scholarly journals Homotopy Analysis Method for Large-Amplitude Free Vibrations of Strongly Nonlinear Generalized Duffing Oscillators

2012 ◽  
Vol 02 (04) ◽  
pp. 167-175
Author(s):  
Youhua Qian ◽  
Dongxu Ren ◽  
Shengmin Chen ◽  
Lin Ping
Keyword(s):  
Homotopy Analysis Method ◽  
Large Amplitude ◽  
Free Vibrations ◽  
Analysis Method ◽  
Strongly Nonlinear ◽  
Homotopy Analysis

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.

scholarly journals Three new approaches for solving a class of strongly nonlinear two-point boundary value problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Monireh Nosrati Sahlan ◽  
Hojjat Afshari
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Homotopy Analysis Method ◽  
Boundary Value ◽  
Scaling Functions ◽  
Point Boundary ◽  
Analysis Method ◽  
Linearization Technique ◽  
Strongly Nonlinear ◽  
Homotopy Analysis

AbstractThree new and applicable approaches based on quasi-linearization technique, wavelet-homotopy analysis method, spectral methods, and converting two-point boundary value problem to Fredholm–Urysohn integral equation are proposed for solving a special case of strongly nonlinear two-point boundary value problems, namely Troesch problem. A quasi-linearization technique is utilized to reduce the nonlinear boundary value problem to a sequence of linear equations in the first method. Second method is devoted to applying generalized Coiflet scaling functions based on the homotopy analysis method for approximating the numerical solution of Troesch equation. In the third method we use an interesting technique to convert the boundary value problem to Urysohn–Fredholm integral equation of the second kind; afterwards generalized Coiflet scaling functions and Simpson quadrature are employed for solving the obtained integral equation. Introduced methods are new and computationally attractive, and applications are demonstrated through illustrative examples. Comparing the results of the presented methods with the results of some other existing methods for solving this kind of equations implies the high accuracy and efficiency of the suggested schemes.

Analytic Approximations for a Strongly Nonlinear Problem of Combined Convective and Radiative Cooling of a Spherical Body

2009 ◽  
Vol 131 (11) ◽  
Author(s):  
A. El-Nahhas
Keyword(s):  
Boundary Condition ◽  
Nonlinear Problem ◽  
Homotopy Analysis Method ◽  
Spherical Body ◽  
Radiative Cooling ◽  
Strong Nonlinearity ◽  
Analysis Method ◽  
Strongly Nonlinear ◽  
Analytic Approximations ◽  
Homotopy Analysis

In this paper, a nonlinear problem for combined convective and radiative cooling of a spherical body is considered. This problem represents a strong nonlinearity in both the governing equation and the boundary condition. Analytic approximations for the solution of this problem are obtained using the homotopy analysis method and via a polynomial exponential basis. Also, the effect of the radiation-conduction parameter Nrc and the Biot number Bi for the temperature on the surface of the spherical body is investigated and discussed.

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