- [1] B. N. Achar, J. W. Hanneken, and T. Clarke, Response characteristics of a fractional oscillator, Phys. A: Stat. Mech. Appl., 309(3-4) (2002), 275–288.
- [2] M. Ahmadi Darani and M. Nasiri, A fractional type of the Chebyshev polynomials for approximation of solution of linear fractional differential equations, Comput. Methods Differ. Equ., 1(2) (2013), 96–107.
- [3] H. A. Alyousef, A. H. Salas, M. R. Alharthi, and S. A. El-Tantawy, Galerkin method, ansatz method, and He’s frequency formulation for modeling the forced damped parametric driven pendulum oscillators, J. Low Freq. Noise Vib. Act. Control., (2022), 14613484221101235
- [4] I. Y. Aref’eva, E. Piskovskiy, and I. Volovich, Oscillations and rolling for duffing’s equation , Quantum Bio- Informatics V, World Scientific, (2013), 37–48.
- [5] P. Baratella and A. P. Orsi, A new approach to the numerical solution of weakly singular volterra integral equations, J. Comput. Appl. Math., 163(2) (2004), 401–418.
- [6] J. Biazar and H. Ebrahimi, Orthonormal bernstein polynomials for volterra integral equations of the second kind, Int. J. Appl. Math. Res., 9(1) (2019), 9–20.
- [7] J. Biazar and H. Ebrahimi, A numerical algorithm for a class of non-linear fractional volterra integral equations via modified hat functions, J. Integral Equ. Appl., 34(3) (2022), 295–316.
- [8] J. Biazar and R. Montazeri, Optimal homotopy asymptotic and multistage optimal homotopy asymptotic methods for solving system of volterra integral equations of the second kind, J. Appl. Math., 2019 (2019).
- [9] T. Chen, X. Cao, and D. Niu, Model modification and feature study of duffing oscillator , J. Low Freq. Noise Vib. Act. Control., 41(1) (2022), 230–243.
- [10] M. Erfanian and A. Mansoori, Solving the nonlinear integro-differential equation in complex plane with rationalized haar wavelet, Math. Comput. Simul., 165 (2019), 223–237.
- [11] X. Han and X. Guo, Cubic hermite interpolation with minimal derivative oscillation, J. Comput. Appl. Math., 331 (2018), 82–87.
- [12] B. Hasani Lichae, J. Biazar, and Z. Ayati, Asymptotic decomposition method for fractional order Riccati differential equation, Comput. Methods Differ. Equ., 9(1) (2022), 63–78.
- [13] J. H. He and Y. O. El-Dib, The reducing rank method to solve third-order duffing equation with the homotopy perturbation, Numer. Methods Partial Differ. Equ., 37(2) (2021), 1800–1808.
- [14] J. H. He, Taylor series solution for a third order boundary value problem arising in architectural engineering, Ain Shams Eng. J., 11(4) (2020), 1411–1414.
- [15] S. Heydary and A. Aminataei, Numerical solution of Drinfel’d–Sokolov system with the Haar wavelets method, Comput. Methods Differ. Equ., (2022).
- [16] K. Issa, B. M. Yisa, and J. Biazar, Numerical solution of space fractional diffusion equation using shifted Gegen- bauer polynomials, Comput. Methods Differ. Equ., 10(2) (2022), 431–444.
- [17] I. Kovacic and M. J. Brennan, The Duffing equation: nonlinear oscillators and their behaviour, John Wiley & Sons, 2011.
- [18] S. Lal and P. Kumari, Approximation of functions with bounded derivative and solution of riccati differential equations by jacobi wavelet operational matrix, Appl. Math. Comput., 394 (2021), 125834.
- [19] C. Li and W. Deng, Chaos synchronization of fractional-order differential systems, Int. J. Mod. Phys. B., 20(07) (2006), 791–803.
- [20] Y. Li and N. Sun, Numerical solution of fractional differential equations using the generalized block pulse opera- tional matrix, Comput. Math. Appl., 62(63) (2011), 1046-1054.
- [21] S. Mockary, A. Vahidi, and E. Babolian, An efficient approximate solution of Riesz fractional advection-diffusion equation, Comput. Methods Differ. Equ., 10(2) (2022), 307–319.
- [22] S. Momani and Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos, Solitons & Fractals, 31 (2007), 1248-1255.
- [23] E. Montagu and J. Norbury, Bifurcation tearing in a forced duffing equation, J. Differ. Equ., 300 (2021), 1–32.
- [24] S. Nourazar and A. Mirzabeigy, Approximate solution for nonlinear duffing oscillator with damping effect using the modified differential transform method, Sci. Iran. 20(2) (2013), 364–368.
- [25] P. Pirmohabbati, A. R. Sheikhani, H. S. Najafi, and A. A. Ziabari, Numerical solution of full fractional duffing equations with cubic-quintic-heptic nonlinearities, AIMS Math, 5(2) (2020), 1621–1641.
- [26] I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, Academic Press New York, 1999.
- [27] J. Rad, S. Kazem, and K. Parand, A numerical solution of the nonlinear controlled duffing oscillator by radial basis functions, Comput. Math. with Appl. 64(6) (2012), 2049–2065.
- [28] M. Rasty and M. Hadizadeh, A product integration approach based on new orthogonal polynomials for nonlinear weakly singular integral equations, Acta Appl. Math., 109(3) (2010), 861–873.
- [29] L. Torkzadeh, Numerical behavior of nonlinear duffing equations with fractional damping, Rom. Rep. Phys., 73 (2021), 113.
- [30] M. P. Tripathi, V. K. Baranwal, R. K. Pandey, and O. P. Singh, A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions, Commun. Nonlinear Sci. Numer. Simul., 18(6) (2013), 1327-1340.
- [31] J. Wang, J. Zhou, and B. Peng, Weak signal detection method based on duffing oscillator, Kybernetes, 38(10) (2009), 1662–1668.
- [32] Z. Yang, Gr¨obner bases for solving multivariate polynomial equations, Computing Equilibria and Fixed Points, Springer, 1999, 265–288.
- [33] E. Yusufo˘glu, Numerical solution of duffing equation by the laplace decomposition algorithm, Appl. Math. Comput., 177(2) (2006), 572–580.
- [34] H. Zhang, Y. Mo, and Z. Wang, A high order difference method for fractional sub- diffusion equations with the spatially variable coefficients under periodic boundary conditions, J. Appl. Anal. Comput., 10(2) (2020), 474–485.
|