Lucas polynomial solution of nonlinear differential equations with variable delays

Year 2020, Volume: 49 Issue: 2, 553 - 564, 02.04.2020

Sevin Gümgüm , Nurcan Baykuş Savaşaneril Ömür Kıvanç Kürkçü , Mehmet Sezer

https://doi.org/10.15672/hujms.460975

Cited By: 22

https://izlik.org/JA67GU63EJ

Abstract

In this study, a novel matrix method based on Lucas series and collocation points has been used to solve nonlinear differential equations with variable delays. The application of the method converts the nonlinear equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Lucas coefficients. The method is tested on three problems to show that it allows both analytical and approximate solutions.

Keywords

Nonlinear delay differential equations , Variable delays , Matrix and collocation methods , Lucas polynomials and series.

References

• [1] W.M. Abd-Elhameed and Y.H. Youssri, A novel operational matrix of Caputo frac- tional derivatives of Fibonacci polynomials: Spectral solutions of fractional differential equations, Entropy, 18 (345), 2016.
• [2] W.M. Abd-Elhameed and Y.H. Youssri, Spectral solutions for fractional differential equations via a novel Lucas operational matrix of fractional derivatives, Rom. J. Phys. 61 (5-6), 795–813, 2016.
• [3] W.M. Abd-Elhameed and Y.H. Youssri, Generalized Lucas polynomial sequence ap- proach for fractional differential equations, Nonlinear Dyn. 89, 1341–1355, 2017.
• [4] W.M. Abd-Elhameed and Y.H. Youssri, Spectral tau algorithm for certain coupled sys- tem of fractional differential equations via generalized Fibonacci polynomial sequence, Iran J. Sci. Technol. A, doi.org/10.1007/s40995-017-0420-9, 2017.
• [5] K.S. Aboodh, R.A. Farah, I.A. Almardy and A.K. Osman, Solving delay differential equations by Aboodh transformation method, Int. J. Appl. Math. Stats. Sci. 7 (1), 21–30, 2018.
• [6] I. Ali, H. Brunner, and T. Tang, A spectral method for pantograph-type delay dif- ferential equations and its convergence analysis, J. Comp. Math. 27 (2-3), 254–265, 2009.
• [7] A. Ardjouni and A. Djoudi, Fixed points and stability in linear neutral differential equations with variable delays, Nonlinear Anal. 74, 2062–2070, 2011.
• [8] A.G. Atta, G.M. Moatimid, and Y.H. Youssri, Generalized Fibonacci operational col- location approach for fractional initial value problems, Int. J. Appl. Comput. Math 5 (9), 2019.
• [9] N. Baykuş-Savaşaneril and M. Sezer, Hybrid Taylor-Lucas collocation method for nu- merical solution of high-order Pantograph type delay differential equations with vari- ables delays, Appl. Math. Inf. Sci. 11 (6), 1795–1801, 2017.
• [10] B. Benhammouda and H. Vazquez-Leal, A new multi-step technique with differential transform method for analytical solution of some nonlinear variable delay differential equations, SpringerPlus 5 (1723), 2016.
• [11] B. Benhammouda, H. Vazquez-Leal and L. Hernandez-Martinez, Procedure for exact solutions of nonlinear pantograph delay differential equations, British J. Math. Comp. Sci. 40 (19), 2738–2751, 2014.
• [12] L. Blanco-Cocom, A.G. Estrella and E. Avila-Vales, Solving delay differential systems with history functions by the Adomian decomposition method, Appl. Math. Comput. 218, 5994–6011 2012.
• [13] B. Cˇaruntu and C. Bota, Analytical approximate solutions for a general class of non- linear delay differential equations, Scientific World J. 2014, (6 pp), 2014.
• [14] G. Chen, L. Dingshi, O.V. Gaans and S.V. Lunel, Stability of nonlinear Neutral delay differential equations with variable delays, Electron. J. Differ. Eq. 2017 (118), 1–14, 2017.
• [15] M. Çetin, M. Sezer and C. Güler, Lucas polynomial approach for system of high-order linear differential equations and residual error estimation, Math. Prob. Eng. 2015, (14 pp), 2015.
• [16] S. Davaeifar and J. Rashidinia, Solution of a system of delay differential equations of multipantograph type, J. Taibah Univ. Sci. 11, 1141–1157, 2017.
• [17] L. Ding, X. Li, and Z. Li, Fixed points and stability in nonlinear equations with variable delays, Fixed Point Theory Appl. 2010, (14pp), 2010.
• [18] J.G. Dix, Asymptotic behavior of solutions to a first-order differential equation with variable delays, Comp. Math. Appl. 50, 1791–1800, 2005.
