scholarly journals He’s frequency formula to fractal undamped Duffing equation

2021 ◽  
pp. 146134842199260
Author(s):  
Guang-Qing Feng
Keyword(s):  
Variational Principle ◽  
Nonlinear Oscillation ◽  
Duffing Equation ◽  
Transform Method ◽  
Simple Method ◽  
Interesting Topic ◽  
Numerical Result ◽  
Fractal Derivative ◽  
Fractal Space ◽  
Approximate Frequency

Nonlinear oscillation is an increasingly important and extremely interesting topic in engineering. This article completely reviews a simple method proposed by Ji-Huan He and successfully establishes a fractal undamped Duffing equation through the two-scale fractal derivative in a fractal space. Its variational principle is established, and the two-scale transform method and the fractal frequency formula are adopted to find the approximate frequency of the fractal oscillator. The numerical result shows that He’s frequency formula is a unique tool for the fractal equations.

scholarly journals A powerful and simple frequency formula to nonlinear fractal oscillators

2020 ◽  
pp. 146134842094783
Author(s):  
Kang-Le Wang ◽  
Chun-Fu Wei
Keyword(s):  
Variational Principle ◽  
Conservation Laws ◽  
Nonlinear Oscillator ◽  
Transform Method ◽  
Inverse Transform ◽  
Simple Tool ◽  
Fractal Derivative ◽  
Fractal Space ◽  
Approximate Frequency

In this work, a fractal nonlinear oscillator is successfully established by fractal derivative in a fractal space, and its variational principle is obtained by semi-inverse transform method. The variational principle can provide conservation laws in an energy form. The approximate frequency of the fractal oscillator is found by a simple fractal frequency formula. An example shows the fractal frequency formula is a powerful and simple tool to fractal oscillators.

scholarly journals Frequency formula for a class of fractal vibration system

2022 ◽  
Vol 3 (1) ◽  
pp. 55-61
Author(s):  
Yi Tian ◽  
Keyword(s):  
Variational Principles ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Vibration System ◽  
Transform Method ◽  
Inverse Transform ◽  
Simple Tool ◽  
Fractal Derivative ◽  
Fractal Space ◽  
Approximate Frequency

Four fractal nonlinear oscillators (The fractal Duffing oscillator, fractal attachment oscillator, fractal Toda oscillator, and a fractal nonlinear oscillator) are successfully established by He’s fractal derivative in a fractal space, and their variational principles are obtained by semi-inverse transform method. The approximate frequency of the four fractal oscillators are found by a simple frequency formula. The results show the frequency formula is a powerful and simple tool to a class of fractal oscillators.

scholarly journals Variational principle for nonlinear fractional wave equation in a fractal space

2021 ◽  
pp. 18-18
Author(s):  
Shao-Wen Yao
Keyword(s):  
Wave Equation ◽  
Variational Principle ◽  
Inverse Method ◽  
Scale Method ◽  
Fractional Wave Equation ◽  
Fractal Derivative ◽  
Fractal Space

The fractal derivative is adopted to describe the nonlinear fractional wave equation in a fractal space. A variational principle is successfully established by the semi-inverse method. The two-scale method and He?s exp-function are usedto solve the equation, and a good result is obtained.

scholarly journals He’s fractal calculus and its application to fractal KdV equation

2021 ◽  
pp. 100-100
Author(s):  
Xue-Si Ma ◽  
Li-Na Zhang
Keyword(s):  
Analytical Solution ◽  
Iteration Method ◽  
Approximate Analytical Solution ◽  
Kdv Equation ◽  
Variational Iteration Method ◽  
Natural Phenomena ◽  
Transform Method ◽  
Fractal Derivative ◽  
Fractal Calculus ◽  
Fractal Space

He?s fractal calculus is a powerful and effective tool to dealing with natural phenomena in a fractal space. In this paper, we study the fractal KdV equation with He?s fractal derivative. We first adopt the two-scale transform method to convert the fractal KdV equation into its traditional partner in acontinuous space. Finally, we successfully use He?s variational iteration method (HVIM) to obtain its approximate analytical solution.

scholarly journals Generalized Variational Principle for the Fractal (2 + 1)-Dimensional Zakharov–Kuznetsov Equation in Quantum Magneto-Plasmas

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1022
Author(s):  
Yan-Hong Liang ◽  
Kang-Jia Wang
Keyword(s):  
Variational Principle ◽  
Variational Formulation ◽  
Inverse Method ◽  
Fractal Theory ◽  
Traveling Wave Solutions ◽  
Generalized Variational Principle ◽  
Fractal Derivative ◽  
Fractal Space ◽  
First Time ◽  
Symmetric Theory

In this paper, we propose the fractal (2 + 1)-dimensional Zakharov–Kuznetsov equation based on He’s fractal derivative for the first time. The fractal generalized variational formulation is established by using the semi-inverse method and two-scale fractal theory. The obtained fractal variational principle is important since it not only reveals the structure of the traveling wave solutions but also helps us study the symmetric theory. The finding of this paper will contribute to the study of symmetry in the fractal space.

scholarly journals A variational approach to a porous catalyst

2021 ◽  
pp. 44-44
Author(s):  
Zhi-Qiang Sun
Keyword(s):  
Approximate Solution ◽  
Variational Principle ◽  
Diffusion Process ◽  
Variational Approach ◽  
Fractal Model ◽  
Porous Electrodes ◽  
Porous Catalyst ◽  
Convection Diffusion ◽  
Fractal Derivative ◽  
Fractal Space

The convection-diffusion process in porous electrodes depends greatly upon the porous structure. A fractal model for porous catalyst in a thin-zone bed reactor is established using He?s fractal derivative, and a variational principle is also established in a fractal space, and an approximate solution is obtained. Additionally an ancient Chinese algorithm is adopted to solve an algebraic equation.

