Browsing by Author "Agrawal, Om. P."

Now showing 1 - 7 of 7

Citation Count: Baleanu, Dumitru; Defterli, Özlem; Agrawal, Om.P., "A central difference numerical scheme for fractional optimal control problems", Journal Of Vibration And Control, Vol.15, No.4, pp.583-597, (2009).
A central difference numerical scheme for fractional optimal control problems
(Sage Publications LTD, 2009) Baleanu, Dumitru; Defterli, Özlem; Agrawal, Om. P.; 56389; 31401
This paper presents a modified numerical scheme for a class of fractional optimal control problems where a fractional derivative (FD) is defined in the Riemann-Liouville sense. In this scheme, the entire time domain is divided into several sub-domains, and a FD at a time node point is approximated using a modified Grunwald-Letnikov approach. For the first-order derivative, the proposed modified Grunwald-Letnikov definition leads to a central difference scheme. When the approximations are substituted into the fractional optimal control equations, it leads to a set of algebraic equations which are solved using a direct numerical technique. Two examples, one time-invariant and the other time-variant, are considered to study the performance of the numerical scheme. Results show that 1) as the order of the derivative approaches an integer value, these formulations lead to solutions for the integer-order system, and 2) as the sizes of the sub-domains are reduced, the solutions converge. It is hoped that the present scheme would lead to stable numerical methods for fractional differential equations and optimal control problems.
Citation Count: Muslih, S.I., Agrawal, Om P., Baleanu, D. (2010). A fractional Dirac equation and its solution. Journal of Physics A-Mathematical and Theoretical, 43(5). http://dx.doi.org/10.1088/1751-8113/43/5/055203
A fractional Dirac equation and its solution
(IOP Publishing LTD, 2010) Muslih, Sami I.; 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
Citation Count: Muslih, S.I., Baleanu, D., Agrawal, O.P. (2010). A fractional schrödinger equation and its solution. International Journal of Theoretical Physics, 49(8), 1746-1752. http://dx.doi.org/ 10.1007/s10773-010-0354-x
A fractional schrödinger equation and its solution
(Springer/Plenum Publishers, 2010) Muslih, Sami I.; 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
Citation Count: Agrawal, O.P.; Baleanu, D., "A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems", Journal Of Vibration And Control, Vol.13, No.9-10, pp.1269-1281, (2007).
A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems
(Sage Publications LTD, 2007) Agrawal, Om. P.; Baleanu, Dumitru; 56389
This paper deals with a direct numerical technique for Fractional Optimal Control Problems (FOCPs). In this paper, we formulate the FOCPs in terms of Riemann-Liouville Fractional Derivatives (RLFDs). It is demonstrated that right RLFDs automatically arise in the formulation even when the dynamics of the system is described using left RLFDs only. For numerical computation, the FDs are approximated using the Grunwald-Letnikov definition. This leads to a set of algebraic equations that can be solved using numerical techniques. Two examples, one time-invariant and the other time-variant, are considered to demonstrate the effectiveness of the formulation. Results show that as the order of the derivative approaches an integer value, these formulations lead to solutions for integer order system. The approach requires dividing of the entire time domain into several sub-domains. Further, as the sizes of the sub-domains are reduced, the solutions converge to unique solutions. However, the convergence is slow. A scheme that improves the convergence rate will be considered in a future paper. Other issues to be considered in the future include formulations using other types of derivatives, nonlinear and stochastic fractional optimal controls, existence and uniqueness of the solutions, and the error analysis.
Citation Count: Baleanu, D., Agrawal, O.P. (2006). Fractional Hamilton formalism within Caputo's derivative. Czechoslovak Journal of Physics, 56(10-11), 1087-1092. http://dx.doi.org/10.1007/s10582-006-0406-x
Fractional Hamilton formalism within Caputo's derivative
(Inst Physics Acad Sci Czech Republic, 2006) Baleanu, Dumitru; Agrawal, Om. P.
In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canonical Hamiltonian are given, and a set of fractional Hamiltonian equations are obtained. Using an example, it is shown that the canonical fractional Hamiltonian and the fractional Euler Lagrange formulations lead to the same set of equations.
Citation Count: Agrawal, O.P., Defterli, Ö., Baleanu, D. (2010). Fractional optimal control problems with several state and control variables. Journal of Vibration and Control, 16(13), 1967-1976. http://dx.doi.org/10.1177/1077546309353361
Fractional optimal control problems with several state and control variables
(Sage Publications LTD, 2010) Agrawal, Om. P.; Defterli, Özlem; Baleanu, Dumitru; 31401
In many applications, fractional derivatives provide better descriptions of the behavior of dynamic systems than other techniques. For this reason, fractional calculus has been used to analyze systems having noninteger order dynamics and to solve fractional optimal control problems. In this study, we describe a formulation for fractional optimal control problems defined in multi-dimensions. We consider the case where the dimensions of the state and control variables are different from each other. Riemann-Liouville fractional derivatives are used to formulate the problem. The fractional differential equations involving the state and control variables are solved using Grunwald-Letnikov approximation. The performance of the formulation is shown using an example
Citation Count: Agrawal, O.P., Muslih, S.I., Baleanu, D. (2011). Generalized variational calculus in terms of multi-parameters fractional derivatives. Communications In Nonlinear Science And Numerical Simulation, 16(12), 4756-4767. http://dx.doi.org/10.1016/j.cnsns.2011.05.002
Generalized variational calculus in terms of multi-parameters fractional derivatives
(Elsevier Science, 2011) Agrawal, Om. P.; 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