Browsing by Author "Muslih, Sami I."

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A fractional Dirac equation and its solution
(IOP Publishing LTD, 2010) Baleanu, Dumitru; Agrawal, Om. P.; Baleanu, Dumitru
This paper presents a fractional Dirac equation and its solution. The fractional Dirac equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives. By applying the variational principle to a fractional action S, we obtain the fractional Euler-Lagrange equations of motion. We present a Lagrangian and a Hamiltonian for the fractional Dirac equation of order a. We also use a fractional Klein-Gordon equation to obtain the fractional Dirac equation which is the same as that obtained using the fractional variational principle. Eigensolutions of this equation are presented which follow the same approach as that for the solution of the standard Dirac equation. We also provide expressions for the path integral quantization for the fractional Dirac field which, in the limit a. 1, approaches to the path integral for the regular Dirac field. It is hoped that the fractional Dirac equation and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics
A fractional schrödinger equation and its solution
(Springer/Plenum Publishers, 2010) Baleanu, Dumitru; Baleanu, Dumitru; Agrawal, Om. P.
This paper presents a fractional Schrodinger equation and its solution. The fractional Schrodinger equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives to obtain the fractional Euler-Lagrange equations of motion. We present the Lagrangian for the fractional Schrodinger equation of order alpha. We also use a fractional Klein-Gordon equation to obtain the fractional Schrodinger equation which is the same as that obtained using the fractional variational principle. As an example, we consider the eigensolutions of a particle in an infinite potential well. The solutions are obtained in terms of the sines of the Mittag-Leffler function
About fractional supersymmetric quantum mechanics
(Inst Physics Acad Sci Czech Republic, 2005) Baleanu, Dumitru; Muslih, Sami I.; 56389
Fractional Euler-Lagrange equations are investigated in the presence of Grassmann variables. The fractional Hamiltonian and the path integral of the fractional supersymmetric classical model are constructed.
About Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives
(Amer Soc Mechanical Engineers, 2005) Baleanu, Dumitru; Muslih, Sami I.; 56389
sRecently, an extension of the simplest fractional problem and the fractional variational problem of Lagrange was obtained by Agrawal. The first part of this study presents the fractional Lagrangian formulation of mechanical systems and introduce the Levy path integral. The second part is an extension to Agrawal's approach to classical fields with fractional derivatives. The classical fields with fractional derivatives are investigated by using the Lagrangian formulation. The case of the fractional Schrodinger equation is presented.
Equations of motion for Einstein’s field in non-integer dimensional space
(Inst Physics Acad Sci Czech Republic, 2006) Baleanu, Dumitru; Muslih, Sami I.; Baleanu, Dumitru
Equations of motion for Einstein's field in fractional dimension of 4 spatial coordinates are obtained. It is shown that time dependent part of Einstein's wave function is single valued for only 4-integer dimensional space
Formulation of Hamiltonian equations for fractional variational problems
(Inst Physics Acad Sci Czech Republic, 2005) Baleanu, Dumitru; Baleanu, Dumitru; 56389
An extension of Riewe's fractional Hamiltonian formulation is presented for fractional constrained systems. The conditions of consistency of the set of constraints with equations of motion are investigated. Three examples of fractional constrained systems are analyzed in details.
Fractional dimensional harmonic oscillator
(Editura Acad Romane, 2011) Baleanu, Dumitru; Muslih, Sami I.; Baleanu, Dumitru; Rabei, E.
The fractional Schrodinger equation corresponding to the fractional oscillator was investigated. The regular singular points and the exact solutions of the corresponding radial Schrodinger equation were reported
Fractional Euler-Lagrange and fractional Hamilton equations for super symmetric classical model
(World Scientific, 2007) Baleanu, Dumitru; Muslih, Sami I.
Fractional Euler-Lagrange equations were investigated in the presence of the elements of Berezin algebra. The super fractional Hessian was defined and the fractional Hamiltonian of the super symmetric classical model was constructed
Fractional Euler-Lagrange equations of motion in fractional space
(Sage Publications Ltd, 2007) Baleanu, Dumitru; Baleanu, Dumitru
Fractional variational principles have gained considerable importance during the last decade due to their various applications in several areas of science and engineering. In this study, the fractional Euler-Lagrange equations corresponding to a prescribed fractional space are obtained. These equations are obtained using the traditional method of calculus of variations adapted to the case of fractional space. The most general fractional Lagrangian is considered and the limit case when the parameters involved in fractional derivatives are equal to one, is obtained. Two examples are investigated in this study, namely the free particle on fractional space and the fractional simple pendulum, and their corresponding fractional Euler-Lagrange equations ar obtained
Fractional Hamilton's equations of motion in fractional time
(De Gruyter Open LTD, 2007) Baleanu, Dumitru; Baleanu, Dumitru; Rabei, Eqab M.; 56389
The Hamiltonian formulation for mechanical systems containing Riemman-Liouville fractional derivatives are investigated in fractional time. The fractional Hamilton's equations are obtained and two examples are investigated in detail.
