Travelling wave solutions for a surface wave equation in fluid mechanics

Abstract

Orig i nal sci en tific pa per

A general sub-equation method to the burgers-like equation

“…For example, in the case s > 1, the solutions are the compounding of five elementary functions exp, arctan, tanh, tan, arcos, etc. These solutions can not be obtained by using other methods [4][5][6][7][8][9][10]. When s < 1 or s = 1, these solutions have included soliton solutions and rational or irrational solutions if we properly take the parameters.…”

“…Because of this, an important research area is connected to the obtaining of exact analytical solutions of such equations. In past decades, the most impressive methods for obtaining exact solutions have been presented such as the inverse scattering method [1], Backlund transformation method [2], the homogeneous balance method [3], tanh-function method [4], F-expansion method [5], sub-ordinary differential equation (ODE) method [6], exp-function method [7,8], (G′/G)-expansion method [9,10], etc.…”

A heuristic review on the homotopy perturbation method for non-conservative oscillators

“…There are alternative methods for nonlinear oscillators; some famous ones include the variational iteration method, [74][75][76][77][78][79] the exp-function method, [80][81][82][83] the variational theory, [82][83][84] the G'/G-expansion method, 85 the Bayesian inference method, 86 the barycentric rational interpolation collocation method, 87 and others. 88 This section focuses itself on a simple method to find the frequency-amplitude relationship of a nonlinear oscillator using He's frequency formulation, [89][90][91] which represents a genius idea in converting a nonlinear equation into a linear equation.…”

Quasi hyperbolic function expansion method and tanh-function method for solving vibrating string equation and elastic rod equation

“…Nonlinear vibration equations are widely appeared in many practical application, [1][2][3] many researches focused on amplitude-frequency relationship, but the wave-like solutions were rarely studied. For example, a bridge vibrating under winds can be expressed by a nonlinear vibration equation; the amplitude and the intrinsic frequency are the main sources of research; however, the bridge vibration behaves sometimes like wave travelling.…”

“…12,13 In this paper, we adopt the quasi hyperbolic function expansion method 14 and modified extended tanh-function (METF) method. 15 where c and k are arbitrary constants to be determined; we can rewrite equation (1) in the following ODE RðU; kU 0 ; ÀckU 0 ; Á Á ÁÞ ¼ 0…”