scholarly journals Analytical accurate solutions of nonlinear oscillator systems via coupled homotopy-variational approach

2021 ◽  
Author(s):  
G.M. Ismail ◽  
M. Abul-Ez ◽  
M. Zayed ◽  
N.M. Farea
Keyword(s):  
Variational Approach ◽  
Nonlinear Oscillator

scholarly journals Higher-Order Approximate Periodic Solutions of a Nonlinear Oscillator with Discontinuity by Variational Approach

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
M. Orhan Kaya ◽  
S. Altay Demirbağ ◽  
F. Özen Zengin
Keyword(s):  
Approximate Solution ◽  
Periodic Solutions ◽  
Relative Error ◽  
Variational Approach ◽  
Nonlinear Oscillator ◽  
Higher Order ◽  
Second Order ◽  
Elastic Force ◽  
Force Term ◽  
The Third

He's variational approach is modified for nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(u). Three levels of approximation have been used. We obtained 1.6% relative error for the first approximate period, 0.3% relative error for the second-order approximate period. The third approximate solution has the accuracy as high as 0.1%.

Dynamic Analysis of Nonlinear Oscillator Equation Arising in Double-Sided Driven Clamped Microbeam-Based Electromechanical Resonator

2012 ◽  
Vol 67 (8-9) ◽  
pp. 435-440 ◽  
Author(s):  
Yasir Khan ◽  
Mehdi Akbarzade
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Variational Approach ◽  
Nonlinear Oscillator ◽  
Analytical Form ◽  
Angular Frequency ◽  
Initial Amplitude ◽  
Hamiltonian Approach ◽  
The Galerkin Method ◽  
The Relationship

In this paper, three different analytical methods have been successfully used to study a nonlinear oscillator equation arising in the microbeam-based electromechanical resonator. These methods are: variational approach, Hamiltonian approach, and amplitude-frequency formulation. The governing equation is based on the Euler-Bernoulli hypothesis and the partial differential equation (PDE) is simplified into an ordinary differential equartion (ODE) by using the Galerkin method. A frequency analysis is carried out, and the relationship between the angular frequency and the initial amplitude is obtained in closed analytical form. A comparison of the present solutions is made with the existing solutions and excellent agreement is noted.

scholarly journals An effective modification of He’s variational approach to a nonlinear oscillator

2019 ◽  
Vol 38 (3-4) ◽  
pp. 1013-1022 ◽  
Author(s):  
Yasir Nawaz ◽  
Muhammad Shoaib Arif ◽  
Mairaj Bibi ◽  
Mehvish Naz ◽  
Rabia Fayyaz
Keyword(s):  
Variational Approach ◽  
Nonlinear Oscillator

scholarly journals Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

2020 ◽  
pp. 2050008
Author(s):  
Shaya Shakerian
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variational Approach ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

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