He–Laplace method for nonlinear vibration systems and nonlinear wave equations

Abstract

This study suggests a new approach for solving telegraph equation, commonly called damped wave equation, arising in electromagnetic waves and propagation of electrical signals. In this paper, He-Laplace method, formulated by He's variational iteration method and Laplace transformation, is used to find the exact solution or a closed approximate solution of differential equations. The most distinct aspect of this method is that there is no need to calculate integration for next iterations in recurrence relations and convolution theorem is kept away to calculate the Lagrange multipliers in Laplace transformation. Moreover, He's polynomials via homotopy perturbation method is introduced to bring down the computational work in nonlinear terms as Laplace transform has some limitation to nonlinear terms. The results obtained by proposed method indicate that this approach is easy to implement and converges rapidly to exact solution. Several problems are illustrated to demonstrate the accuracy and stability of this method.

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.…”

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.…”

Solving system of partial differential equations using variational iteration method with He's polynomials

“…To illustrate the basic concept of homotopy perturbation method, consider the following non-linear functional equation [15,17] A…”