The max–min approach to a relativistic equation

The Variational Approach Coupled with an Ancient Chinese Mathematical Method to the Relativistic Oscillator

“…Hereby we will introduce He Chengtian's interpolation [14] to approximate the above integral. He Chengtian's interpolation, an ancient Chinese mathematics, was developed to the max-min method [18][19][20][21][22] for nonlinear oscillators. By a simple analysis, we know that…”

Analysis of highly nonlinear oscillation systems using He’s max-min method and comparison with homotopy analysis and energy balance methods

“…Many nonlinear problems do not contain such perturbation quantity, so to overcome the shortcomings, many new techniques have appeared in open literature such as; harmonic balance (Hamdan & Shabaneh 1997), homotopy perturbation (Barari et al 2008;He 2004;Özis & Yildirim 2007a), modified Lindstedt-Poincaré (He 2001;He 2002b;Özis & Yildirim 2007b;Mohammadi et al 2009), Adomian decomposition (Dehghan & Shakeri 2008), amplitude-frequency formulation , parameterexpansion , parameterized perturbation (He 1999a), multiple scale (Okuizumi & Kimura 2004), energy balance (He 2006a;Momeni et al 2010), variational approach (He 2007;Xu 2008;Tolou et al 2009;Hashemi Kachapi et al 2009), Variational Iteration (He et al 2010;Herisanu & Marinca 2010), Newton-harmonic balancing (Lai et al 2009), differential transformation (Catal 2008), max-min (He 2008;Zeng & Lee 2009;Zeng 2009;Shen & Mo 2009).…”

On the Approximate Analytical Solution to Non-Linear Oscillation Systems

“…For most real-life nonlinear problems, it is not always possible and sometimes not even advantageous to express exact solutions of nonlinear differential equations explicitly in terms of elementary functions or independent spatial and/or temporal variables; however, it is possible to find approximate solutions. In recent years, many ingenious analytical methods have been developed for solving different kinds of strongly nonlinear equations, such as homotopy perturbation method [7,8], energy balance method [9][10][11][12], variational iteration method [13,14], variational approach [15][16][17][18], iteration perturbation method [19], Hamiltonian Approach [20], max-min approach [21][22][23][24], parameter expansion method [25], and other analytical and numerical methods [26][27][28][29][30].…”