Analytical solutions to nonlinear conservative oscillator with fifth-order nonlinearity

Abstract

This paper describes analytical and numerical methods to analyze the steady state periodic response of an oscillator with symmetric elastic and inertia nonlinearity. A new implementation of the homotopy perturbation method (HPM) and an ancient Chinese method called the max-min approach are presented to obtain an approximate solution. The major concern is to assess the accuracy of these approximate methods in predicting the system response within a certain range of system parameters by examining their ability to establish an actual (numerical) solution. Therefore, the analytical results are compared with the numerical results to illustrate the effectiveness and convenience of the proposed methods.

Non-linear vibration of Euler-Bernoulli beams

“…In recent years, some promising approximate analytical solutions have been proposed, such as Frequency Amplitude Formulation [13], Variational Iteration [5,6,14,17], Homotopy-Perturbation [3,4,7,24], Parametrized-Perturbation [18], Max-Min [15,19,29], Differential Transform Method [16], Adomian Decomposition Method [22], Energy Balance [23,30], etc.…”

Semi-analytical Investigation of Momentum and Heat Transfer of a Non-Newtonian Fluid Flow for Specific Turbine Cooling Application Using AGM

“…The main reason of widespread usage of analytical and semi-analytical methods is that by utilizing these approaches, researches would obtain a unique function in their solving procedures which could be used in other fields such as designing a control system for heat transfer or fluid mechanics appliances. Therefore, for the purpose of achieving the afore-mentioned fact, many researchers have tried to reach acceptable solution for differential equations due to their nonlinearity by utilizing analytical and semi-analytical methods such as: Perturbation Method [9], Homotopy Perturbation Method [10][11][12], Variational Iteration Method [13,14], Optimal Variational Iteration Method using Adomian's Polynomials (OVIMAP) [15], Homotopy Analysis Method [14,16,17], Parameterized Perturbation Method (PPM) [18], Collocation Method (CM) [19], Adomian Decomposition Method [20,21], Variation of Parameters Method (VPM) [22] and so on.…”

Analyzing the nonlinear heat transfer equation by AGM

“…Recently, many new numerical techniques have been widely applied to the nonlinear problems. Some of these methods are Perturbation Method (PM) [1], Homotopy Perturbation Method (HPM) [2,3,4,5,6], Variational Iteration Method (VIM) [7,8,9], Homotopy Analysis Method (HAM) [10,11], Differential Transform Method (DTM) [12] and Adomian Decomposition Method (ADM) [13,14].…”