Adapted block hybrid method for the numerical solution of Duffing equations and related problems

Ridwanulahi Iyanda Abdulganiy; Shiping Wen; Yuming Feng; Wei Zhang; Ning Tang; Ridwanulahi Iyanda Abdulganiy; Shiping Wen; Yuming Feng; Wei Zhang; Ning Tang

doi:10.3934/math.2021810

AIMS Mathematics

2021, Volume 6, Issue 12: 14013-14034. doi: 10.3934/math.2021810

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Research article

Adapted block hybrid method for the numerical solution of Duffing equations and related problems

1.
Key Laboratory of Intelligent Information Processing and Control, Chongqing Three Gorges University, Wanzhou, Chongqing 404100, China
2.
Department of Education, Distance Learning Institute, University of Lagos, Akoka, Nigeria
3.
Australian AI Institute, University of Technology Sydney, Ultimo NSW 2007, Australia
4.
Chongqing Engineering Research Center of Internet of Things and Intelligent Control Technology, Chongqing Three Gorges University, Wanzhou, Chongqing 404100, China
5.
School of Three Gorges Artificial Intelligence, Chongqing Three Gorges University, Wanzhou, Chongqing 404100, China

Received: 26 June 2021 Accepted: 21 September 2021 Published: 28 September 2021
MSC : 65L03, 65L05, 65L50

Problems of non-linear equations to model real-life phenomena have a long history in science and engineering. One of the popular of such non-linear equations is the Duffing equation. An adapted block hybrid numerical integrator that is dependent on a fixed frequency and fixed step length is proposed for the integration of Duffing equations. The stability and convergence of the method are demonstrated; its accuracy and efficiency are also established.
- adapted method,
- convergence,
- Duffing equation,
- hybrid,
- non-linear equation
Citation: Ridwanulahi Iyanda Abdulganiy, Shiping Wen, Yuming Feng, Wei Zhang, Ning Tang. Adapted block hybrid method for the numerical solution of Duffing equations and related problems[J]. AIMS Mathematics, 2021, 6(12): 14013-14034. doi: 10.3934/math.2021810

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Abstract

Problems of non-linear equations to model real-life phenomena have a long history in science and engineering. One of the popular of such non-linear equations is the Duffing equation. An adapted block hybrid numerical integrator that is dependent on a fixed frequency and fixed step length is proposed for the integration of Duffing equations. The stability and convergence of the method are demonstrated; its accuracy and efficiency are also established.