• [19] B. Dorociaková and R. Olach, Some notes to existence and stability of the positive periodic solutions for a delayed nonlinear differential equations, Open Math. 14, 361– 369, 2016.
• [20] S. Gümgüm, N. Baykuş-Savaşaneril, Ö.,K. Kürkçü and M. Sezer, A numerical tech- nique based on Lucas polynomials together with standard and Chebyshev-Lobatto col- location points for solving functional integro-differential equations involving variable delays, Sakarya Univ. J. Sci. 22 (6), 2018.
• [21] P. Ha, Analysis and numerical solutions of delay differential-slgebraic equations, PhD Thesis, Berlin, 4-9, 2015.
• [22] F. Ismail, R.A. Al-Khasawneh, A.S. Lwin and M. Suleiman, Numerical treatment of delay differential equations by Runge-Kutta method using Hermite interpolation, Matematika 18 (2), 79–90, 2002.
• [23] C. Jin and J. Luo, Fixed points and stability in neutral differential equations with variable delay, Proc. Amer. Math. Soc. 136, 909–918, 2008.
• [24] S. Karimi-Vanani and A. Aminataei, On the numerical solution of nonlinear delay differential equations, J. Concrete Applicable Math. 8 (4), 568–576, 2010.
• [25] M.M. Khader, Numerical and theoretical treatment for solving linear and nonlinear delay differential equations using variational iteration method, Arab J. Math. Sci. 19 (2), 243–256, 2013.
• [26] E. Lucas, Theorie de fonctions numeriques simplement periodiques, Amer. J. Math. 1, 184–240; 289–321, 1878.
• [27] J.A. Martín and O. García, Variable multistep methods for delay differential equations, Math. Comput. Model. 35, 241–257, 2002.
• [28] F. Mirzaee and L. Latifi, Numerical solution of delay differential equations by differ- ential transform method, J. Sci. I. A. U. 20 (78/2), 83–88, 2011.
• [29] S.T. Mohyud-Dina and A. Yıldırım, Variational iteration method for delay differential equations using He’s polynomials, Z. Naturforsch. 65a, 1045–1048, 2010.
• [30] H.J. Oberle and H.J. Pesch, Numerical treatment of delay differential equations by Hermite interpolation, Numer. Math. 37, 235–255, 1981.
• [31] F.O. Ogunfiditimi, Numerical solution of delay differential equations using the Ado- mian decomposition method, Int. J. Eng. Sci. 4 (5), 18–23, 2015.
• [32] M. Özel, M. Tarakçı and M. Sezer, A numerical approach for a nonhomogeneous differential equation with variable delays, Math. Sci. 12, 145–155, 2018.
• [33] L. Pezza and F. Pitolli, A multiscale collocation method for fractional differential problems, Math. Comp. Simul. 174, 210–219, 2018.
• [34] A.S.V. Ravi-Kanth and P. Murali-Mohan Kumar, A numerical technique for solving nonlinear singularly perturbed delay differential equations, Math. Model. Anal. 23 (1), 64–78, 2018.
• [35] F. Shakeri and M. Dehghan, Solution of delay differential equations via a homotopy perturbation method, Math. Comput. Model. 48, 486–498, 2008.
• [36] S.C. Shiralashetti, B.S. Hoogar and S. Kumbinarasaiah, Hermite wavelet based method for the numerical solution of linear and nonlinear delay differential equations, Int. J. Eng. Sci. Math. 6 (8), 71–79, 2017.
• [37] A.G. Stephen and Y. Kuang, A stage structured predator-prey model and its depen- dence on maturation delay and death rate, J. Math. Biol. 49, 188-200, 2004.
• [38] A.G. Stephen and Y. Kuang, A delay reaction-diffusion model of the spread of bacte- riophage infection, SIAM J. Appl. Math. 65 (2), 550–566, 2005.
• [39] O.A. Taiwo and O.S. Odetunde, On the numerical approximation of delay differential equations by a decomposition method, Asian J. Math. Stat. 3 (4), 237–243, 2010.
• [40] Z.Q. Wang and L.L. Wang, A Legendre-Gauss collocation method for nonlinear delay differential equations, Discrete Continuous Dyn. Syst. Ser. B 13 (3), 685–708, 2010.
• [41] A. Yıldırım, H. Kocak, and S. Tutkun, Reliable analysis for delay differential equations arising in mathematical biology, J. King Saud Univ. Sci. 24, 359–365, 2012.
• [42] H.Y. Youssri, A new operational matrix of Caputo fractional derivatives of Fermat polynomials: an application for solving the Bagley-Torvik equation, Adv. Diff. Eq. 2017 (73), 2017.
• [43] H.Y. Youssri and W.M. Abd-Elhameed, Spectral solutions for multi-term fractional initial value problems using a new Fibonacci operational matrix of fractional integra- tion, Progr. Fract. Differ. Appl. 2 (2), 141–151, 2016.
• [44] B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Anal. 63, 233–242, 2005.