scholarly journals Homotopy Analysis Method for the Rayleigh Equation Governing the Radial Dynamics of a Multielectron Bubble

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
F. A. Godínez ◽  
M. A. Escobedo ◽  
M. Navarrete
Keyword(s):  
Dynamical Systems ◽  
Liquid Helium ◽  
Analytical Solutions ◽  
Homotopy Analysis Method ◽  
Nonlinear Dynamical Systems ◽  
Rayleigh Equation ◽  
Analysis Method ◽  
Nonlinear Dynamical ◽  
Strongly Nonlinear ◽  
Homotopy Analysis

The homotopy analysis method is used to obtain analytical solutions of the Rayleigh equation for the radial oscillations of a multielectron bubble in liquid helium. The small order approximations for amplitude and frequency fit well with those computed numerically. The results confirm that the homotopy analysis method is a powerful and manageable tool for finding analytical solutions of strongly nonlinear dynamical systems.

Homotopy Analysis Method for Multi-Degree-of-Freedom Nonlinear Dynamical Systems

2010 ◽  
Author(s):  
W. Zhang ◽  
Y. H. Qian ◽  
M. H. Yao ◽  
S. K. Lai
Keyword(s):  
Dynamical Systems ◽  
Homotopy Analysis Method ◽  
Nonlinear Dynamical Systems ◽  
Analytical Approximation ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Nonlinear Dynamical ◽  
Homotopy Analysis ◽  
Practical Engineering ◽  
Proof Of Convergence

In reality, the behavior and nature of nonlinear dynamical systems are ubiquitous in many practical engineering problems. The mathematical models of such problems are often governed by a set of coupled second-order differential equations to form multi-degree-of-freedom (MDOF) nonlinear dynamical systems. It is extremely difficult to find the exact and analytical solutions in general. In this paper, the homotopy analysis method is presented to derive the analytical approximation solutions for MDOF dynamical systems. Four illustrative examples are used to show the validity and accuracy of the homotopy analysis and modified homotopy analysis methods in solving MDOF dynamical systems. Comparisons are conducted between the analytical approximation and exact solutions. The results demonstrate that the HAM is an effective and robust technique for linear and nonlinear MDOF dynamical systems. The proof of convergence theorems for the present method is elucidated as well.

Periodic Solutions for Coupled Van Der Pol Oscillators of Two-Degree-of-Freedom Solved by Homotopy Analysis Method

2009 ◽  
Author(s):  
Wei Zhang ◽  
Youhua Qian ◽  
Qian Wang
Keyword(s):  
Dynamical Systems ◽  
Analytical Solutions ◽  
Nonlinear Dynamical Systems ◽  
Degree Of Freedom ◽  
Analysis Method ◽  
Van Der Pol ◽  
Nonlinear Dynamical ◽  
Homotopy Analysis ◽  
Van Der Pol Oscillators ◽  
Two Degree Of Freedom

Innumerable engineering problems can be described by multi-degree-of-freedom (MDOF) nonlinear dynamical systems. The theoretical modelling of such systems is often governed by a set of coupled second-order differential equations. Albeit that it is extremely difficult to find their exact solutions, the research efforts are mainly concentrated on the approximate analytical solutions. The homotopy analysis method (HAM) is a useful analytic technique for solving nonlinear dynamical systems and the method is independent on the presence of small parameters in the governing equations. More importantly, unlike classical perturbation technique, it provides a simple way to ensure the convergence of solution series by means of an auxiliary parameter ħ. In this paper, the HAM is presented to establish the analytical approximate periodic solutions for two-degree-of-freedom coupled van der Pol oscillators. In addition, comparisons are conducted between the results obtained by the HAM and the numerical integration (i.e. Runge-Kutta) method. It is shown that the higher-order analytical solutions of the HAM agree well with the numerical integration solutions, even if time t progresses to a certain large domain in the time history responses.

One Dimensional Model of Martensitic Transformation Solved by Homotopy Analysis Method

2012 ◽  
Vol 67 (5) ◽  
pp. 230-238 ◽  
Author(s):  
Chen Xuan ◽  
Cheng Peng ◽  
Yongzhong Huo
Keyword(s):  
Phase Transition ◽  
Martensitic Transformation ◽  
Analytical Solutions ◽  
Homotopy Analysis Method ◽  
Lagrange Equation ◽  
Nonlinear Ordinary Differential Equation ◽  
Physical Parameters ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
The One

The homotopy analysis method (HAM) is applied to solve a nonlinear ordinary differential equation describing certain phase transition problem in solids. Both bifurcation conditions and analytical solutions are obtained simultaneously for the Euler-Lagrange equation of the martensitic transformation. HAM is capable of providing an analytical expression for the bifurcation condition to judge the occurrence of the phase transition, while other numerical techniques have difficulties in bifurcation analysis. The convergence of the analytical solutions on the one hand can be adjusted by the auxiliary parameter and on the other hand is always obtainable for all relevant physical parameters satisfying the bifurcation condition.

scholarly journals On the Solutions of Fractional Cauchy Problem Featuring Conformable Derivative

2018 ◽  
Vol 22 ◽  
pp. 01045 ◽  
Author(s):  
Mehmet Yavuz ◽  
Necati Özdemir
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Exact Solutions ◽  
Detailed Analysis ◽  
Analytical Solutions ◽  
Homotopy Analysis Method ◽  
Analysis Method ◽  
Conformable Derivative ◽  
Homotopy Analysis ◽  
Fractional Cauchy Problem

In this study, we have obtained analytical solutions of fractional Cauchy problem by using q-Homotopy Analysis Method (q-HAM) featuring conformable derivative. We have considered different situations according to the homogeneity and linearity of the fractional Cauchy differential equation. A detailed analysis of the results obtained in the study has been reported. According to the results, we have found out that our obtained solutions approach very speedily to the exact solutions.

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