zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
26A33Fractional derivatives and integrals (real functions)
34A08Fractional differential equations
45J05Integro-ordinary differential equations
65L05Initial value problems for ODE (numerical methods)
Full Text: DOI
[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) · Zbl 0924.34008
[3] Mainardi, F.: Fractional calculus: ’some basic problems in continuum and statistical mechanics’, Fractals and fractional calculus in continuum mechanics, 291-348 (1997) · 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, No. 1, 119-132 (2007) · Zbl 1144.26009
[5] Achar, B. N. Narahari; Hanneken, J. W.; Enck, T.; Clarke, T.: Dynamics of the fractional oscillator, Physica A 297, 361-367 (2001) · Zbl 0969.70511 · doi:10.1016/S0378-4371(01)00200-X
[6] Achar, B. N. Narahari; Hanneken, J. W.; Enck, T.; Clarke, T.: Response characteristics of a fractional oscillator, Physica A 309, 275-288 (2002) · Zbl 0995.70017 · doi:10.1016/S0378-4371(02)00609-X
[7] Zhou, J. K.: Differential transformation and its applications for electrical circuits, (1986)
[8] Pukhov, G. E.: Computational structure for solving differential equations by Taylor transformations, Cybernetics and systems analysis 14, No. 3, 383-390 (1978)
[9] Arikoglu, A.; Ozkol, I.: Solution of fractional differential equations by using differential transform method, Chaos solitons and fractals 34, No. 5, 1473-1481 (2007) · Zbl 1152.34306 · doi:10.1016/j.chaos.2006.09.004
[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 · doi:10.1016/j.cam.2007.03.029
[11] Dehghan, M.; Shakourifar, M.; Hamidid, A.: The solution of linear and nonlinear systems of Volterra functional equations using Adomian--Padé technique, Chaos, solitons and fractals 39, 2509-2521 (2009) · Zbl 1197.65223 · doi:10.1016/j.chaos.2007.07.028
[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) · Zbl 1159.78319 · doi:10.1088/0031-8949/78/06/065004
[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 · doi:10.1016/j.mcm.2006.07.010