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

Ismail, Gamal; Abul-Ez, Mahmoud; Zayed, Mohra; Farea, N.M.

doi:10.1016/j.aej.2021.09.021

Cited by 7 publications

(5 citation statements)

References 35 publications

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“…So, equation (6) is the solution of equation (3), where the approximate solution ρ N (τ) can be obtained by truncated equation ( 6),…”

Section: Description Of the Methodsmentioning

confidence: 99%

“…Some natural phenomena can be modeled by linear and non-linear systems of integral and differential equations, which are commonly used in fields such as biology, chemistry, and physics [1][2][3][4][5][6][7][8][9][10][11]. Many numerical methods, such as collocation boundary value methods, discontinuous Galerkin approximations, Euler matrix method, spectral element method, Chebyshev wavelets approach, and Radial basis Functions, have been provided to solve linear and nonlinear Volterra integral equations [12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

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Application of the reproducing kernel method for solving linear Volterra integral equations with variable coefficients

Amoozad,

Allahviranloo,

Abbasbandy

et al. 2024

Phys. Scr.

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This article proposes a new approach for solving linear Volterra integral equations with variable coefficients using the Reproducing Kernel Method (RKM). This method eliminates the need for the Gram-Schmidt process. However, the accuracy of RKM is influenced by various factors, including the selection of points, bases, space, and implementation method. The main objective of this article is to introduce a generalized method based on the Reproducing Kernel, which is successful in solving a special type of singular weakly nonlinear boundary value problems (BVPs). The easy implementation, elimination of the Gram-Schmidt process, fewer calculations, and high accuracy of the present method are interesting. The conformity of numerical results, including tables and figures, with theorems related to error analysis and convergence order, confirms the practicality of the present method.

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“…So, equation (6) is the solution of equation (3), where the approximate solution ρ N (τ) can be obtained by truncated equation ( 6),…”

Section: Description Of the Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Application of the reproducing kernel method for solving linear Volterra integral equations with variable coefficients

Amoozad,

Allahviranloo,

Abbasbandy

et al. 2024

Phys. Scr.

View full text Add to dashboard Cite

show abstract

“…In literature, a lot of efforts have been paid on studying them and there are many techniques (analytical and numerical) that have been introduced and used to solve such systems. [1][2][3][4][5][6][7][8][9] Some researchers have presented several techniques for solving analytically a second order differential equation with various strong nonlinear terms. For example, Belhaq and Lakrad 6 used the multiple scales method for solving strongly nonlinear oscillators, and the variational approach and the Hamiltonian approach to nonlinear oscillators are systematically discussed in Refs.…”

Section: Introductionmentioning

confidence: 99%

Power series approach to nonlinear oscillators

As’ad

Asad

2023

Journal of Low Frequency Noise, Vibration and Active Control

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In this article, we introduce a nonlinear oscillator equation containing two strong linear terms. An approximate solution was obtained using power series approach. Furthermore, by introducing a parameter to the original equation, we fined the fixed points of the modified nonlinear oscillator equation and study stability analysis of these fixed points. On the other hand, we simulate the solution of the nonlinear oscillator equation and introduced many plots for different initial conditions. Finally, we make some plots concerning the phase portrait for different cases.

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“…were also used by many researchers to solve strong nonlinear problems. Coupling techniques between the homotopy perturbation method and the variational method 44 and between the Lindested–Poincare method and the homotopy perturbation method 45 were also used for solving the nonlinear oscillators. Harmonic balance method and different modified forms of the harmonic balance method have been widely used for solving nonlinear oscillators from last few decades.…”

Section: Introductionmentioning

confidence: 99%

A simple modified harmonic balance method for strongly nonlinear oscillator with cubic non-linearity and harmonic restoring force

Sharif,

Molla,

Alim

2023

Journal of Low Frequency Noise, Vibration and Active Control

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In this article, a very simple modified form of the harmonic balance method is used to solve a strongly nonlinear oscillator with cubic nonlinearity and harmonic restoring force. Taylor series expansion up to third term is considered for the harmonic restoring force. The first approximate solutions of the present method pleasantly agree with the numerical solution obtained by Runge–Kutta fourth order method. Accuracy and simplicity of the present method solution is established when compared with the other method solutions. The present method can be utilized to other nonlinear oscillators.

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

Cited by 7 publications

References 35 publications

Application of the reproducing kernel method for solving linear Volterra integral equations with variable coefficients

Application of the reproducing kernel method for solving linear Volterra integral equations with variable coefficients

Power series approach to nonlinear oscillators

A simple modified harmonic balance method for strongly nonlinear oscillator with cubic non-linearity and harmonic restoring force

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