Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method

MERDAN, Mehmet; GÖKDOĞAN, AHMET

Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method

Mehmet MERDAN, (Gümüşhane Üniversitesi, Harita Mühendisliği Bölümü, Gümüşhane, Türkiye)

Ahmet GÖKDOĞAN (Gümüşhane Üniversitesi, Yazılım Mühendisliği Bölümü, Gümüşhane, Türkiye)

Mathematical and Computational Applications

3 1

Yıl: 2011 Cilt: 16 Sayı: 3 Sayfa Aralığı: 761 - 772 Metin Dili: İngilizce İndeks Tarihi: 29-07-2022

Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method

Öz:

In this paper, an aproximate analytical method called the differential transform method (DTM) is used as a tool to give approximate solutions of nonlinear oscillators with fractional nonlinearites. The differential transformation method is described in a nuthsell. DTM can simply be applied to linear or nonlinear problems and reduces the required computational effort. The proposed scheme is based on the differential transform method (DTM), Laplace transform and Padé approximants. The results to get the differential transformation method (DTM) are applied Padé approximants. The reliability of this method is investigated by comparison with the classical fourth-order Runge–Kutta (RK4) method and Cos-AT and Sine-AT method. Our the presented method showed results to analytical solutions of nonlinear ordinary differential equation. Some plots are gived to shows solutions of nonlinear oscillators with fractional nonlinearites for illustrating the accurately and simplicity of the methods.

Anahtar Kelime:

Konular: Matematik

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık

1. J.K.Zhou, Differential Transformation and its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986.
2. G.E. Pukhov, Computational structure for solving differential equations by Taylor transformations, Cybernetics and Systems Analysis 14(3),383–390, 1978.
3. G.A. Baker, Essentials of Pade´ Approximants, Academic Press, London, 1975.
4. S.H.Chang, I.L.Chang, A New Algorithm for Calculating One-Dimensional Differential Transform of Nonlinear Functions, Appl Math Comput 195(2),799–808, 2008.
5. D.W.Jordan, P.Smith, Nonlinear Ordinary Differential Equations, Oxford University Press, 1999.
6. A.E.H. Ebaid, A reliable aftertreatment for improving the differential transformation method and its application to nonlinear oscillators with fractional nonlinearities, Commun Sci Numer Simulation 16, 528-536, 2011.
7. S.H. Ho Shing and C.K. Chen, Analysis of general elastically end restrained non- uniform beams using differential transform, Appl Math Model 22, 219–34, 1998.
8. C.K. Chen and S.H. Ho, Transverse vibration of a rotating twisted Timoshenko beam under axial loading using differential transform, Int JMech Sci 41, 1339–56,1999.
9. M.J. Jang, C.L. Chen and Y.C. Liy, On solving the initial-value problems using the differential transformation method, Appl Math Comput 115, 145–60, 2000.
10. M. Köksal and S.Herdem, Analysis of nonlinear circuits by using differential Taylor transform, Comput Electr Eng 28, 513–25, 2002.
11. A.S.V.K. Ravi, K. Aruna. Solution of singular two-point boundary value problems using differential transformation method, Phys Lett A 372, 4671–3, 2008.
12. A.E.H. Ebaid, Approximate periodic solutions for the non-linear relativistic harmonic oscillator via differential transformation method, Commun Nonlinear Sci Numer Simul 15, 1921–7, 2010.
13. F. Ayaz, Solutions of the system of differential equations by differential transform Method, Appl Math Comput 147, 547–67, 2004.
14. S.H. Chang and I.L. Chang, A new algorithm for calculating one-dimensional differential transform of nonlinear functions, Appl Math Comput 195, 799–808, 2008.
15. A.S.V.K. Ravi, K. Aruna, Two-dimensional differential transform method for solving linear and non-linear Schrödinger equations, Chaos Solitons Fractals 41, 2277-2281, 2009.
16. A.S.V.K. Ravi, K. Aruna, Differential transform method for solving the linear and nonlinear Klein–Gordon equation, Comput Phys Commun 180, 708–11, 2009.
17. S.N. Venkatarangan and K. Rajakshmi, A modification of adomian’s solution for nonlinear oscillatory systems, Comput Math Appl, 29, 67–73, 1995.
18. Y.C. Jiao, Y. Yamamoto, C. Dang and Y. Hao, An aftertreatment technique for improving the accuracy of Adomian’s decomposition method, Comput Math Appl 43, 783–98, 2002.
19. M.El-Shahed, Application of differential transform method to non-linear oscillatory systems, Commun Nonlinear Sci Numer Simul 13, 1714–20, 2008.
20. H.S. Da, The homotopy perturbation method for nonlinear oscillators, ComputMath App 58, 2456–9, 2009.
21. C.W. Lim and B.S.Wu, A new analytical approach to the Duffing-harmonic oscillator, Phys Lett A 311, 365–73, 2003.
22. X.C. Cai and W.Y. Wu, He’s frequency formulation for the relativistic harmonic Oscillator, Comput Math Appl 58, 2358–9, 2009..
23. C.L.Chen and Y.C.Liu, Solution of two point boundary value problems using the differential transformation method, J Op. Theory Appl 99, 23–35, 1998.
24. M.J. Jang, C.L.Chen and Y.C.Liu, Two-dimensional differential transform for partial differential equations, Appl Math Comput 121, 261–70, 2001.
25. M.J. Jang, C.L.Chen and Y.C.Liu, On solving the initial value problems using the differential transformation method, Appl Math Comput 115,145–60, 2000.
26. M.J. Jang, C.L.Chen and Y.C.Liu, Analysis of the response of a strongly nonlinear damped system using a differential transformation technique, Appl Math Comput 88, 137-51, 1997.
27. S.Momani and V.S.Ertürk, Solutions of non-linear oscillators by the modified differential transform method, Comput Math Appl 55(4), 833- 842, 2008.
28. S. Momani, Analytical approximate solutions of non-linear oscillators by the modified decomposition method, Int J Modern Phys C 15(7), 967–979, 2004.

