Commun. Theor. Phys. 61 (2014) 221–225 Vol. 61, No. 2, February 1, 2014 Fractional Bateman–Feshbach Tikochinsky Oscillator * Dumitru Baleanu, 1,2,3,† Jihad H. Asad, 4 and Ivo Petras 5 1 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia 2 Department of Mathematics and Computer Science, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey 3 Institute of Space Sciences, P.O. Box, MG-23, 76900, Magurele, Bucharest, Romania 4 Department of Physics, College of Arts and Sciences, Palestine Technical University, P.O. Box 7, Tulkarm, Palestine 5 BERG Faculty, Technical University of Kosice, B. Nemcovej 3, 04200 Kosice, Slovakia (Received July 1, 2013; revised manuscript received October 8, 2013) Abstract In the last few years the numerical methods for solving the fractional differential equations started to be applied intensively to real world phenomena. Having these thinks in mind in this manuscript we focus on the fractional Lagrangian and Hamiltonian of the complex Bateman–Feshbach Tikochinsky oscillator. The numerical analysis of the corresponding fractional Euler-Lagrange equations is given within the Gr¨ unwald–Letnikov approach, which is power series expansion of the generating function. PACS numbers: 11.10.Ef Key words: Riemann–Liouville derivatives, Bateman–Feshbach Tikochinsky oscillator, fractional Hamiltonian equations, Gr¨ unwald–Letnikov approach 1 Introduction One of the new directions in fractional calculus and its applications is to investigate the numerical so- lutions of fractional Euler-Lagrange and Hamiltonian equations. [1-7] These types of equations are new and they involved both left and right derivatives (see for more de- tails Refs. [8–11] and the references therein). The fractional Hamiltonians are non-local and they are associated with dissipative systems. We recall that Bateman suggested the time-dependent Hamiltonian to describe the dissipative systems. [12] Also, we mention the fact that the time dependent Hamiltonian describing the damped oscillation was introduced by Caldirola [13] (see for more details Refs. [14] and [15]). Bateman suggested a variational principle for equations of motion containing a friction linear term in velocity. [12] After more than half century it was find out that the frictional models can be treated naturally within the fractional calculus, [1-6] which studies derivatives and integrals of non-integer order. Constructing a complete description for non-conservative systems can be considered as one of promising applications of fractional calculus. The results reported in Refs. [16– 17] are considered as the beginning of the fractional calcu- lus of variations with a deep impact for non-conservative and dissipative processes. Besides, in Ref. [8] it was in- vestigated a Lagrangian formulation for variation prob- lems with both the right and the left fractional derivatives within Riemann–Liouville sense as well as the Lagrangian and Hamiltonian fractional sequential mechanics. Recently, the numerical methods are used intensively and successfully to solve the fractional nonlinear differen- tial equations fractional calculus. [4] We have used the decomposition method to study the fractional Euler–Lagrange equations for some important three different physical systems, [11,18-20] and we have ob- tained a numerical solution for the corresponding equa- tions. In two of these references [18-19] we considered the Lagrangian of a Harmonic oscillators, where in Ref. [18] the considered model (i.e., Pais–Uhlenbeck oscillator) is interesting by itself and in connection with gravity since it involves a differential equation of order higher than two, whereas in Ref. [19] we considered a Harmonic Oscillator whose mass depends on time. In the last work [20] we con- sidered the Lagrangian of a two-electric pendulum. Bearing in mind the above mentioned facts, in this manuscript, we study the fractional Euler-Lagrange equa- tions for the fractional Bateman–Feshbach–Tikochinsky oscillator, which is a non-conservative dissipative system. We mention that the corresponding fractional differential equations contain both the left and the right derivatives and the study of this type of equations is still at the be- ginning of its development. The plan of this manuscript is given below. In Sec. 2, we introduce briefly the basic definitions of the fractional derivatives as well as their basic properties. In Sec. 3, we study the fractional Bateman–Feshbach–Tikochinsky oscillator. In Sec. 4, we investigate numerically the frac- * Supported in part by the Slovak Grant Agency for Science under Grants VEGA: 1/0497/11, 1/0746/11, 1/0729/12, and by the Slovak Research and Development Agency under Grant No. APVV-0482-11 † Corresponding author, E-mail: dumitru@cankaya.edu.tr c  2013 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/EJ/journal/ctp http://ctp.itp.ac.cn