Fractional Hamiltonian analysis of higher order derivatives systems
(American Institute of Physics, 2006) Baleanu, Dumitru; Taş, Kenan; Taş, Kenan; 4971
The fractional Hamiltonian analysis of 1+1 dimensional field theory is investigated and the fractional Ostrogradski’s formulation is obtained. The fractional path integral of both simple harmonic oscillator with an acceleration-squares part and a damped oscillator are analyzed. The classical results are obtained when fractional derivatives are replaced with the integer order derivatives
Fractional lagrangian and hamiltonian mechanics with memory
(2019) Baleanu, Dumitru; Muslih, Sami I.; 56389
Fractional variational principles are very important for science and engineering. Within this field of study, the fractional Lagrangian and Hamiltonian equations are challenging ones from the viewpoint of mathematics. During the last fifteen years, the field of fractional variational principles was continuously improved and developed. In this chapter, the fractional variational principles-with and without delay-will be briefly reviewed. Several illustrative examples from mechanics are presented. © 2019 Walter de Gruyter GmbH, Berlin/Boston.
Fractional mechanics on the extended phase space
(Amer Soc Mechanical Engineers, 2010) Baleanu, Dumitru; Muslih, Sami I.; Golmankhaneh, Alireza K.; Golmankhaneh, Ali K.; Rabei, Eqab M.; 56389
Fractional calculus has gained a lot of importance and potential applications in several areas of science and engineering. The fractional dynamics and the fractional variational principles started to be used intensively as an alternative tool in order to describe the physical complex phenomena. In this paper we have discussed the fractional extension of the classical dynamics. The fractional Hamiltonian is constructed and the fractional generalized Poisson's brackets on the extended phase space is established.
Fractional multipoles in fractional space
(Pergamon-Elsevier ,Science ltd, 2007) Baleanu, Dumitru; Baleanu, Dumitru
Gauss’ law in -dimensional fractional space is investigated. The electrostatic potential with th-order fractional multipole is obtained in -dimensionally fractional space.
Fractional time action and perturbed gravity
(World Scientific, 2011) Baleanu, Dumitru; Muslih, Sami I.; Baleanu, Dumitru; Rabei, Eqab
In this paper, we used the scaling concepts of Mandelbrot of fractals in variational problems of mechanical systems in order to re-write the action integral function as an integration over the fractional time. In addition, by applying the variational principle to this new fractional action, we obtained the modified Euler-Lagrange equations of motion in any fractional time of order 0 < alpha <= 1. Two examples are investigated in detail
Fractional WKB approximation
(Springer, 2009) Baleanu, Dumitru; Altarazi, İbrahim M. A.; Muslih, Sami I.; Baleanu, Dumitru
Wentzel-Kramer-Brillouin (WKB) approximation for fractional systems is investigated in this paper using the fractional calculus. In the fractional case, the wave function is constructed such that the phase factor is the same as the Hamilton's principle function S. To demonstrate our proposed approach, two examples are investigated in detail
Generalized variational calculus in terms of multi-parameters fractional derivatives
(Elsevier Science, 2011) Baleanu, Dumitru; Muslih, Sami I.; Baleanu, Dumitru
In this paper, we briefly introduce two generalizations of work presented a few years ago on fractional variational formulations. In the first generalization, we consider the Hilfer's generalized fractional derivative that in some sense interpolates between Riemann-Liouville and Caputo fractional derivatives. In the second generalization, we develop a fractional variational formulation in terms of a three parameter fractional derivative. We develop integration by parts formulas for the generalized fractional derivatives which are key to developing fractional variational calculus. It is shown that many derivatives used recently and their variational formulations can be obtained by setting different parameters to different values. We also define fractional generalized momenta and provide fractional Hamiltonian formulations in terms of the new generalized derivatives. An example is presented to show applications of the formulations presented here. Some possible extensions of this research are also discussed
Gravitational potential in fractional space
(Versita, 2007) Baleanu, Dumitru; Baleanu, Dumitru; Rabei, Eqab M.
In this paper the gravitational potential with beta-th order fractional mass distribution was obtained in a dimensionally fractional space. We show that the fractional gravitational universal constant G(alpha) is given by G(alpha) = 2 Gamma(alpha/2)/Pi(alpha/2-1)(alpha-2) G, where G is the usual gravitational universal constant and the dimensionality of the space is alpha > 2
Hamilton formulation for continuous systems with second order derivatives
(Springer/Plenum Publishers, 2008) Baleanu, Dumitru; Muslih, Sami I.; Rabei, Eqab M.; Baleanu, Dumitru
In this paper the Hamilton formulation for continuous systems with second order derivatives has been developed. We generalized the Hamilton formulation for continuous systems with second order derivatives and apply this new formulation to Podolsky generalized electrodynamics, comparing with the results obtained through Dirac's method
Hamilton-Jacobi and fractional like action with time scaling
(Springer, 2011) Baleanu, Dumitru; Muslih, Sami I.; Baleanu, Dumitru; Rabei, Eqab M.
This paper represents the Hamilton-Jacobi formulation for fractional variational problem with fractional like action written as an integration over a time scaling parameter. Also we developed the fractional Hamiltonian formulation for the fractional like action. In all the given calculations, the most popular Riemann-Liouville (RL) and Caputo fractional derivatives are employed. An example illustrates our approach