References

[1]	S. N. Jator, H. B. Oladejo, Block Nyström method for singular differential equations of the Lane-Emdem Type and problems with highly oscillatory solutions, Int. J. Appl. Comput. Math., 3 (2017), 1385–1402. doi: 10.1007/s40819-017-0376-7
[2]	T. Monovasilis, Z. Kalogiratou, H. Ramos, T. E. Simos, Modified two-step hybrid methods for the numerical integration of oscillatory problems, Math. Method Appl. Sci., 40 (2017), 5286–5294. doi: 10.1002/mma.4386
[3]	J. Sunday, Y. Skwane, M. R. Odekunle, A continuous block integrator for the solution of stiff and oscillatory differential equations, IOSR J. Math., 8 (2013), 75–80.
[4]	W. H. Enright, Second derivative multistep method for stiff ODEs, SIAM J. Numer. Anal., 11 (1974), 321–331. doi: 10.1137/0711029
[5]	E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag, Berlin, 1993.
[6]	J. D. Lambert, I. A. Watson, Symmetric multistep methods for periodic initial value problems, IMA J. Appl. Math., 18 (1976), 189–202. doi: 10.1093/imamat/18.2.189
[7]	S. N. Jator, Solving second order initial value problems by a hybrid multistep method without predictors, Appl. Math. Comput., 277 (2010), 4036–4046.
[8]	J. M. Franco, An embedded pair of exponentially fitted explicit Runge-Kutta methods, J. Comput. Appl. Math., 149 (2002), 407–414. doi: 10.1016/S0377-0427(02)00485-5
[9]	J. M. Franco, Exponentially-fitted explicit Runge-Kutta-Nyström methods, J. Comput. Appl. Math., 167 (2003), 1–19.
[10]	W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math., 3 (1961), 381–397. doi: 10.1007/BF01386037
[11]	B. Neta, C. H. Ford, Families of methods for ordinary differential equations based on trigonometric polynomials, J. Comp. Appl. Math., 10 (1984), 33–38. doi: 10.1016/0377-0427(84)90066-9
[12]	B. Neta, Families of backward differentiation methods based on trigonometric polynomials, Int. J. Comput. Math., 20 (1986), 67–75. doi: 10.1080/00207168608803532
[13]	B. B. Sanugi, D. J. Evans, The numerical solution of oscillatory problems, Int. J. Comput. Math., 31 (1989), 237–255.
[14]	S. C. Duxbury, Mixed Collocation Methods for $y'' = f(x, y)$, Durham Theses, Durham University, 1999.
[15]	J. P. Coleman, S. C. Duxbury, Mixed collocation methods for $y'' = f(x, y)$, J. Comput. Appl. Math., 126 (2000), 47–75. doi: 10.1016/S0377-0427(99)00340-4
[16]	J. M. Franco, A class of explicit two-step hybrid methods for second-order IVPs, J. Comput. Appl. Math., 187 (2006), 41–57. doi: 10.1016/j.cam.2005.03.035
[17]	J. M. Franco, Exponentially fitted explicit Runge-Kutta-Nyström methods, J. Comput. Appl. Math., 167 (2004), 1–19. doi: 10.1016/j.cam.2003.09.042
[18]	Y. Fang, X. Wu, A trigonometrically fitted explicit Numerov-type method for second order initial value problems with oscillating solutions, Appl. Numer. Math., 58 (2008), 341–351. doi: 10.1016/j.apnum.2006.12.003
[19]	Y. Fang, Y. Song, X. Wu, A robust trigonometrically fitted embedded pair for perturbed oscillators, J. Comput. Appl. Math., 225 (2009), 347–355. doi: 10.1016/j.cam.2008.07.053
[20]	L. Gr. Ixaru, G. Vanden Berghe, M. Van Daele, Frequency evaluation in exponentially-fitted algorithms for ODEs, J. Comput. Appl. Math., 140 (2002), 423–434. doi: 10.1016/S0377-0427(01)00474-5
[21]	G. Vanden Berhe, M.Van Daele, Exponentially-fitted Numerov methods, J. Comput. Appl. Math., 200 (2007), 140–153. doi: 10.1016/j.cam.2005.12.022
[22]	J. Martin-Vaquero, J. Vigo-Aguiar, Exponential fitted Gauss, Radau and Lobatto methods of low order, Numer. Algorithms, 48 (2008), 327–346. doi: 10.1007/s11075-008-9202-y
[23]	F. F. Ngwane, S. N. Jator, Solving oscillatory problems using a block hybrid trigonometrically fitted method with two off-step points, Electron. J. Differ. Eq., 20 (2013), 119–132.
[24]	X. You, B. Chen, Symmetric and symplectic exponentially-Fitted Runge-Kutta-Nyström methods for Hamiltonian Problems, Math. Comput. Simulat., 94 (2013), 76–95. doi: 10.1016/j.matcom.2013.05.010
