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

Uddin, Marjan; Ullah, M Wali; Bipasha, Rehana Sultana

doi:10.3329/jbas.v39i1.23664

Cited by 3 publications

(3 citation statements)

References 13 publications

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“…In such cases, the original nonlinear oscillatory systems must be solved straightway. Several researchers have developed analytical techniques to handle nonlinear oscillatory systems using various methods (Belendez et al, 2012;Kovacic and Mickens, 1986;Krylov and Bogoliubov, 1947;Mondal et al, 2019;Nayfeh, 1981;Liu, 2005;Lim and Lai, 2006;Guo and Leung, 2010;Guo and Ma, 2014;He, 1998He, , 1999He, , 2006Uddin al et., 2011Uddin al et., , 2012Uddin al et., , 2015Mishara et al, 2016;Khan, 2019;Yeasmin et al, 2020;Uddin and Sattar, 2010;Ullah et al, 2021). Perturbation techniques (Kovacic and Mickens, 1986;Krylov and Bogoliubov, 1947;Nayfeh, 1981;Uddin and Sattar, 2010), which are extensively applied tools.…”

Section: Introductionmentioning

confidence: 99%

An analytical technique for handling forced Van der Pol vibration equation

Ullah¹,

Uddin

Rahman

2022

J Bangladesh Acad Sci

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Based on the modified harmonic balance method, an analytical method has been developed for handling the forced Van der Pol vibration equation. Usually, a system of nonlinear algebraic equations arises within the unfamiliar coefficients in several harmonics terms and the frequency of the forcing term. A numerical technique has been applied to handle those nonlinear algebraic equations in the classical harmonic balance method. In our study, a system of linear algebraic equations is calculated with the aid of a single nonlinear one. The solutions attained by the suggested scheme have been likened to the results acquired by the well-known Runge-Kutta method, and these results display very nice harmony with the result obtained by the mentioned method. Also, it is noticed that the proposed technique is straightforward and gives the desired results in the whole solution domain. J. Bangladesh Acad. Sci. 45(2); 231-240: December 2021

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Section: Introductionmentioning

confidence: 99%

An analytical technique for handling forced Van der Pol vibration equation

Ullah¹,

Uddin

Rahman

2022

J Bangladesh Acad Sci

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“…Scientists, physicists, applied mathematicians and engineers are forced to find the approximate solutions of the nonlinear physical problems rather than the numerical solutions. A lot of researchers have developed analytical procedures to handle nonlinear oscillatory problems (Alam 2002, 2003, Alam et al 2006, Belendezet al 2012, Guo and Ma 2014, Guo and Leung 2010, Uddin and Sattar 2010, He 1998, Krylov and Bogoliubov 1947, Kovacic and Mickens 2012, Khan 2019, Liu et al 2007, Mondal et al 2019, Mishara et al 2016, Nayfeh 1981, Uddin et al 2011, 2015, Ullah et al 2021a, 2021b, 2021c, 2022, Yeasmin 2020. Among of them, perturbation techniques (Alam 2002, 2003, Alam et al 2006, Uddin and Sattar 2010, Krylov and Bogoliubov 1947, Kovacic and Mickens 2012, Nayfeh 1981) are well-established and most popular methods.…”

Section: Introductionmentioning

confidence: 99%

“…Numerous approximation procedures have been developed to tackle strongly nonlinear oscillators. These techniques include the modified Lindstedt-Poincare method (Cheung et al 1991, Liu 2005, Nayfeh 1981), Homotopy perturbation method (He 1998, Uddin et al 2011, 2015, Ghosh and Uddin 2021, harmonic balance method (Alam et al 2016, Mickens 1986, Rahman et al 2010, residual harmonic balance method (Guo and Ma 2014), iterative harmonic balance method (Guo and Leung 2010) etc. in order to describe the action of the heart, Zduniak et al (2014) created a modified Van der Pol equation with delay.…”

Section: Introductionmentioning

confidence: 99%

Analytical Solution of Modified Forced Van Der Pol Vibration Equation Using Modified Harmonic Balance Method

Ullah¹,

Uddin

Rahman

2022

Khulna Univ. Stud.

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For obtaining analytical solutions to the modified forced Van der Pol equation, the modified harmonic balance method has been developed. In classical harmonic balance method, the numerical procedure is used for solving a set of nonlinear algebraic equations. But it requires laborious computational attempt and accurate primary guesses values which makes it very hard-working to calculate. According to our method, a set of nonlinear algebraic equations has been converted to a set of linear algebraic equations by using a nonlinear one instead of a set. As a result, it reduces the massive computational work. This method provides not only better results than the existing harmonic balance method but also provides very close solutions to the corresponding numerical results. It is noticeable that there is substantial similarity between the approximate and the numerical results attained by the fourth order Runge-Kutta method. Moreover, the method is facile and straightforward. This technique may play great role to handle strongly nonlinear damped systems with external forces.

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Some Novel Analytical Approximations to the (Un)damped Duffing–Mathieu Oscillators

Alyousef

Salas

Alkhateeb

et al. 2022

Journal of Mathematics

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Some novel exact solutions and approximations to the damped Duffing–Mathieu-type oscillator with cubic nonlinearity are obtained. This work is divided into two parts: in the first part, some exact solutions to both damped and undamped Mathieu oscillators are obtained. These solutions are expressed in terms of the Mathieu functions of the first kind. In the second part, the equation of motion to the damped Duffing–Mathieu equation (dDME) is solved using some effective and highly accurate approaches. In the first approach, the nonintegrable dDME with cubic nonlinearity is reduced to the integrable dDME with linear term having undermined optimal parameter (maybe called reduced method). Using a suitable technique, we can determine the value of the optimal parameter and then an analytical approximation is obtained in terms of the Mathieu functions. In the second approach, the ansatz method is employed for deriving an analytical approximation in terms of trigonometric functions. In the third approach, the homotopy perturbation technique with the extended Krylov–Bogoliubov–Mitropolskii (HKBM) method is applied to find an analytical approximation to the dDME. Furthermore, the dDME is solved numerically using the Runge–Kutta (RK) numerical method. The comparison between the analytical and numerical approximations is carried out. All obtained approximations can help a large number of researchers interested in studying the nonlinear oscillations and waves in plasma physics and many other fields because many evolution equations related to the nonlinear waves and oscillations in a plasma can be reduced to the family of Mathieu-type equation, Duffing-type equation, etc.

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

Cited by 3 publications

References 13 publications

An analytical technique for handling forced Van der Pol vibration equation

An analytical technique for handling forced Van der Pol vibration equation

Analytical Solution of Modified Forced Van Der Pol Vibration Equation Using Modified Harmonic Balance Method

Some Novel Analytical Approximations to the (Un)damped Duffing–Mathieu Oscillators

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