scholarly journals Homotopy Analysis Method for a Conservative Nonlinear Oscillator with Fractional Power

2021 ◽  
Vol 09 (01) ◽  
pp. 31-40
Author(s):  
Huaxiong Chen ◽  
Yanyan Wang
Keyword(s):  
Homotopy Analysis Method ◽  
Nonlinear Oscillator ◽  
Fractional Power ◽  
Analysis Method ◽  
Homotopy Analysis

Approximate solution of a piecewise linear–nonlinear oscillator using the homotopy analysis method

2017 ◽  
Vol 24 (19) ◽  
pp. 4551-4562 ◽  
Author(s):  
Jixiong Fei ◽  
Bin Lin ◽  
Shuai Yan ◽  
Xiaofeng Zhang
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Homotopy Analysis Method ◽  
Nonlinear Oscillator ◽  
Piecewise Linear ◽  
Dynamical Behavior ◽  
Nonlinear Damping ◽  
Analysis Method ◽  
Bifurcation Diagrams ◽  
Homotopy Analysis

Most of the piecewise oscillators in engineering fields include nonlinear damping or stiffness and the contained damping or stiffness is strongly nonlinear, but to the authors’ knowledge little attention has been paid to those systems. Thus, in the present paper, a sinusoidal excited piecewise linear–nonlinear oscillator is analyzed. The mathematical model of the oscillator is described by a combination of a linear and a nonlinear differential equation which contains strong nonlinear terms of stiffness. An approximate solution for the oscillator is proposed by using the homotopy analysis method and matching method. The validity of the proposed solution is verified by comparing it with the exact solution. It is found that the approximate solution is in good agreement with the exact solution. The influence of some system parameters on the dynamical behavior of the oscillator is also investigated by the bifurcation diagrams of these parameters. From these bifurcation diagrams, one can observe the motion of the oscillator directly.

Limit Cycles of Nonlinear Oscillator Equations with Absolute Value by Means of the Homotopy Analysis Method

2015 ◽  
Vol 70 (3) ◽  
pp. 193-202 ◽  
Author(s):  
Jifeng Cui ◽  
Zhiliang Lin ◽  
Yinlong Zhao
Keyword(s):  
Limit Cycles ◽  
Homotopy Analysis Method ◽  
Nonlinear Oscillator ◽  
Fourier Expansion ◽  
Analysis Method ◽  
Absolute Value ◽  
Oscillating Systems ◽  
Highly Nonlinear ◽  
Homotopy Analysis ◽  
Oscillator Equations

AbstractAn analytic approach based on the homotopy analysis method is proposed to obtain the limit cycles of highly nonlinear oscillating equations with absolute value terms. The non-smoothness of the absolute value terms is handled by means of an iteration approach with Fourier expansion. Two typical examples are employed to illustrate the validity and flexibility of this approach. It has general meanings and thus can be used to solve many other highly nonlinear oscillating systems with this kind of non-smoothness.

HOMOTOPY ANALYSIS METHOD TO SOLVE BOUSSINESQ EQUATIONS

2015 ◽  
Vol 10 (3) ◽  
pp. 2825-2833
Author(s):  
Achala Nargund ◽  
R Madhusudhan ◽  
S B Sathyanarayana
Keyword(s):  
Homotopy Analysis Method ◽  
Degrees Of Freedom ◽  
Initial Approximation ◽  
Boussinesq Equations ◽  
Convergent Series ◽  
Boussinesq System ◽  
Analysis Method ◽  
Analytical Technique ◽  
Homotopy Analysis ◽  
Region Of Convergence

In this paper, Homotopy analysis method is applied to the nonlinear coupleddifferential equations of classical Boussinesq system. We have applied Homotopy analysis method (HAM) for the application problems in [1, 2, 3, 4]. We have also plotted Domb-Sykes plot for the region of convergence. We have applied Pade for the HAM series to identify the singularity and reflect it in the graph. The HAM is a analytical technique which is used to solve non-linear problems to generate a convergent series. HAM gives complete freedom to choose the initial approximation of the solution, it is the auxiliary parameter h which gives us a convenient way to guarantee the convergence of homotopy series solution. It seems that moreartificial degrees of freedom implies larger possibility to gain better approximations by HAM.

Approximate solutions of singular two-point BVPs by modified homotopy analysis method

2008 ◽  
Vol 372 (22) ◽  
pp. 4062-4066 ◽  
Author(s):  
A. Sami Bataineh ◽  
M.S.M. Noorani ◽  
I. Hashim
Keyword(s):  
Homotopy Analysis Method ◽  
Approximate Solutions ◽  
Analysis Method ◽  
Homotopy Analysis

scholarly journals Dynamics with Cattaneo–Christov heat and mass flux theory of bioconvection Oldroyd-B nanofluid

2020 ◽  
Vol 12 (8) ◽  
pp. 168781402093046 ◽  
Author(s):  
Noor Saeed Khan ◽  
Qayyum Shah ◽  
Arif Sohail
Keyword(s):  
Mass Transfer ◽  
Porous Media ◽  
Differential Equations ◽  
Mass Flux ◽  
Homotopy Analysis Method ◽  
Two Dimensional ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Transfer Flow ◽  
Insight Into

Entropy generation in bioconvection two-dimensional steady incompressible non-Newtonian Oldroyd-B nanofluid with Cattaneo–Christov heat and mass flux theory is investigated. The Darcy–Forchheimer law is used to study heat and mass transfer flow and microorganisms motion in porous media. Using appropriate similarity variables, the partial differential equations are transformed into ordinary differential equations which are then solved by homotopy analysis method. For an insight into the problem, the effects of various parameters on different profiles are shown in different graphs.

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