Approximate Solutions by Perturbation Methods

2000 ◽  
pp. 513-576
Author(s):  
J. Kevorkian
Keyword(s):  
Perturbation Methods ◽  
Approximate Solutions

A Perturbation Method for Low-Frequency Fluid-Structure Interaction Problems

1973 ◽  
Vol 40 (2) ◽  
pp. 388-394 ◽  
Author(s):  
Y. K. Lou
Keyword(s):  
Wave Equation ◽  
Perturbation Method ◽  
Electromagnetic Scattering ◽  
Perturbation Methods ◽  
Separation Of Variables ◽  
Fluid Structure Interaction ◽  
Low Frequency ◽  
Approximate Solutions ◽  
Fluid Structure ◽  
Structure Interaction

Perturbation methods have been used for electromagnetic scattering and diffraction problems in recent years. A similar method suitable for low-frequency fluid-structure interaction problems is presented. The essence of the method lies in the fact that approximate solutions for fluid-structure interaction problems can be obtained from a set of Poisson’s equations, rather than from the reduced wave equation. The method is particularly useful for those problems where the Poisson’s equation may be solved by the method of separation of variables while the reduced wave equation cannot. As an illustrative example, the vibrations of a submerged spherical shell is studied using the perturbation method and the accuracy of the method is demonstrated.

Assessment of Perturbation and Homotopy-Perturbation Methods for Solving Nonlinear Oscillator Equations

2013 ◽  
Vol 376 ◽  
pp. 207-215
Author(s):  
Maziar Ramezani ◽  
Thomas Rainer Neitzert
Keyword(s):  
Nonlinear Systems ◽  
Perturbation Method ◽  
Nonlinear Oscillator ◽  
Perturbation Methods ◽  
Approximate Solutions ◽  
Parameter Method ◽  
Van Der Pol ◽  
Homotopy Perturbation ◽  
Comparison Of The Results ◽  
Oscillator Equations

In this paper, two modified perturbation methods, namely, artificial parameter method (APM) and homotopy perturbation method (HPM) have been successfully implemented to find the solution of van der Pol nonlinear oscillator equation. Different from classical perturbation method, APM and HPM do not require small parameter and therefore, obtained approximate solutions may be uniformly valid for both weak nonlinear systems and strong nonlinear systems. Comparison of the results obtained by the proposed methods reveals that APM and HPM are more effective compared to classical perturbation method and with only a few terms, approximate the exact solution with a fairly reasonable error.

Analysis of Nonlinear Active Noise Behavior of Fuzzy Controller Using Non-Perturbation Methods

2020 ◽  
pp. 2150033
Author(s):  
Monika Rani ◽  
Rakesh Goyal ◽  
Vikramjeet Singh
Keyword(s):  
Homotopy Perturbation Method ◽  
Perturbation Methods ◽  
Fuzzy Controller ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Riccati Differential Equation ◽  
Homotopy Perturbation ◽  
Active Noise ◽  
Proposed Model ◽  
Adomian Decomposition

In this paper, a Fuzzy controller model has been converted into a time-dependent nonlinear model and then quadratic Riccati differential equation was analyzed to satisfy the solution of the nonlinear active noise behavior of Fuzzy controller. Further, the approximate solutions of this equation using non-perturbation methods i.e., adomian decomposition method (ADM), variational iterational method (VIM) and homotopy perturbation method (HPM) were investigated. A comparison of these methods has also been given with tabular and graphical presentations. Our results reveal that VIM provides the closest approximate solution and fast convergence for the proposed model as compare to ADM and HPM.

On the Perturbation Methods for Vibration Analysis of Linear Time-Varying Systems

2016 ◽  
Vol 08 (03) ◽  
pp. 1650035 ◽  
Author(s):  
Xiao-Dong Yang ◽  
Ming Liu ◽  
Wei Zhang ◽  
Ying-Jing Qian ◽  
Roderick V. N. Melnik
Keyword(s):  
Perturbation Methods ◽  
Multiple Scales ◽  
Linear Time ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Beam Model ◽  
Time Varying ◽  
Analytical Approximations ◽  
Time Varying Systems ◽  
New Perspective

Some perturbation methods in the studying vibrations of the linear time-varying (LTV) system are discussed. Three classical perturbation methods, namely, averaging method, harmonic balance method, and multiple scales method with linear scales, have been used from a new perspective based on analytical approximations to the corresponding LTV ordinary differential equations. The deploying beam model has been taken as an example to validate the explicit approximate solutions obtained by these perturbation methods. It is demonstrated that such approximate solutions have good agreement with numerical and exact solutions, excluding the vicinity of the turning point.

