scholarly journals Approximate Periodic Solution for the Nonlinear Helmholtz-Duffing Oscillator via Analytical Approaches

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
A. Mirzabeigy ◽  
M. K. Yazdi ◽  
M. H. Nasehi
Keyword(s):  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Duffing Oscillator ◽  
Analytical Techniques ◽  
Second Order ◽  
Global Error ◽  
Error Minimization ◽  
Minimization Method ◽  
Engineering Problems ◽  
Analytical Approaches

The conservative Helmholtz-Duffing oscillator is analyzed by means of three analytical techniques. The max-min, second-order of the Hamiltonian, and the global error minimization approaches are applied to achieve natural frequencies. The obtained results are compared with the homotopy perturbation method and numerical solutions. The results show that second-order of the global error minimization method is very accurate, so it can be widely applicable in engineering problems.

scholarly journals An approximate analytical technique for solving second order strongly nonlinear generalized duffing equation with small damping

2015 ◽  
Vol 39 (1) ◽  
pp. 103-114
Author(s):  
M Alhaz Uddin ◽  
M Wali Ullah ◽  
Rehana Sultana Bipasha
Keyword(s):  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Second Order ◽  
Extended Form ◽  
Analytical Technique ◽  
Strongly Nonlinear ◽  
Homotopy Perturbation ◽  
Kbm Method ◽  
Method Accuracy ◽  
Academy Of Sciences

In this paper, He’s homotopy perturbation method has been extended for obtaining the analytical approximate solution of second order strongly nonlinear generalized duffing oscillators with damping based on the extended form of the Krylov-Bogoliubov-Mitropolskii (KBM) method. Accuracy and validity of the solutions obtained by the presented method are compared with the corresponding numerical solutions obtained by the well-known fourth order Rangue-Kutta method. The method has been illustrated by examples.Journal of Bangladesh Academy of Sciences, Vol. 39, No. 1, 103-114, 2015

Global Error Minimization method for nonlinear oscillators with highly nonlinearity

2013 ◽  
Vol 7 (4) ◽  
pp. 1557-1561
Author(s):  
Muhammad Aslam Noor ◽  
Waseem Asghar Khan
Keyword(s):  
Nonlinear Oscillators ◽  
Global Error ◽  
Error Minimization ◽  
Minimization Method

Analytical study of the vibrating double-sided quintic nonlinear nano-torsional actuator using higher-order Hamiltonian approach

2021 ◽  
pp. 146134842110320
Author(s):  
Gamal Mohamed Ismail ◽  
Mahmoud Abul-Ez ◽  
Hijaz Ahmad ◽  
Nadia Mohamed Farea
Keyword(s):  
Analytical Study ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Higher Order ◽  
Second Order ◽  
Hamiltonian Approach ◽  
Homotopy Perturbation ◽  
The Relationship ◽  
Better Than

In this work, we investigate and apply higher-order Hamiltonian approach (HA) as one of the novelty techniques to find out the approximate analytical solution for vibrating double-sided quintic nonlinear nano-torsional actuator. Periodic solutions are analytically verified, and consequently, the relationship between the initial amplitude and the natural frequency are obtained in a novel analytical way. The HA is then extended to the second-order to find more accurate results. To show the accuracy and applicability of the technique, the approximated results are compared with the homotopy perturbation method and numerical solution. According to the numerical results, it is highly remarkable that the second-order approximate solutions produce better than previously existing results and almost similar in comparing with the numerical solutions.

Out-of-plane static compression and energy absorption of paper hierarchical corrugation sandwich panels

2021 ◽  
pp. 030932472110085
Author(s):  
Huineng Wang ◽  
Yanfeng Guo ◽  
Yungang Fu ◽  
Dan Li
Keyword(s):  
Yield Strength ◽  
Bearing Capacity ◽  
Energy Absorption ◽  
Sandwich Panel ◽  
Second Order ◽  
Compression Rate ◽  
Static Compression ◽  
First Order ◽  
Out Of Plane ◽  
Analytical Approaches

This study introduces the opinion of the corrugation hierarchy to develop the second-order corrugation paperboard, and explore the deformation characteristics, yield strength, and energy absorbing capacity under out-of-plane static evenly compression loading by experimental and analytical approaches. On the basis of the inclined-straight strut elements of corrugation unit and plastic hinge lines, the yield and crushing strengths of corrugation unit were analyzed. This study shows that as the compressive stress increases, the second-order corrugation core layer is firstly crushed, and the first-order corrugation structures gradually compacted until the failure of entire structure. The corrugation type has an obvious influence on the yield strength of the corrugation sandwich panel, and the yield strength of B-flute corrugation sandwich panel is wholly higher than that of the C-flute structure. At the same compression rate, the flute type has a significant impact on energy absorption, and the C-flute second-order corrugation sandwich panel has better bearing capacity than the B-flute structure. The second-order corrugation sandwich panel has a better bearing capacity than the first-order structure. The static compression rate has little effect on the yield strength and deformation mode. However, with the increase of the static compression rate, the corrugation sandwich panel has a better cushioning energy absorption and material utilization rate.

scholarly journals A Comparative Analysis of Fractional-Order Gas Dynamics Equations via Analytical Techniques

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1735
Author(s):  
Shuang-Shuang Zhou ◽  
Nehad Ali Shah ◽  
Ioannis Dassios ◽  
S. Saleem ◽  
Kamsing Nonlaopon
Keyword(s):  
Homotopy Perturbation Method ◽  
Graphical Representation ◽  
Analytical Techniques ◽  
Variational Iteration Method ◽  
Computational Techniques ◽  
System Of Nonlinear Equations ◽  
Gas Dynamics Equations ◽  
Homotopy Perturbation ◽  
Fractional System ◽  
Modified Forms

This article introduces two well-known computational techniques for solving the time-fractional system of nonlinear equations of unsteady flow of a polytropic gas. The methods suggested are the modified forms of the variational iteration method and the homotopy perturbation method by the Elzaki transformation. Furthermore, an illustrative scheme is introduced to verify the accuracy of the available techniques. A graphical representation of the exact and derived results is presented to show the reliability of the suggested approaches. It is also shown that the findings of the current methodology are in close harmony with the exact solutions. The comparative solution analysis via graphs also represents the higher reliability and accuracy of the current techniques.

scholarly journals Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Darae Jeong ◽  
Yibao Li ◽  
Chaeyoung Lee ◽  
Junxiang Yang ◽  
Yongho Choi ◽  
...  
Keyword(s):  
Parabolic Equations ◽  
Convergence Rates ◽  
Numerical Solutions ◽  
Second Order ◽  
Order Convergence ◽  
Numerical Convergence ◽  
Verification Method ◽  
First Order ◽  
Convergence Test ◽  
Second Order Convergence

In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

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