Sevin GÜMGÜM, ÖMÜR KIVANÇ KÜRKÇÜ, Nurcan BAYKU S SAVA SANER IL, Mehmet SEZER

Lucas Polynomial Approach for Second Order Nonlinear Differential Equations

This paper presents the Lucas polynomial solution of second-order nonlinearordinary differential equations with mixed conditions. Lucas matrix method is based oncollocation points together with truncated Lucas series. The main advantage of the methodis that it has a simple structure to deal with the nonlinear algebraic system obtained frommatrix relations. The method is applied to four problems. In the first two problems, exactsolutions are obtained. The last two problems, Bratu and Duffing equations are solvednumerically; the results are compared with the exact solutions and some other numericalsolutions. It is observed that the application of the method results in either the exact oraccurate numerical solutions.

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[1]Danaila, I., Joly, P., Kaber, S.M., Postel, M. 2007.Nonlinear Differential Equations: Application toChemical Kinetics. An Introduction to Scientific Com-puting, Springer, New York, NY.

[2]Fay, T.H., Graham, S.D. 2003. Coupled spring equa-tions, Int. J. Math. Educ. Sci. Technol., 34(1), 65–79.

[3]Bostancı, B., Karahan, M.M.F. 2018. Nonlinear Os-cillations of a Mass Attached to Linear and NonlinearSprings in Series Using Approximate Solutions, CelalBayar Univ. J. Sci., 14(2), 201–207.

[4]Cruz, H., Schuch, D., Casta ̃nos, O., Rosas-Ortiz,O. 2015. Time-evolution of quantum systems viaa complex nonlinear Riccati equation. I. Conser-vative systems with time-independent Hamiltonian,arXiv:1505.02687v1 [quant-ph].

[5]Ganji, D.D., Nourollahi, M., Mohseni, E. 2007. Ap-plication of He’s methods to nonlinear chemistry prob-lems, Computers Math. with Appl., 54(7-8), 1122–1132.

[6]Martens, P.C.H. 1984. Applications of nonlinear meth-ods in astronomy, Physics Reports (Review Sectionof Physics Letters) 115(6), 315–378, North Holland,Amsterdam.

[7]Ilea, M., Turnea, M., Rotariu, M. 2012. Ordinarydifferential equations with applications in molecularbiology, Rev. Med. Chir. Soc. Med. Nat. Iasi., 116(1),347–52.

[8]Gümgüm, S., Bayku ̧s-Sava ̧saneril, N., Kürkçü, Ö.K.,Sezer, M. 2018. A numerical technique based on Lucaspolynomials together with standard and Chebyshev-Lobatto collocation points for solving functionalintegro-differential equations involving variable de-lays, Sakarya Univ. J. Sci., 22(6), 1659–1668.

[9]Gümgüm, S., Bayku ̧s Sava ̧saneril, N., Kürkçü, Ö.K.,Sezer, M. 2019.Lucas polynomial solution ofnonlinear differential equations with variable delays,Hacettepe J. Math. Stat. 1–12. DOI: 10.15672/hujms.460975.

[10]Ascher, U.M., Matheij, R., Russell, R.D. 1995. Nu-merical Solution of Boundary Value Problems for Or-dinary Differential Equations. Society for Industrialand Applied Mathematics (SIAM), Philadelphia.

[11]Chandrasekhar, S. 1967. Introduction to the Study ofStellar Structure. Dover, New York.

[12]Aregbesola, Y. 2003. Numerical solution of Bratuproblem using the method of weighted residual, Elec-tronic J. Southern African Math. Sci. Assoc., 3, 1–7.

[13]Wazwaz, A.M. 2005.Adomian decompositionmethod for a reliable treatment of the Bratu-type equa-tions, Appl. Math. Comput., 166, 652–663.

[14]Vahidi, A.R., Hasanzade, M. 2012. Restarted Ado-mian’s Decomposition Method for the Bratu-TypeProblem. Appl. Math. Sci. 6(10), 479–486.

[15]Mohsen, A.2014. A simple solution of the Bratuproblem, Comput. Math. Appl., 67, 26–33.

[16]Deeba, E., Khuri, S.A., Xie, S. 2000. An Algo-rithm for Solving Boundary Value Problems, J. Com-put. Phy., 159, 125–138.

[17]Venkatesh, S.G., Ayyaswamy, S.K., Balachandar,S.R. 2012. The Legendre wavelet method for solvinginitial value problems of Bratu-type, Comput. Math.Appl. 63, 1287–1295.

[18]Kazemi Nasab, A., Pashazadeh Atabakan, Z., Kılıç-man, A. 2013. An Efficient Approach for SolvingNonlinear Troesch’s and Bratu’s Problems by WaveletAnalysis Method, Math. Problems Eng., 2013, 1–10.

[19]Caglar, H., Caglar, N., Özer, M., Valarıstos, A., Anag-nostopoulos, A.N. 2010. B-spline method for solvingBratu’s problem, Int. J. Computer Math., 87(8), 1885–1891.

[20]Doha, E.H., Bhrawy, A.H., Baleanud, D., Hafez, R.M.2013. Efficient Jacobi-Gauss Collocation Method forSolving Initial Value Problems of Bratu Type, Comput.Math. Math. Phys., 53(9), 1292–1302.

[21]Khuri, S.A. 2004. Laplace transform decompositionnumerical algorithm is introduced for solving Bratu’sproblem, Appl. Math. Comput. 147, 131–136.

[22]Batiha, B. 2010. Numerical solution of Bratu-typeequations by the variational iteration model, HacettepeJ. Math Stat., 39(1), 23–29.

[23]Saravi, M., Hermann, M., Kaiser, D. 2013. Solu-tion of Bratu’s Equation by He’s Variational IterationMethod, American J. Comput. Appl. Math., 3(1), 46–48.

[24]Zauderer, E. 1983. Partial Differential Equations ofApplied Mathematics. Wiley, New York.

[25]Al-Jawary, M.A., Abd-Al-Razaq, S.G. 2016. Ana-lytic and numerical solution for Duffing equations, Int.J. Basic Appl. Sci., 5(2), 115–119.

[26]Bülbül, B., Sezer, M. 2013. Numerical Solution ofDuffing Equation by Using an Improved Taylor MatrixMethod, J. Appl. Math., 2013, 1–6.

[27]Liu, G.R., Wu, T.Y. 2000. Numerical solution fordifferential equations of Duffing-type non-linearityusing the generalized quadrature rule, J. Sound Vib.,237(5), 805–817.

[28]Anapalı, A., Yalçın, Ö., Gülsu, M. 2015. NumericalSolutions of Duffing Equations Involving Linear Inte-gral with Shifted Chebyshev Polynomials, AKU J. Sci.Eng., 15, 1–11.

[29]Kaminski, M., Corigliano, A. 2015. Numerical solu-tion of the Duffing equation with random coefficients,Mechanica, 50(7), 1841–1853.

[30]Yusufo ̆glu, E. 2006. Numerical solution of Duff-ing equation by the Laplace decomposition algorithm.Appl. Math. Comput., 177(2), 572–580.

[31]Constandache, A., Das, A., Toppan, F. 2002. Lucaspolynomials and a standart Lax representation for thepolyropic gas dynamics, Lett. Math. Phys., 60(3), 197–209.

[32]Lucas, E. 1878. Theorie de fonctions numeriquessimplement periodiques, Amer. J. Math. 1, 184–240;289–321.