representations ◽  
Mathematical Ecology ◽  
Sis Epidemic Model ◽  
Stochastic Sis Epidemic Model ◽  
Sis Epidemic

In this paper, we applied a fractional multi-step differential transformed method, which is a generalization of the multi-step differential transformed method, to find approximate solutions to one of the most important epidemiology and mathematical ecology, fractional stochastic SIS epidemic model with imperfect vaccination, subject to appropriate initial conditions. The fractional derivatives are described in the Caputo sense. Numerical results coupled with graphical representations indicate that the proposed method is robust and precise which can give new interpretations for various types of dynamical systems.

scholarly journals Deterministic Chaos Theory: Basic Concepts

2016 ◽  
Vol 39 (1) ◽  
Author(s):  
Mauro Cattani ◽  
Iberê Luiz Caldas ◽  
Silvio Luiz de Souza ◽  
Kelly Cristiane Iarosz
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Chaos Theory ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Deterministic Chaos ◽  
Chaotic Motions ◽  
And Mathematics ◽  
Irregular Behavior ◽  
Mathematics And Physics

This article was written to students of mathematics, physics and engineering. In general, the word chaos may refer to any state of confusion or disorder and it may also refer to mythology or philosophy. In science and mathematics it is understood as irregular behavior sensitive to initial conditions. In this article we analyze the deterministic chaos theory, a branch of mathematics and physics that deals with dynamical systems (nonlinear differential equations or mappings) with very peculiar properties. Fundamental concepts of the deterministic chaos theory are briefly analyzed and some illustrative examples of conservative and dissipative chaotic motions are introduced. Complementarily, we studied in details the chaotic motion of some dynamical systems described by differential equations and mappings. Relations between chaotic, stochastic and turbulent phenomena are also commented.

scholarly journals The Novel Integral Homotopy Expansive Method

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1204
Author(s):  
Uriel Filobello-Nino ◽  
Hector Vazquez-Leal ◽  
Jesus Huerta-Chua ◽  
Jaime Ramirez-Angulo ◽  
Darwin Mayorga-Cruz ◽  
...  
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
The Novel ◽  
Practical Applications ◽  
Linear And Nonlinear ◽  
Scientific Phenomena ◽  
Analytical Expressions

This work proposes the Integral Homotopy Expansive Method (IHEM) in order to find both analytical approximate and exact solutions for linear and nonlinear differential equations. The proposal consists of providing a versatile method able to provide analytical expressions that adequately describe the scientific phenomena considered. In this analysis, it is observed that the proposed solutions are compact and easy to evaluate, which is ideal for practical applications. The method expresses a differential equation as an integral equation and expresses the integrand of the equation in terms of a homotopy. As a matter of fact, IHEM will take advantage of the homotopy flexibility in order to introduce adjusting parameters and convenient functions with the purpose of acquiring better results. In a sequence, another advantage of IHEM is the chance to distribute one or more of the initial conditions in the different iterations of the proposed method. This scheme is employed in order to introduce some additional adjusting parameters with the purpose of acquiring accurate analytical approximate solutions.

scholarly journals A Unified KBM Method for Obtaining the Second Approximate Solution of a Third Order Weakly Non-linear Differential System with Strong Damping and Slowly Varying Coefficients

2011 ◽  
Vol 35 (1) ◽  
pp. 77-89
Author(s):  
M Alhaz Uddin ◽  
MAM Talukder ◽  
M Hasanuzzaman ◽  
MST Mumtahinah
Keyword(s):  
Approximate Solution ◽  
Differential System ◽  
Initial Conditions ◽  
Numerical Procedure ◽  
Oscillatory Process ◽  
Third Order ◽  
Strong Damping ◽  
Varying Coefficients ◽  
Kbm Method ◽  
Linear Differential

To obtain the second order approximate solution of a third order weakly nonlinear ordinary differential system with strong damping and slowly varying coefficients modeling a damped oscillatory process is considered based on the extension of a unified Krylov-Bogoliubov- Mitropolskii (KBM) method. The asymptotic solution for different initial conditions shows a good coincidence with those obtained by the numerical procedure for obtaining the transient’s response. The method is illustrated by an example.DOI: http://dx.doi.org/10.3329/jbas.v35i1.7973Journal of Bangladesh Academy of Sciences, Vol.35, No.1, 77-89, 2011

scholarly journals Analytical solution of system of differential equations by variational iteration method

BIBECHANA ◽  
2015 ◽  
Vol 13 ◽  
pp. 77-86
Author(s):  
Jamshad Ahmed ◽  
Faizan Hussain
Keyword(s):  
Linear System ◽  
Analytical Solution ◽  
Differential Equations ◽  
Iteration Method ◽  
Graphical Representation ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Numerical Results ◽  
System Of Differential Equations ◽  
Non Linear System

In this paper, Varitational Iteration Method using He’s Polynomials is used to construct the exact as well as approximate solutions of differential equations. From the obtained numerical results, it has been observed that this proposed technique is very efficient and reliable for the solution of the linear and non-linear system of differential equations. Numerical results and graphical representation reflect the accuracy and effectiveness of the proposed modification.BIBECHANA 13 (2016) 77-86

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