[25]	J. P. Coleman, L. G. Ixaru, P-stability and exponential fitting methods for $y'' = f(x, y)$, IMA J. Numer. Anal., 16 (1996), 179–199. doi: 10.1093/imanum/16.2.179
[26]	A. Konguetsof, T. E. Simos, An exponentially-fitted and trigonometrically-fitted methods for the numerical integration of periodic initial value problems, Comput. Math. Appl., 45 (2003), 547–554. doi: 10.1016/S0898-1221(03)80036-6
[27]	S. N. Jator, S. Swindell, R. D. French, Trigonmetrically fitted block numerov type method for $y{''} = f \left(x, y, y{'} \right)$, Numer. Algorithms, 62 (2013), 13–26. doi: 10.1007/s11075-012-9562-1
[28]	R. I. Abdulganiy, O. A. Akinfenwa, S. A. Okunuga, Maximal order block trigonometrically fitted scheme for the numerical treatment of second order initial value problem with oscillating solutions, IJMSO, 2017 168–186.
[29]	F. F. Ngwane, S. N. Jator, Trigonometrically-fitted second derivative method for oscillatory problems, SpringerPlus, 3 (2014).
[30]	M. A. Razzaq, An analytical approximate technique for solving cubic-quintic Duffing oscillator, Alex. Eng. J., 55 (2016), 2959–2965. doi: 10.1016/j.aej.2016.04.036
[31]	P. L. Ndukum, T. A. Biala, S. N. Jator, R. B. Adeniyi, On a family of trigonometrically fitted extended backward differentiation formulas for stiff and oscillatory initial value problems, Numer. Algorithms, 74 (2017), 267–287. doi: 10.1007/s11075-016-0148-1
[32]	H. Ramos, J. Vigo-Aguiar, On the frequency choice in trigonometrically fitted methods, Appl. Math. Lett., 23 (2010), 1378–1381. doi: 10.1016/j.aml.2010.07.003
[33]	L. Schovanec, J. T. White, A power series method for solving initial value problems utilizing computer algebra systems, Int. J. Comput. Math., 47 (1993), 181–189. doi: 10.1080/00207169308804175
[34]	S. Nourazar, A. Mirzabeigy, Approximate solution for nonlinear Duffing oscillator with damping effect using the modified differential transform method, Sci. Iran., 20 (2013), 364–368.
[35]	F. F. Ngwane, S. N. Jator, A family of trigonometrically fitted Enright second derivative methods for stiff and oscillatory initial value problems, J. Appl. Math., 2015 (2015) 1–17.
[36]	J. Vigo-Aguiar, H. Ramos, On the choice of the frequency in trigonometrically fitted methods for periodic problems, J. Comput. Appl. Math., 277 (2015), 94–105. doi: 10.1016/j.cam.2014.09.008
[37]	R. I. Abdulganiy, O. A. Akinfenwa, S. A. Okunuga, G. O. Oladimeji, A robust block hybrid trigonometric method for the numerical integration of oscillatory second order nonlinear initial value problems, Adv. Modell. Anal. A, 54 (2017), 497–518.
[38]	A. Belendez, C. Pascual, M. Ortuno, T. Belendez, S. Gallego, Application of a modified He's homotopy perturbation method to obtain higher-order aroximations to a nonlinear oscillator with discontinuities, Nonlinear Anal-Real., 10 (2009), 601–610. doi: 10.1016/j.nonrwa.2007.10.015
[39]	H. S. Nguyen, R. B. Sidje, N. H. Cong, Analysis of trigonometric implicit Runge-Kutta methods, J. Comput. Appl. Math., 198 (2007), 187–207. doi: 10.1016/j.cam.2005.12.006
[40]	S. N. Jator, A. O. Akinfenwa, S. A. Okunuga, A. B. Sofoluwe, High-order continuous third derivative formulas with block extension for $y{''} = f \left(x, y, y{'} \right)$, Int. J. Comput. Math., 90 (2003), 1899–1914.
[41]	J. D. Lambert, Computational Methods in Ordinary Differential System, the Initial Value Problem, New York: John Wiley & Sons, 1973.
[42]	S. O. Fatunla, Numerical Methods for Initial Value Problems in Ordinary Differential Equations, Cambridge: Academic Press Inc., 1988.
[43]	J. He, The homotopy perturbation method for nonlinear oscillators with discontinuous, Appl. Math. Comput., 151 (2004), 287–292.
