Solutions of a fractional oscillator by using differential transform method. (English) Zbl 1189.34068

Summary: We present an efficient algorithm for solving a fractional oscillator using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of a fractional oscillator. The method provides the solution in the form of a rapidly convergent series. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method.


34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
45J05 Integro-ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
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[1] Caputo, M., Linear models of dissipation whose Q is almost frequency independent II, Geophysical journal of royal astronomical society, 13, 529-539, (1967)
[2] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[3] Mainardi, F., Fractional calculus: ‘some basic problems in continuum and statistical mechanics’, (), 291-348 · Zbl 0917.73004
[4] Momani, Shaher; Ibrahim, Rabha, Analytical solutions of a fractional oscillator by the decomposition method, International journal of pure and applied mathematics, 37, 1, 119-132, (2007) · Zbl 1144.26009
[5] Narahari Achar, B.N.; Hanneken, J.W.; Enck, T.; Clarke, T., Dynamics of the fractional oscillator, Physica A, 297, 361-367, (2001) · Zbl 0969.70511
[6] Narahari Achar, B.N.; Hanneken, J.W.; Enck, T.; Clarke, T., Response characteristics of a fractional oscillator, Physica A, 309, 275-288, (2002) · Zbl 0995.70017
[7] Zhou, J.K., Differential transformation and its applications for electrical circuits, (1986), Huazhong University Press Wuhan, China, (in Chinese)
[8] Pukhov, G.E., Computational structure for solving differential equations by Taylor transformations, Cybernetics and systems analysis, 14, 3, 383-390, (1978)
[9] Arikoglu, A.; Ozkol, I., Solution of fractional differential equations by using differential transform method, Chaos solitons and fractals, 34, 5, 1473-1481, (2007) · Zbl 1152.34306
[10] Erturk, V.S.; Momani, S., Solving systems of fractional differential equations using differential transform method, Journal of computational and applied mathematics, 215, 142-151, (2008) · Zbl 1141.65088
[11] Dehghan, M.; Shakourifar, M.; Hamidid, A., The solution of linear and nonlinear systems of Volterra functional equations using adomian – pade technique, Chaos, solitons and fractals, 39, 2509-2521, (2009) · Zbl 1197.65223
[12] Dehghan, M.; Shakeri, F., The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics, Physica scripta, 78, (2008), 111. Article No. 065004 · Zbl 1159.78319
[13] Tatari, M.; Dehghan, M.; Razzaghi, M., Application of the Adomian decomposition method for the fokker – planck equation, Mathematical and computer modelling, 45, 639-650, (2007) · Zbl 1165.65397
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