الفهرس 
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Abstract
Di¤erential equation is one of the linchpins of modern mathematics which,
along with matrices, is essential for analyzing and solving complex problems
in engineering, physics, economics, biology, and even business.
The study of nonlinear oscillators equations in engineering and applied
mathematics have been a topic to intensive research for many years [Nayfeh
and Mook, 1979; Hagedorn, 1988]. Lately, many authors used various an-
alytical methods for solving nonlinear oscillation systems. Some of these
well-known methods are such as: iteration perturbation method [Özis and
Yildirim, 2009; Bayat et al, 2011a; Bayat et al, 2012 ], variational iteration
method [He, 2007a; Rafei et al, 2007; Barari et al, 2008; Bayat et al, 2012],
modi ed Lindstedt-Poincaré methods [He, 2002a; Özis and Yildirim, 2007a;
Shou and He, 2007], amplitude frequency formulation [He, 2008a; He, 2008b],
homotopy perturbation method [He, 2004a; Ganji, 2006; Bayat et al, 2012],
energy balance method [Mehdipour et al, 2010; Pakar and Bayat, 2011 ], and
others.
The application of micro-scale devices is continuously growing and the
microelectromechanical systems (MEMS) have become the interesting area
of research in recent years.
Du¢ ng equations describe many kinds of nonlinear oscillatory system in
physics, mechanics and engineering [Nayfeh, 1973; Nayfeh and Mook, 1979;
Mickens, 1996]. They are famous in nonlinear dynamic and have been pre-
sented considering various types of nonlinearity.
The Du¢ ng equation is a well-Known nonlinear di¤erential equation which
is related to many practical engineering systems such as the classical nonlinear
spring system with odd nonlinear restoring characteristics [Nayfeh and Mook,
1979] and more recently in di¤erent physical phenomena [Xie and Gao, 2009].
The variational iteration method proposed by He [He, 1999] used to obtain
an approximate analytical solutions for nonlinear problems. VIM in most
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