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Approximate periodic solutions for the non-linear relativistic harmonic oscillator via differential transformation method. (English) Zbl 1222.65084
Summary: The relativistic harmonic oscillator equation is a nonlinear ordinary differential equation given by x '' +(1-x ' 2 ) 3/2 x=0. In this paper, the differential transformation method (DTM) and a relatively new technique, known as aftertreatment technique, are proposed to obtain new approximate periodic solutions for the relativistic harmonic oscillator equation under the initial conditions x(0)=0, x ' (0)=β.
MSC:
65L99Numerical methods for ODE
34A45Theoretical approximation of solutions of ODE
34C25Periodic solutions of ODE
References:
[1]Penfield, R.; Zatzkis, H.: The relativistic linear harmonic oscillator, J franklin inst 262, 121 (1956)
[2]Gold, Louis: Note on the relativistic harmonic oscillator, J franklin inst, 25-27 (1957)
[3]Mickens, R. E.: Periodic solutions of the relativistic harmonic oscillator, J sound vib 212, 905-908 (1998)
[4]Mickens, R. E.: Oscillations in planar dynamic systems, (1996)
[5]Zhou, J. K.: Differential transformation and its applications for electrical circuits, (1986)
[6]Ho, Shing Huei; Chen, Cha’o Kuang: Analysis of general elastically end restrained non-uniform beams using differential transform, Appl math model 22, 219-234 (1998)
[7]Chen, Cha’o Kuang; Ho, Shing Huei: Transverse vibration of a rotating twisted Timoshenko beams under axial loading using differential transform, Int J mech sci 41, 1339-1356 (1999) · Zbl 0945.74029 · doi:10.1016/S0020-7403(98)00095-2
[8]Jang, Ming-Jyi; Chen, Chieh-Li; Liy, Yung-Chin: On solving the initial-value problems using the differential transformation method, Appl math comput 115, 145-160 (2000) · Zbl 1023.65065 · doi:10.1016/S0096-3003(99)00137-X
[9]Köksal, Muhammet; Herdem, Saadetdin: Analysis of nonlinear circuits by using differential Taylor transform, Comput electr eng 28, 513-525 (2002) · Zbl 1006.68600 · doi:10.1016/S0045-7906(00)00066-5
[10]Hassan, I. H. Abdel-Halim: Different applications for the differential transformation in the differential equations, Appl math comput 129, 183-201 (2002) · Zbl 1026.34010 · doi:10.1016/S0096-3003(01)00037-6
[11]Arikoglu, Aytac; Ozkol, Ibrahim: Solution of boundary value problems for integro-differential equations by using differential transform method, Appl math comput 168, 1145-1158 (2005) · Zbl 1090.65145 · doi:10.1016/j.amc.2004.10.009
[12]Kanth, A. S. V. Ravi; Aruna, K.: Solution of singular two-point boundary value problems using differential transformation method, Phys lett A 372, 4671-4673 (2008) · Zbl 1221.34060 · doi:10.1016/j.physleta.2008.05.019
[13]Chen, Cha’o Kuang; Ho, Shing Huei: Solving partial differential equations by two-dimensional differential transform method, Appl math comput 41, 171-179 (1999) · Zbl 1028.35008 · doi:10.1016/S0096-3003(98)10115-7
[14]Jang, Ming-Jyi; Chen, Chieh-Li; Liy, Yung-Chin: Two-dimensional differential transform for partial differential equations, Appl math comput 121, 261-270 (2001) · Zbl 1024.65093 · doi:10.1016/S0096-3003(99)00293-3
[15]Ayaz, Fatma: On two-dimensional differential transform method, Appl math comput 143, 361-374 (2003) · Zbl 1023.35005 · doi:10.1016/S0096-3003(02)00368-5
[16]Ayaz, Fatma: Solutions of the system of differential equations by differential transform method, Appl math comput 147, 547-567 (2004) · Zbl 1032.35011 · doi:10.1016/S0096-3003(02)00794-4
[17]Chang, Shih-Hsiang; Chang, I-Ling: A new algorithm for calculating one-dimensional differential transform of nonlinear functions, Appl math comput 195, 799-808 (2008) · Zbl 1132.65062 · doi:10.1016/j.amc.2007.05.026
[18]Jiao, Y. C.; Yamamoto, Y.; Dang, C.; Hao, Y.: An aftertreatment technique for improving the accuracy of Adomian’s decomposition method, Comput math appl 43, 783-798 (2002) · Zbl 1005.34006 · doi:10.1016/S0898-1221(01)00321-2
[19]El-Shahed, Moustafa: Application of differential transform method to non-linear oscillatory systems, Commun nonlinear sci numer simul 13, 1714-1720 (2008)
[20]Momani, Shaher; Ertürk, Vedat Suat: Solutions of non-linear oscillators by the modified differential transform method, Comput math appl 55, 833-842 (2008) · Zbl 1142.65058 · doi:10.1016/j.camwa.2007.05.009
[21]Ravi Kanth ASV, Aruna K. Two-dimensional differential transform method for solving linear and non-linear Schrdinger equations. Chaos Solitons Fract. doi:10.1016/j.chaos.2008.08.037.
[22]Kanth, A. S. V. Ravi; Aruna, K.: Differential transform method for solving the linear and nonlinear Klein – Gordon equation, Comput phys commun 180, 708-711 (2009)
[23]Al-Sawalha, M. Mossa; Noorani, M. S. M.: Application of the differential transformation method for the solution of the hyperchaotic Rössler system, Commun nonlinear sci numer simul 14, 1509-1514 (2009)
[24]Kuo, Bor-Lih; Lo, Cheng-Ying: Application of the differential transformation method to the solution of a damped system with high nonlinearity, Nonlinear anal 70, 1732-1737 (2009) · Zbl 1168.34301 · doi:10.1016/j.na.2008.02.056
[25]Mei, C.: Application of differential transformation technique to free vibration analysis of a centrifugally stiffened beam, Comput struct 86, 1280-1284 (2008)
[26]Chu, Hsin-Ping; Chen, Chieh-Li: Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem, Commun nonlinear sci numer simul 13, 1605-1614 (2008) · Zbl 1221.80019 · doi:10.1016/j.cnsns.2007.03.002