APA	MERDAN M, GÖKDOĞAN A (2011). Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method. , 761 - 772.
Chicago	MERDAN Mehmet,GÖKDOĞAN AHMET Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method. (2011): 761 - 772.
MLA	MERDAN Mehmet,GÖKDOĞAN AHMET Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method. , 2011, ss.761 - 772.
AMA	MERDAN M,GÖKDOĞAN A Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method. . 2011; 761 - 772.
Vancouver	MERDAN M,GÖKDOĞAN A Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method. . 2011; 761 - 772.
IEEE	MERDAN M,GÖKDOĞAN A "Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method." , ss.761 - 772, 2011.
ISNAD	MERDAN, Mehmet - GÖKDOĞAN, AHMET. "Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method". (2011), 761-772.

APA	MERDAN M, GÖKDOĞAN A (2011). Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method. Mathematical and Computational Applications, 16(3), 761 - 772.
Chicago	MERDAN Mehmet,GÖKDOĞAN AHMET Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method. Mathematical and Computational Applications 16, no.3 (2011): 761 - 772.
MLA	MERDAN Mehmet,GÖKDOĞAN AHMET Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method. Mathematical and Computational Applications, vol.16, no.3, 2011, ss.761 - 772.
AMA	MERDAN M,GÖKDOĞAN A Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method. Mathematical and Computational Applications. 2011; 16(3): 761 - 772.
Vancouver	MERDAN M,GÖKDOĞAN A Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method. Mathematical and Computational Applications. 2011; 16(3): 761 - 772.
IEEE	MERDAN M,GÖKDOĞAN A "Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method." Mathematical and Computational Applications, 16, ss.761 - 772, 2011.
ISNAD	MERDAN, Mehmet - GÖKDOĞAN, AHMET. "Solution of nonlinear oscillators with fractional nonlinearities by using the modified differential transformation method". Mathematical and Computational Applications 16/3 (2011), 761-772.