Lucas Polynomial Approach for Second Order Nonlinear Differential Equations

Yıl 2020, Cilt: 24 Sayı: 1, 230 - 236, 20.04.2020

Sevin Gümgüm , Nurcan Baykuş-savaşaneril Ömür Kıvanç Kürkçü , Mehmet Sezer

https://doi.org/10.19113/sdufenbed.546847

Cited By: 5

Öz

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

Anahtar Kelimeler

Lucas polynomial, Operational matrices, Collocation points

İkinci Mertebeden Doğrusal Olmayan Diferansiyel Denklemler için Lucas Polinom Yaklaşımı

Öz

Bu makale, karışık koşullar altında ikinci mertebeden doğrusal olmayan adi diferansiyel denklemlerin Lucas polinom çözümünü oluşturur. Lucas matris yöntemi sıralama noktaları ile birlikte sınırlandırılmış Lucas serisine dayanmaktadır. Yöntemin en büyük avantajı matris bağıntılarında elde edilen doğrusal olmayan cebirsel sistemi ele almak için basit bir yapıya sahip olmasıdır. Yöntem dört probleme uygulanır. İlk iki problemde, tam çözümler elde edilir. Son iki problemde Bratu ve Duffing denklemleri sayısal olarak çözülür; sonuçlar, tam çözümler ve diğer bazı sayısal çözümler ile karşılaştırılır. Yöntemin uygulanması, tam ve doğru sayısal çözümler vermesine yol açtığı gözlemlenmektedir.

Anahtar Kelimeler

Lucas polinomu, İşlevsel matrisler, Sıralama noktaları

Kaynakça

• [1] Danaila, I., Joly, P., Kaber, S.M., Postel, M. 2007. Nonlinear Differential Equations: Application to Chemical Kinetics. An Introduction to Scientific Computing, Springer, New York, NY.
• [2] Fay, T.H., Graham, S.D. 2003. Coupled spring equations, Int. J. Math. Educ. Sci. Technol., 34(1), 65–79.
• [3] Bostancı, B., Karahan, M.M.F. 2018. Nonlinear Oscillations of a Mass Attached to Linear and Nonlinear Springs in Series Using Approximate Solutions, Celal Bayar Univ. J. Sci., 14(2), 201–207.
• [4] Cruz, H., Schuch, D., Casta ˜ nos, O., Rosas-Ortiz, O. 2015. Time-evolution of quantum systems via a complex nonlinear Riccati equation. I. Conservative systems with time-independent Hamiltonian, arXiv:1505.02687v1 [quant-ph].
• [5] Ganji, D.D., Nourollahi, M., Mohseni, E. 2007. Application of He’s methods to nonlinear chemistry problems, Computers Math. with Appl., 54(7-8), 1122–1132.
• [6] Martens, P.C.H. 1984. Applications of nonlinear methods in astronomy, Physics Reports (Review Section of Physics Letters) 115(6), 315–378, North Holland, Amsterdam.
• [7] Ilea, M., Turnea, M., Rotariu, M. 2012. Ordinary differential equations with applications in molecular biology, 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 Lucas polynomials together with standard and Chebyshev-Lobatto collocation points for solving functional integro-differential equations involving variable delays, 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 of nonlinear 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. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia.
• [11] Chandrasekhar, S. 1967. Introduction to the Study of Stellar Structure. Dover, New York. 235 S. Gümgüm et al. / Lucas Polynomial Approach
• [12] Aregbesola, Y. 2003. Numerical solution of Bratu problem using the method of weighted residual, Electronic J. Southern African Math. Sci. Assoc., 3, 1–7.
• [13] Wazwaz, A.M. 2005. Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comput., 166, 652–663.
• [14] Vahidi, A.R., Hasanzade, M. 2012. Restarted Adomian’s Decomposition Method for the Bratu-Type Problem. Appl. Math. Sci. 6(10), 479–486.
• [15] Mohsen, A.2014. A simple solution of the Bratu problem, Comput. Math. Appl., 67, 26–33.
• [16] Deeba, E., Khuri, S.A., Xie, S. 2000. An Algorithm for Solving Boundary Value Problems, J. Comput. Phy., 159, 125–138.
• [17] Venkatesh, S.G., Ayyaswamy, S.K., Balachandar, S.R. 2012. The Legendre wavelet method for solving initial value problems of Bratu-type, Comput. Math. Appl. 63, 1287–1295.
• [18] Kazemi Nasab, A., Pashazadeh Atabakan, Z., Kılıçman, A. 2013. An Efficient Approach for Solving Nonlinear Troesch’s and Bratu’s Problems by Wavelet Analysis Method, Math. Problems Eng., 2013, 1–10.
• [19] Caglar, H., Caglar, N., Özer, M., Valarıstos, A., Anagnostopoulos, A.N. 2010. B-spline method for solving Bratu’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 for Solving Initial Value Problems of Bratu Type, Comput. Math. Math. Phys., 53(9), 1292–1302.
• [21] Khuri, S.A. 2004. Laplace transform decomposition numerical algorithm is introduced for solving Bratu’s problem, Appl. Math. Comput. 147, 131–136.
• [22] Batiha, B. 2010. Numerical solution of Bratu-type equations by the variational iteration model, Hacettepe J. Math Stat., 39(1), 23–29.
• [23] Saravi, M., Hermann, M., Kaiser, D. 2013. Solution of Bratu’s Equation by He’s Variational Iteration Method, American J. Comput. Appl. Math., 3(1), 46–48.
• [24] Zauderer, E. 1983. Partial Differential Equations of Applied Mathematics. Wiley, New York.
• [25] Al-Jawary, M.A., Abd-Al-Razaq, S.G. 2016. Analytic and numerical solution for Duffing equations, Int. J. Basic Appl. Sci., 5(2), 115–119.
• [26] Bülbül, B., Sezer, M. 2013. Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method, J. Appl. Math., 2013, 1–6.
• [27] Liu, G.R., Wu, T.Y. 2000. Numerical solution for differential equations of Duffing-type non-linearity using the generalized quadrature rule, J. Sound Vib., 237(5), 805–817.
• [28] Anapalı, A., Yalçın, Ö., Gülsu, M. 2015. Numerical Solutions of Duffing Equations Involving Linear Integral with Shifted Chebyshev Polynomials, AKU J. Sci. Eng., 15, 1–11.
• [29] Kaminski, M., Corigliano, A. 2015. Numerical solution of the Duffing equation with random coefficients, Mechanica, 50(7), 1841–1853.
• [30] Yusufoğlu, E. 2006. Numerical solution of Duffing equation by the Laplace decomposition algorithm. Appl. Math. Comput., 177(2), 572–580.
• [31] Constandache, A., Das, A., Toppan, F. 2002. Lucas polynomials and a standart Lax representation for the polyropic gas dynamics, Lett. Math. Phys., 60(3), 197– 209.
• [32] Lucas, E. 1878. Theorie de fonctions numeriques simplement periodiques, Amer. J. Math. 1, 184–240; 289–321.