scholarly journals Stability Analysis of Nonlocal Elastic Columns with Initial Imperfection

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
S. P. Xu ◽  
M. R. Xu ◽  
C. M. Wang
Keyword(s):  
Size Effect ◽  
Homotopy Perturbation Method ◽  
Large Deflection ◽  
Initial Imperfection ◽  
Nonlocal Elasticity Theory ◽  
Equilibrium Path ◽  
Postbuckling Behavior ◽  
Homotopy Perturbation ◽  
Parametric Studies ◽  
Elastic Column

Investigated herein is the postbuckling behavior of an initially imperfect nonlocal elastic column, which is simply supported at one end and subjected to an axial force at the other movable end. The governing nonlinear differential equation of the axially loaded nonlocal elastic column experiencing large deflection is first established within the framework of Eringen's nonlocal elasticity theory in order to embrace the size effect. Its semianalytical solutions by the virtue of homotopy perturbation method, as well as the successive approximation algorithm, are determined in an explicit form, through which the postbuckling equilibrium loads in terms of the end rotation angle and the deformed configuration of the column at this end rotation are predicted. By comparing the degenerated results with the exact solutions available in the literature, the validity and accuracy of the proposed methods are numerically substantiated. The size effect, as well as the initial imperfection, on the buckled configuration and the postbuckling equilibrium path is also thoroughly discussed through parametric studies.

Nonlinear Vibration Analysis of Functionally Graded Nanobeam Using Homotopy Perturbation Method

2016 ◽  
Vol 9 (1) ◽  
pp. 144-156 ◽  
Author(s):  
Majid Ghadiri ◽  
Mohsen Safi
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equation ◽  
Beam Theory ◽  
Nonlocal Elasticity Theory ◽  
Functionally Graded ◽  
Simply Supported ◽  
Homotopy Perturbation ◽  
Functionally Graded Nanobeam

AbstractIn this paper, He's homotopy perturbation method is utilized to obtain the analytical solution for the nonlinear natural frequency of functionally graded nanobeam. The functionally graded nanobeam is modeled using the Eringen's nonlocal elasticity theory based on Euler-Bernoulli beam theory with von Karman nonlinearity relation. The boundary conditions of problem are considered with both sides simply supported and simply supported-clamped. The Galerkin's method is utilized to decrease the nonlinear partial differential equation to a nonlinear second-order ordinary differential equation. Based on numerical results, homotopy perturbation method convergence is illustrated. According to obtained results, it is seen that the second term of the homotopy perturbation method gives extremely precise solution.

scholarly journals Approximations for Large Deflection of a Cantilever Beam under a Terminal Follower Force and Nonlinear Pendulum

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
H. Vázquez-Leal ◽  
Y. Khan ◽  
A. L. Herrera-May ◽  
U. Filobello-Nino ◽  
A. Sarmiento-Reyes ◽  
...  
Keyword(s):  
Perturbation Method ◽  
Cantilever Beam ◽  
Homotopy Perturbation Method ◽  
Large Deflection ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Follower Force ◽  
Homotopy Perturbation ◽  
Solution Methods ◽  
Nonlinear Pendulum

In theoretical mechanics field, solution methods for nonlinear differential equations are very important because many problems are modelled using such equations. In particular, large deflection of a cantilever beam under a terminal follower force and nonlinear pendulum problem can be described by the same nonlinear differential equation. Therefore, in this work, we propose some approximate solutions for both problems using nonlinearities distribution homotopy perturbation method, homotopy perturbation method, and combinations with Laplace-Padé posttreatment. We will show the high accuracy of the proposed cantilever solutions, which are in good agreement with other reported solutions. Finally, for the pendulum case, the proposed approximation was useful to predict, accurately, the period for an angle up to179.99999999∘yielding a relative error of 0.01222747.

scholarly journals Homotopy perturbation method for Ozone decomposition of the second order in aqueous solutions

2015 ◽  
Vol 11 (1) ◽  
pp. 63-72
Author(s):  
Z. Ayati ◽  
J. Biazar ◽  
M. Partovi
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Reaction Order ◽  
Analytical Approximation ◽  
Second Order ◽  
Ozone Decomposition ◽  
System Of Differential Equations ◽  
Homotopy Perturbation ◽  
Hydroxyl Ion ◽  
Parametric Studies

Abstract In this article the problem of mass transfer of ozone of the second order from a gaseous phase into an aqueous phase has been studied. Homotopy perturbation method is employed to derive an analytical approximation to the solutions of the system of differential equations governing on the problem. Some parametric studies have been included. The effects of the temperature and hydroxyl ion reaction order to the solutions are illustrated by some plots

On Spectral-Homotopy Perturbation Method Solution of Nonlinear Differential Equations in Bounded Domains

2013 ◽  
Vol 1 (1) ◽  
pp. 25-37
Author(s):  
Ahmed A. Khidir
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Iteration Algorithm ◽  
Test Problems ◽  
Nonlinear Boundary Value Problems ◽  
Homotopy Perturbation ◽  
Spectral Technique ◽  
Homotopy Analysis

In this study, a combination of the hybrid Chebyshev spectral technique and the homotopy perturbation method is used to construct an iteration algorithm for solving nonlinear boundary value problems. Test problems are solved in order to demonstrate the efficiency, accuracy and reliability of the new technique and comparisons are made between the obtained results and exact solutions. The results demonstrate that the new spectral homotopy perturbation method is more efficient and converges faster than the standard homotopy analysis method. The methodology presented in the work is useful for solving the BVPs consisting of more than one differential equation in bounded domains. 

A new modification of the homotopy perturbation method

2021 ◽  
pp. 095745652199987
Author(s):  
Magaji Yunbunga Adamu ◽  
Peter Ogenyi
Keyword(s):  
Comparative Analysis ◽  
Error Analysis ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Oscillators ◽  
Absolute Error ◽  
New Method ◽  
Optimal Analysis ◽  
Homotopy Perturbation ◽  
Accuracy And Precision

This study proposes a new modification of the homotopy perturbation method. A new parameter alpha is introduced into the homotopy equation in order to improve the results and accuracy. An optimal analysis identifies the parameter alpha, aimed at improving the solutions. A comparative analysis of the proposed method reveals that the new method presents results with higher degree of accuracy and precision than the classic homotopy perturbation method. Absolute error analysis shows the convenience of the proposed method, providing much smaller errors. Two examples are presented: Duffing and Van der pol’s nonlinear oscillators to demonstrate the efficiency, accuracy, and applicability of the new method.

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