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Journal ArticleDOI

Accurate approximate analytical solutions for multi-degree-of-freedom coupled van der Pol-Duffing oscillators by homotopy analysis method

TL;DR: In this article, the homotopy analysis method (HAM) is presented to establish the accurate approximate analytical solutions for multi-degree-of-freedom (MDOF) nonlinear coupled oscillators.
About: This article is published in Communications in Nonlinear Science and Numerical Simulation.The article was published on 2010-10-01. It has received 26 citations till now. The article focuses on the topics: Homotopy analysis method & Homotopy.
Citations
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Journal ArticleDOI
TL;DR: In this article, a survey of recent developments in asymptotic techniques and new developments, which are valid not only for weakly nonlinear equations, but also for strongly ones, is presented.
Abstract: This review features a survey of some recent developments in asymptotic techniques and new developments, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the achieved approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modified perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to over-come the shortcomings.In this review we have applied different powerful analytical methods to solve high nonlinear problems in engineering vibrations. Some patterns are given to illustrate the effectiveness and convenience of the methodologies.

69 citations

Journal ArticleDOI
TL;DR: In this article, a two-dimensional magnetohydrodynamic boundary layer flow of the Eyring-Powell fluid on a stretching surface in the presence of thermal radiation and Joule heating is analyzed.
Abstract: A two-dimensional magnetohydrodynamic boundary layer flow of the Eyring–Powell fluid on a stretching surface in the presence of thermal radiation and Joule heating is analyzed. The Soret and Dufour effects are taken into account. Partial differential equations are reduced to a system of ordinary differential equations, and series solutions of the resulting system are derived. Velocity, temperature, and concentration profiles are obtained. The skin friction coefficient and the local Nusselt and Sherwood numbers are computed and analyzed.

46 citations

01 Aug 2003
TL;DR: In this paper, the existence of limit cycles of coupled van der Pol equations by using S1-degree theory due to Dylawerski et al. was considered, and it was shown that limit cycles can be found in the limit cycle of the coupled Van der Pol equation.
Abstract: Abstract In this paper, we consider the existence of limit cycles of coupled van der Pol equations by using S1-degree theory due to Dylawerski et al. (see Ann. Polon. Math. 62 (1991) 243).

36 citations

Journal ArticleDOI
TL;DR: In this article, the residue harmonic balance is developed for coupled systems exemplified by the damped Duffing resonator driven by a van der Pol oscillator, which combines the features of harmonic balance and parameter bookkeeping to obtain approximate solutions to any desired accuracy.

27 citations

Journal ArticleDOI
TL;DR: The residue harmonic balance method is presented to find periodic solutions of Duffing–van der Pol oscillators having damping terms described by fractional derivative and time delay respectively to investigate the damping effects of these two oscillators.

25 citations

References
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27 Oct 2003
TL;DR: In this paper, a simple bifurcation of a nonlinear problem multiple solutions of a Nonlinear Problem Nonlinear Eigenvalue Problem Thomas-Fermi Atom Model Volterra's Population Model Free Oscillation Systems with Odd Nonlinearity Free oscillations with Quadratic nonlinearity Limit Cycle in a Multidimensional System Blasius' viscous flow Boundary-layer Flow Boundarylayer Flow with Exponential Property Boundary Layer Flow with Algebraic Property Von Karman Swirling Flow Nonlinear Progressive Waves in Deep Water BIBLIOGR
Abstract: PART I BASIC IDEAS Introduction Illustrative Description Systematic Description Relations to Some Previous Analytic Methods Advantages, Limitations, and Open Questions PART II APPLICATIONS Simple Bifurcation of a Nonlinear Problem Multiple Solutions of a Nonlinear Problem Nonlinear Eigenvalue Problem Thomas-Fermi Atom Model Volterra's Population Model Free Oscillation Systems with Odd Nonlinearity Free Oscillation Systems with Quadratic Nonlinearity Limit Cycle in a Multidimensional System Blasius' viscous Flow Boundary-layer Flow with Exponential Property Boundary-layer Flow with Algebraic Property Von Karman Swirling Flow Nonlinear Progressive Waves in Deep Water BIBLIOGRAPHY INDEX

2,831 citations

Journal ArticleDOI
TL;DR: A powerful, easy-to-use analytic tool for nonlinear problems in general, namely the homotopy analysis method, is further improved and systematically described through a typical nonlinear problem, i.e. the algebraically decaying viscous boundary layer flow due to a moving sheet.

1,589 citations

Journal ArticleDOI
TL;DR: In this article, the basic ideas and current developments of the homotopy analysis method, an analytic approach to get convergent series solutions of strongly nonlinear problems, which recently attracts interests of more and more researchers, are described.

835 citations

Journal ArticleDOI
TL;DR: In this paper, an optimal homotopy analysis approach is described by means of the nonlinear Blasius equation as an example, which can be used to get fast convergent series solutions of different types of equations with strong nonlinearity.

822 citations

Journal ArticleDOI
TL;DR: In this paper, the homotopy analysis method (HAM) is compared with the numerical and HPM in the heat transfer file and the auxiliary parameter ℏ, which provides a simple way to adjust and control the convergence region of solution series.

643 citations