Approximate analytical solution for the one-dimensional nonlinear Boussinesq equation

2015 ◽  
Vol 25 (4) ◽  
pp. 831-840 ◽  
Author(s):  
Hossein Aminikhah
Keyword(s):  
Exact Solution ◽  
Laplace Transform ◽  
Arbitrary Function ◽  
Perturbation Methods ◽  
Boussinesq Equation ◽  
Approximate Solutions ◽  
Content Type ◽  
One Dimensional ◽  
Homotopy Perturbation ◽  
The One

Purpose – The purpose of this paper is to provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. Combination of the Laplace transform and homotopy perturbation methods (LTHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation. Design/methodology/approach – The authors present the solution of nonlinear Boussinesq equation by combination of Laplace transform and new homotopy perturbation methods. An important property of the proposed method, which is clearly demonstrated in example, is that spectral accuracy is accessible in solving specific nonlinear nonlinear Boussinesq equation which has analytic solution functions. Findings – The authors proposed a combination of Laplace transform method and homotopy perturbation method to solve the one-dimensional Boussinesq equation. The results are found to be in excellent agreement. The results show that the LTNHPM is an effective mathematical tool which can play a very important role in nonlinear sciences. Originality/value – The authors provide closed-form approximate solutions to the one-dimensional Boussinesq equation for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function of time. In this work combination of Laplace transform and new homotopy perturbation methods (LTNHPM) are considered as an algorithm which converges rapidly to the exact solution of the nonlinear Boussinesq equation.

scholarly journals Nonlinear Vibration of a Micro Piezoelectric Precision Drive System

Micromachines ◽  
2019 ◽  
Vol 10 (3) ◽  
pp. 159
Author(s):  
Chong Li ◽  
Wei Zhong ◽  
Jiwen Fang ◽  
Lining Sun
Keyword(s):  
Nonlinear Dynamic ◽  
Perturbation Methods ◽  
Approximate Solutions ◽  
Drive System ◽  
Dynamic Equations ◽  
Nonlinear Phenomenon ◽  
Nonlinear Dynamic Model ◽  
Output Torque ◽  
Working Principle ◽  
Large Output

A micro piezoelectric precision drive system is proposed, which is advantageous due its small size, large transmission ratio, and large output torque. The working principle of the proposed piezoelectric precision drive system is presented, and the nonlinear dynamic model and equations of the system are established. Using the Linz Ted-Poincaré and perturbation methods, the nonlinear approximate solutions of the dynamic equations are calculated. The results indicate that the nonlinear intensity of the drive system is inversely proportional to the number of meshing movable teeth. It was also noted that the rotor is most affected by the nonlinear phenomenon. These results can be utilized both to optimize the dimensions of the piezoelectric precision drive system and to reduce the intensity of vibrations during operation.

A Note on One-Term Approximate Solutions for Non-Linear Vibration Problems

1958 ◽  
Vol 62 (570) ◽  
pp. 450-451
Author(s):  
S. Mahalingam
Keyword(s):  
Perturbation Methods ◽  
Approximate Solutions ◽  
Successive Approximations ◽  
Force Characteristic ◽  
Mathematical Form ◽  
Linear Vibration ◽  
Restoring Force ◽  
Vibration Problems ◽  
Nonlinear Elastic ◽  
Elastic Characteristics

Several methods are available for the solution of problems of forced vibration of systems with nonlinear elastic characteristics. Of these, the Martienssen, Den Hartog and Rauscher methods may be applied even if the restoring force characteristic is only known graphically, while the Duffing and Perturbation methods are only applicable when the restoring force characteristic is expressed in a convenient mathematical form. Successive approximations are used in the Duffing, Perturbation and Rauscher methods and therefore any desired degree of accuracy can be obtained. The Den Hartog and Martienssen methods give a two-term and one-term solution respectively.

Interactive Graphics for the Analysis of Tires

1985 ◽  
Vol 13 (3) ◽  
pp. 127-146 ◽  
Author(s):  
R. Prabhakaran
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Approximate Solutions ◽  
Interactive Graphics ◽  
Element Analysis ◽  
Analysis Process ◽  
Physical Problems ◽  
The Finite Element Analysis ◽  
Analysis Computer ◽  
Discretization Technique

Abstract The finite element method, which is a numerical discretization technique for obtaining approximate solutions to complex physical problems, is accepted in many industries as the primary tool for structural analysis. Computer graphics is an essential ingredient of the finite element analysis process. The use of interactive graphics techniques for analysis of tires is discussed in this presentation. The features and capabilities of the program used for pre- and post-processing for finite element analysis at GenCorp are included.

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