Application of He's Max-Min Approach to a Generalized Nonlinear Discontinuity Equation

He’s frequency formulation for nonlinear oscillators using a golden mean location

“…Another ancient Chinese method, called the He Chengtian inequality method, was also developed to solve nonlinear oscillators; see Refs. [11][12][13]. Zhong [14] and Lan and Yang [15] asserted that some complex nonlinear equations can be easily solved by ancient Chinese mathematical methods.…”

The modified homotopy perturbation method with an auxiliary term for the nonlinear oscillator with discontinuity

“…Setting p ¼ 1 in equation (11) and substituting for a 2 from equation (37) delivers the second-order approximate angular frequency…”

“…The periodic solution of a generalized nonlinear discontinuity equation using the max-min approach was discussed by Demirba g and Kaya. 11 Next, we review a class of iterative methods where higher order approximations can deliver more accurate solutions. The approaches in this class include, for example, the homotopy perturbation method (HPM), [12][13][14][15] the modified Lindstedt-Poincare´method, [16][17][18][19][20] the parameter-expansion method, [21][22][23][24] energy balance method, 25,26 and the variational iteration method.…”

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…”