In the current work, a new reproducing kernel method (RKM) for solving nonlinear forced Duffing equations with integral boundary conditions is developed. The proposed collocation technique is based on the idea of RKM and the orthonormal Bernstein polynomials (OBPs) approximation together with the quasi-linearization method. In our method, contrary to the classical RKM, there is no need to use the Gram-Schmidt orthogonalization procedure and only a few nodes are used to obtain efficient numerical results. Three numerical examples are included to show the applicability and efficiency of the suggested method. Also, the obtained numerical results are compared with some results in the literature.
[1] S. Abbasbandy and H. R. Khodabandehlo, Application of reproducing kernel Hilbert space method for generalized 1-D linear telegraph equation, International Journal of Nonlinear Analysis and Applications, 13 (2022), 487–497.
[2] B. Ahmad and A. Alsaedi, Existence of approximate solutions of the forced Duffing equation with discontinuous type integral boundary conditions, Nonlinear Analysis: Real World Applications, 10 (2009), 358–367.
[3] B. Ahmad, S. K. Ntouyas, and A. Alsaedi, On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions, Chaos Solitons and Fractals, 83 (2016), 234–241.
[4] A. Alipanah, M. Pendar, and K. Sadeghi, Integrals involving product of polynomials and Daubechies scale functions, Mathematics Interdisciplinary Research, 6 (2021), 275–291.
[5] R. E. Bellman and R. E. Kalaba, Quasilinearization and Nonlinear Boundary-value Problems, Elsevier publishing company, New York, 1965.
[6] M. A. Bellucci, On the explicit representation of orthonormal Bernstein polynomials, arXiv:1404.2293v2, (2014).
[7] A. Boucherif, Second-order boundary value problems with integral boundary conditions, Nonlinear Analysis: Theory, Methods and Applications, 70 (2009), 364–371.
[8] M. Chai and L. Ba, Application of EEG signal recognition method based on Duffing equation in psychological stress analysis, Advances in Mathematical Physics, 2021 (2021), article ID 1454547.
[9] Z. Chen, W. Jiang, and H. Du, A new reproducing kernel method for Duffing equations, International Journal of Computer Mathematics, 98 (2021), 2341–2354.
[10] A. A. Cherevko, E. E. Bord, A. K. Khe, V. A. Panarin, and K. J. Orlov, The analysis of solutions behaviour of Van der Pol Duffing equation describing local brain hemodynamics, Journal of Physics: Conference Series, 894 (2017), 012012.
[11] M. Cui and Y. Lin, Nonlinear numerical analysis in the reproducing kernel space, Nova Science Publishers, Inc., 2009.
[12] F. Deutsch , Best Approximation in Inner Product Spaces, Springer, New York, 2001.
[13] M. R. Doostdar, M. Kazemi, and A. Vahidi, A numerical method for solving the Duffing equation involving both integral and non-integral forcing terms with separated and integral boundary conditions, Computational Methods for Differential Equations, 11 (2023), 241–253.
[14] J. Du and M. Cui, Solving the forced Duffing equation with integral boundary conditions in the reproducing kernel space, International Journal of Computer Mathematics, 87 (2010), 2088–2100.
[15] S. Farzaneh Javan, S. Abbasbandy, and M. A. Fariborzi Araghi, Application of reproducing kernel Hilbert space method for solving a class of nonlinear integral equations, Mathematical Problems in Engineering, 2017 (2017), article ID 7498136.
[16] F. Z. Geng and M. Cui, New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions, Journal of Computational and Applied Mathematics, 233 (2009), 165–172.
[17] H. Jafari and H. Tajadodi, Electro-spunorganic nanofibers elaboration process investigations using BPs operational matrices, Iranian Journal of Mathematical Chemistry, 7 (2016), 19-27.
[18] M. Khaleghi, M. T. Moghaddam, E. Babolian, and S. Abbasbandy, Solving a class of singular two-point boundary value problems using new effective reproducing kernel technique, Applied Mathematics and Computation, 331 (2018), 264–273.
[19] V. Lakshmikantham and A. S. Vatsala, Generalized Quasilinearization for Nonlinear Problems, MIA, Kluwer Academic Publishers, Dordrecht, 1998.
[20] M. Lotfi and A. Alipanah, Legendre spectral element method for solving Volterra-integro differential equations, Results in Applied Mathematics, 7 (2020), 100116.
[21] V. B. Mandelzweig and F. Tabakin, Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs, Computer Physics Communications, 141 (2001), 268–281.
[22] S. Mashayekhi, Y. Ordokhani, and M. Razzaghi, A hybrid functions approach for the Duffing equation, Physica Scripta, 88 (2013), 025002.
[23] R. Najafi and B. Nemati Saray, Numerical solution of the forced Duffing equations using Legendre multiwavelets, Computational Methods for Differential Equations, 5 (2017), 43–55.
[24] J. Niu, M. Xu, Y. Lin, and Q. Xue, Numerical solution of nonlinear singular boundary value problems, Journal of Computational and Applied Mathematics, 331 (2018), 42–51.
[25] K. Parand, Y. Lotfi, and J. A. Rad, An accurate numerical analysis of the laminar two-dimensional flow of an incompressible Eyring-Powell fluid over a linear stretching sheet, The European Physical Journal Plus, 132 (2017), 1–21.
[26] A. Saadatmandi, A. Ghasemi-Nasrabady, and A. Eftekhari, Numerical study of singular fractional Lane-Emden type equations arising in astrophysics, Journal of Astrophysics and Astronomy, 40 (2019), 1–12.
[27] A. Saadatmandi and S. Yeganeh, New approach for the Duffing equation involving both integral and non-integral forcing terms, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 79 (2017), 46–52.
[28] R. Srebro, The Duffing oscillator: a model for the dynamics of the neuronal groups comprising the transient evoked potential, Electroencephalography and clinical Neurophysiology, 96 (1995), 561–573.
[29] J. J. Stokes, Nonlinear Vibrations, Intersciences, New York, 1950.
[30] M. Xu and E. Tohidi, A Legendre reproducing kernel method with higher convergence order for a class of singular two-point boundary value problems, Journal of Applied Mathematics and Computing, 67 (2021), 405–421.
[31] W. Yulan, T. Chaolu, and P. Jing, New algorithm for second-order boundary value problems of integro-differential equation, Journal of Computational and Applied Mathematics, 229 (2009), 1–6.
Ghasemi, A., & Saadatmandi, A. (2024). A new Bernstein-reproducing kernel method for solving forced Duffing equations with integral boundary conditions. Computational Methods for Differential Equations, 12(2), 329-337. doi: 10.22034/cmde.2023.57413.2401
MLA
Azam Ghasemi; Abbas Saadatmandi. "A new Bernstein-reproducing kernel method for solving forced Duffing equations with integral boundary conditions". Computational Methods for Differential Equations, 12, 2, 2024, 329-337. doi: 10.22034/cmde.2023.57413.2401
HARVARD
Ghasemi, A., Saadatmandi, A. (2024). 'A new Bernstein-reproducing kernel method for solving forced Duffing equations with integral boundary conditions', Computational Methods for Differential Equations, 12(2), pp. 329-337. doi: 10.22034/cmde.2023.57413.2401
VANCOUVER
Ghasemi, A., Saadatmandi, A. A new Bernstein-reproducing kernel method for solving forced Duffing equations with integral boundary conditions. Computational Methods for Differential Equations, 2024; 12(2): 329-337. doi: 10.22034/cmde.2023.57413.2401