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

Abstract

In engineering, a fast estimation of the periodic property of a nonlinear oscillator is much needed. This paper reviews some simplest methods for nonlinear oscillators, including He's frequency formulation, the max-min approach and the homotopy perturbation method. A mathematical insight into He's frequency formulation is given, and the weighted average is introduced to further improve the estimated accuracy of the frequency. Fractional oscillators are also discussed.

New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle

“…(37) where α and β are two-scale fractal dimensions in moving direction and time, respectively. Before proceeding further, we first give some definitions and theorems on fractal calculus for easy understanding [7,[61][62][63][64].…”

“…This transform makes the fractal calculus extremely simple in view of traditional calculus. Now the fractal calculus has been applied to non-linear vibration [62], biomechanics [63][64][65], electrochemical arsenic sensor [66], tsunami model [67], thermal insulation [68], fractal rate model [69], biomimic design [70,71], fractal diffusion [72], fractal filtration [73], and nanotechnology [74][75][76][77][78].…”

Numerical solution of second order Painlevé differential equation

“…The overhead strategy for finding the exact solution is known as variational iteration algorithm-I (VIA-I) proposed for the first time by a Chinese mathematicians Ji-Huan He [17][18][19]. The basic concept was taken from the general Lagrange multiplier method of Inokuti et al [23].…”

Solutions of sum‐type singular fractional q integro‐differential equation with m‐point boundary value problem using quantum calculus

“…Nowadays, it has some applications in the fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber reinforced concrete pillar, fractal variational theory for one-dimensional compressible flow in a microgravity space, electro-spinning process, and convection-diffusion equation for E-reaction arising in rotating disk electrodes. [1][2][3][4][5][6] It became an important tool for the mathematical modeling of the complex transport phenomena such as anomalous diffusion, flows with heat and mass transfer of viscoelastic materials, vibration equation, and variational principle for the generalized Korteweg-de Vries-Burgers equation with fractal derivatives for shallow water wave. The variational iteration method is used to introduce the definition of fractional derivatives.…”