A fast insight into the nonlinear oscillation of nano-electro mechanical resonators considering the size effect and the van der Waals force

2021 ◽  
Author(s):  
Kang-Jia Wang
Keyword(s):  
Variational Principle ◽  
Size Effect ◽  
Nonlinear Oscillation ◽  
Van Der Waals ◽  
Mechanical Systems ◽  
Van Der Waals Force ◽  
Mechanical Resonators ◽  
Simple Method ◽  
One Step ◽  
Frequency Relationship

Abstract Nano/micro actuators are widely used in micro/ nano electro mechanical systems (NEMS/MEMS) and the study on its nonlinear oscillation is of great significance. This paper begins with a wrong variational principle ([19] Appl Nanosci, 2016, 6: 309-317) of the reduced governing partial differential equation of the resonator which is used to describe the nonlinear oscillation of nano-electro mechanical resonators that takes into account the size effect and the van der Waals force. By using the Semi-inverse method, we establish the genuine variational principle. Then a simple method so called He’s frequency formulation is applied to solve the problem, where it only needs one-step to get the approximate amplitude-frequency relationship. Comparing with the existing method, it shows that the proposed method is simple but effective, which is helpful to be of significance to the study of the nonlinear oscillation in micro/nano electro mechanical systems.

Dynamics research of Fangzhu’s nanoscale surface

2021 ◽  
pp. 146134842110527
Author(s):  
Pinxia Wu ◽  
Weiwei Ling ◽  
Xiumei Li ◽  
Xichun He ◽  
Liangjin Xie
Keyword(s):  
Energy Balance ◽  
Analytical Solutions ◽  
Numerical Experiments ◽  
Fractal Model ◽  
Balance Method ◽  
Transform Method ◽  
Energy Balance Method ◽  
Collection Rate ◽  
Fractal Derivative ◽  
Approximate Analytical Solutions

In this paper, we mainly focus on a fractal model of Fangzhu’s nanoscale surface for water collection which is established through He’s fractal derivative. Based on the fractal two-scale transform method, the approximate analytical solutions are obtained by the energy balance method and He’s frequency–amplitude formulation method with average residuals. Some specific numerical experiments of the model show that these two methods are simple and effective and can be adopted to other nonlinear fractal oscillators. In addition, these properties of the obtained solution reveal how to enhance the collection rate of Fangzhu by adjusting the smoothness of its surfaces.

A FRACTAL VARIATIONAL PRINCIPLE FOR THE TELEGRAPH EQUATION WITH FRACTAL DERIVATIVES

Fractals ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050058 ◽  
Author(s):  
KANG-LE WANG ◽  
SHAO-WEN Yao ◽  
YAN-PING LIU ◽  
LI-NA ZHANG
Keyword(s):  
Variational Principle ◽  
Perturbation Method ◽  
Inverse Method ◽  
Homotopy Perturbation Method ◽  
Telegraph Equation ◽  
Transform Method ◽  
Homotopy Perturbation

A fractal modification of the telegraph equation with fractal derivatives is given, and its variational principle is established by the semi-inverse method. The two-scale transform method and He’s homotopy perturbation method are successfully adopted to solve the fractal equation.

Solution of Free Vibration Equations of Euler-Bernoulli Cracked Beams by Using Differential Transform Method

2011 ◽  
Vol 110-116 ◽  
pp. 4532-4536 ◽  
Author(s):  
K. Torabi ◽  
J. Nafar Dastgerdi ◽  
S. Marzban
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Differential Transform Method ◽  
Beam Model ◽  
Cracked Beam ◽  
Transform Method ◽  
Simple Method ◽  
Differential Transform

In this paper, free vibration differential equations of cracked beam are solved by using differential transform method (DTM) that is one of the numerical methods for ordinary and partial differential equations. The Euler–Bernoulli beam model is proposed to study the frequency factors for bending vibration of cracked beam with ant symmetric boundary conditions (as one end is clamped and the other is simply supported). The beam is modeled as two segments connected by a rotational spring located at the cracked section. This model promotes discontinuities in both vertical displacement and rotational due to bending. The differential equations for the free bending vibrations are established and then solved individually for each segment with the corresponding boundary conditions and the appropriated compatibility conditions at the cracked section by using DTM and analytical solution. The results show that DTM provides simple method for solving equations and the results obtained by DTM converge to the analytical solution with much more accurate for both shallow and deep cracks. This study demonstrates that the differential transform is a feasible tool for obtaining the analytical form solution of free vibration differential equation of cracked beam with simple expression.

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