He–Laplace method for general nonlinear periodic solitary solution of vibration equations

Abstract

This paper suggests a universal approach to solve nonlinear periodic vibration equations by He-Laplace method, a coupling of He's perturbation method and Laplace transform. The nonlinear periodic solitary solution of vibration equation is used as an example to elucidate the effectiveness and simplicity of the method, only few iterations are needed to obtain an extremely accurate solution.

The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators

“…Nonlinear oscillations arise everywhere in our everyday life and engineering. As an exact solution might be too complex to be used for a practical application, many analytical methods have been used in open literature, for example, the variational iteration method, [1][2][3][4][5][6][7] the homotopy perturbation method, [8][9][10][11][12][13][14][15][16][17][18][19][20] He-Laplace method, [21][22][23] the variational approach [24][25][26][27][28][29] and the Hamiltonian approach. 30,31 The most important property of a nonlinear oscillator is the relationship between the frequency and its amplitude, the simplest method to estimate the frequency-amplitude relationship might be He's frequency formulation [32][33][34] and the max-min approach, 35,36 which are still under development and many modifications were proposed to improve the accuracy.…”

Approximate analytical solution for Phi-four equation with He’s fractal derivative

“…Our example shows that the proposed method is a simple and reliable algorithm. As the Laplace transform is widely coupled with the homotopy perturbation method or the variational iteration method to solve various nonlinear equations including fractional differential equations [29][30][31][32],and the He-Laplace transform sheds a bright light on fractal calculus.…”

Frequency and solution of an oscillator with a damping

“…The couple of the Laplace transform with either the variational iteration method [16][17][18] or the homotopy perturbation method [23][24][25] now is called as He-Laplace method [26][27][28][29] and it has obvious advantages in the simple solution process. Both methods discussed in this paper are also powerful tools for fractal calculus and fractional calculus.…”