Hybridization of homotopy perturbation method and Laplace transformation for the partial differential equations

Abstract

Homotopy perturbation method is combined with Laplace transformation to obtain approximate analytical solutions of non-linear differential equations. An example is given to elucidate the solution process and confirm reliability of the method. The result indicates superiority of the method over the conventional homotopy perturbation method due its flexibility in choosing its initial approximation.

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.…”

The study of heat transfer phenomena by using modified homotopy perturbation method coupled by Laplace transform

“…Thus, [1] reported a combination of homotopy perturbation method (HPM) and LT methods (LT-HPM) in order to obtain highly accurate approximate solutions for these equations. At this time, it is clafiried that the coupling of LT and HPM is known by another name in the literature: A modified homotopy perturbation method coupled by Laplace transform (MHPMLT) [5][6][7]. As a matter of fact, [5] named this modification as He-Laplace method for simplicity.…”

“…Non-linear problems frequently arise in science and engineering, whereby, it is very important to search on differential equations that describe them. In recent years, there have been proposed several methods focused to find approximate solutions to non-linear differential equations; such as those based on: variational approaches [12], tanh method [13], exp-function [14], Adomian's decomposition method (ADM) [15], parameter expansion [16], homotopy analysis method (HAM) [2,3], perturbation method [17], and HPM [1,[5][6][7][8][9][10][11][18][19][20][21][22][23], since the main solution process of this article is HPM, next we briefly mention some of the last developments of this method; such as the coupling of HPM and Frobenius method [20], multiple scales HPM method [21], parametrized HPM [22], non-linearities distribution HPM used to find the solution of Troesch problem [23], among many others.…”

The harmonic balance method for Yao–Cheng oscillator

“…Equation (1) can be effectively solved by the homotopy perturbation method, 12 which is generally effective for oscillators without a damping term. [13][14][15][16][17][18] The harmonic balance method…”