An improved homotopy perturbation method for solving local fractional nonlinear oscillators

Abstract

Local fractional nonlinear oscillators are studied by a modification of the homotopy perturbation method coupled with the variational iteration method. In the solution process, the fractional variational iteration method is adopted to transform the nonlinear oscillator equation into an integral equation, which is then decomposed into a series of equations, which are solved by the homotopy perturbation method. Two examples are given to show that the proposed method is simpler and more flexible than the classical homotopy perturbation method. It is also shown that the key to use this method is to select the appropriate decomposition of the initial value series.

A new fractal nonlinear Burgers' equation arising in the acoustic signals propagation

“…20 The travelling-wave transformation of nondifferentiable type was given to solve the fractal Korteweg-de Vries equation. 21 Making use of the fractal-time-space and fractal-space diffusion, we easily give the nonlinear local fractional Burgers' equation, eg,…”

“…The fractal damped wave equation was considered in Ait Touchent et al The fractal Klein‐Gordon equation was proposed in Kumar et al The fractal‐time‐space wave equation was presented in Hemeda et al The fractal Laplace equation was investigated in Ziane et al The fractal finite‐dimensional heat‐conduction equation was proposed in Debbouche and Antonov . The travelling‐wave transformation of nondifferentiable type was given to solve the fractal Korteweg‐de Vries equation . Making use of the fractal‐time‐space and fractal‐space diffusion, we easily give the nonlinear local fractional Burgers' equation, eg, which is considered as the description for the acoustic signals propagation in the fractal stratified media, where θ is a constant, and ∂ 2 β / ∂u β ∂φ β , ∂ β / ∂u β , ∂ 2 β / ∂γ 2 β and ∂ 3 β / ∂u 3 β are the local fractional partial derivative operators (for the definitions of the local fractional partial derivative operators, see Section ).…”

Non-differentiable solutions of a family of modified Korteweg-de Vries equations within local fractional derivative

“…The results show that all these MKdV equations even with the terms of higher derivatives have common solutions. This property is important in onlinear scienence and solitary theory, and the results can be extended to other fractional differential equations of fractal differential equations [12][13][14][15][16][17][18][19].…”

The extended variational iteration method for local fractional differential equation

“…The local fractional calculus was firstly introduced by Yang [14], which could be used as a powerful tool to describing the motion of a fluid in a porous medium, and this calculus currently has a wide range of physical applications, such as [15][16][17][18][19][20].…”