In the current study, a one-step numerical algorithm is presented to solve strongly non-linear full fractional duffing equations. A new fractional-order operational matrix of integration via quasi-hat functions (QHFs) is introduced. Utilizing the operational matrices of QHFs, the main problem will be transformed into a number of univariate polynomial equations. Absolute errors of the results in approximations and convergence analysis are addressed. Ultimately, five examples are provided to illustrate the capabilities of this algorithm. The numerical results are illustrated in some Tables and Figures, for different values of the parameters $\alpha~ and~ \beta$.
[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.
Biazar, J., & Ebrahimi, H. (2024). A one-step algorithm for strongly non-linear full fractional duffing equations. Computational Methods for Differential Equations, 12(1), 117-135. doi: 10.22034/cmde.2023.53596.2256
MLA
Jafar Biazar; Hamed Ebrahimi. "A one-step algorithm for strongly non-linear full fractional duffing equations". Computational Methods for Differential Equations, 12, 1, 2024, 117-135. doi: 10.22034/cmde.2023.53596.2256
HARVARD
Biazar, J., Ebrahimi, H. (2024). 'A one-step algorithm for strongly non-linear full fractional duffing equations', Computational Methods for Differential Equations, 12(1), pp. 117-135. doi: 10.22034/cmde.2023.53596.2256
VANCOUVER
Biazar, J., Ebrahimi, H. A one-step algorithm for strongly non-linear full fractional duffing equations. Computational Methods for Differential Equations, 2024; 12(1): 117-135. doi: 10.22034/cmde.2023.53596.2256
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