[44]	A. Bel$\acute{e}$ndez, A. Hern$\acute{a}$ndez, T. Bel$\acute{e}$ndez, E. Fern$\acute{a}$ndez, M. L. $\acute{A}$lvarez, C. Neipp, Application of He's homotopy perturbation method to the duffing-harmonic oscillator, Int. J. Nonlin. Sci. Num., 8 (2007), 79–88.
[45]	S. O. Fatunla, Block methods for second order ODEs, Int. J. Comput. Math., 41 (1991), 55–63. doi: 10.1080/00207169108804026
[46]	R. I. Abdulganiy, O. A. Akinfenwa, S. A. Okunuga, Construction of L stable second derivative trigonometrically fitted block backward differentiation formula for the solution of oscillatory initial value problems, Afr. J. Sci. Technol. In., 10 (2018), 411–419.
[47]	D. V. V. Wend, Existence and uniqueness of solution of ordinary differential equations, P. Am. Math. Soc., 23 (1969), 27–33. doi: 10.1090/S0002-9939-1969-0245879-4
[48]	D. V. V. Wend, Uniqueness of solution of ordinary differential equations, Am. Math. Mon., 74 (1967), 948–950. doi: 10.1080/00029890.1967.12000056
[49]	C. Liu, W. Jhao, The power series method for a long-term solution of Duffing oscillator, Commun. Numer. Anal., 2014 (2014), 1–14.
[50]	N. Senu, M. Suleimon, F. Ismail, M. Othman, A new diagonally implicit Runge-Kutta-Nyströ m method for periodic IVPs, WSEAS Trans. Math., 9 (2010), 679–688.
[51]	J. Li, M. Lu, X. Qi, Trigonometrically fitted multi-step hybrid methods for oscillatory special second-order initial value problems, Int. J. Comput. Math., 95, (2018), 979–997.
[52]	I. Kovacic, M. J. Brennan, The Duffing Equation: Nonlinear Oscillators and Their Behaviour, Chichester: John Wiley & Sons, 2011.
[53]	R. I. Abdulganiy, O. A. Akinfenwa, S. A. Okunuga, A Simpson type trigonometrically fitted block scheme for numerical integration of oscillatory problems, UJMST, 5 (2017), 25–36.
[54]	R. I. Abdulganiy, Trigonometrically fitted block backward differentiation methods for first order initial value problems with periodic solution, Adv. Math. Comput., 28 (2018), 1–14.
[55]	R. I. Abdulganiy, O. A. Akinfenwa, O. A. Yusuff, O. E. Enobabor, S. A. Okunuga, Block third derivative trigonometrically-fitted methods for stiff and periodic problems, J. Niger. Soc. Phys. Sci., 2 (2020), 12–25.
[56]	T. Ozis, A. Yildirim, A study of nonlinear oscillators with $u^{1/3}$ force by He's variational iteration method, J. Sound Vib., 306 (2007), 372–376. doi: 10.1016/j.jsv.2007.05.021
[57]	Ch. Tsitouras, Explicit eight order two step methods with nine stages for integrating oscillatory problems, Int. J. Mod. Phys. C, 17 (2006), 861–876. doi: 10.1142/S0129183106009357
[58]	D. Younesian, H. Askari, Z. Saadatnia, M. K. Yazdi, Periodic solutions for nonlinear oscillation of a centrifugal governor system using the He's frequency-amplitude formulation and He's energy balance method, Nonlinear Sci. Lett. A, 2 (2011), 143–148.
[59]	F. Samat, E. S. Ismail, A two-step modified explicit hybrid method with step-size-dependent parameters for oscillatory problems, J. Math., 2020 (2020), Article ID 5108482482, 7 pages.
[60]	V. Marinca, N. Herişanu, Explicit and exact solutions to cubic Duffing and double-well Duffing equations, Math. Comput. Model., 53 (2011), 604–609. doi: 10.1016/j.mcm.2010.09.011
[61]	Z. Gholam-Ali, Y. Emmanuel, Exact solutions of a generalized autonomous Duffing-type equation, Appl. Math. Model., 39 (2015), 4607–4616. doi: 10.1016/j.apm.2015.04.027
[62]	X. Li, J. Shen, H. Akca, R. Rakkiyappan, LMI-based stability for singularly perturbed nonlinear impulsive differential systems with delays of small parameter, Appl. Math. Comput., 250 (2015), 798–804.
[63]	Y. Guo, A. C. J. Luo, Periodic motions in a double-well Duffing oscillator under periodic excitation through discrete implicit mappings, Int. J. Dyn. Control, 5 (2017), 1–16. doi: 10.1007/s40435-016-0251-0
[64]	A. C. J. Luo, Discretization and Implicit Mapping Dynamics: Nonlinear Physical Science, Springer: Higher Education Press, 2015.

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