arXiv subjects
Publications and source records attributed to Dumitru Baleanu.
This article deals with variational optimal-control problems on time scales in the presence of delay in the state variables. The problem is considered on a time scale unifying the discrete, the continuous and the quantum cases. Two examples in the discrete and quantum cases are analyzed to illustrate our results.
Given the solution $f$ of the sequential fractional differential equation $_{a}D_{t}^α(_{a}D_{t}^αf)+P(t)f=0$, $t\in[b,c]$, where $-\infty<a<b<c<+\infty$, $α\in({1/2},1)$ and $P:[a,+\infty)\to[0,P_{\infty}]$, $P_{\infty}<+\infty$, is continuous, assume that there exist $t_1,t_2\in[b,c]$ such that $f(t_1)=(_{a}D_{t}^αf)(t_2)=0$. Then, we establish here a positive lower bound for $c-a$ which depends solely on $α,P_{\infty}$. Such a result might be useful in discussing disconjugate fractional differential equations and fractional interpolation, similarly to the case of (integer order) ordinary differential equations.
This paper presents a modified numerical scheme for a class of Fractional Optimal Control Problems (FOCPs) formulated in Agrawal (2004) 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 fractional derivative (FDs) at a time node point is approximated using a modified Grünwald-Letnikov approach. For the first order derivative, the proposed modified Grünwald-Letnikov definition leads to a central difference scheme. When the approximations are substituted into the Fractional Optimal Control (FCO) 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 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.
Fractional mechanics describes both conservative and non-conservative systems. The fractional variational principles gained importance in studying the fractional mechanics and several versions are proposed. In classical mechanics the equivalent Lagrangians play an important role because they admit the same Euler-Lagrange equations. By adding a total time derivative of a suitable function to a given classical Lagrangian or by multiplying with a constant the Lagrangian we obtain the same equations of motion. In this study, the fractional discrete Lagrangians which differs by a fractional derivative are analyzed within Riemann-Liouville fractional derivatives. As a consequence of applying this procedure the classical results are reobtained as a special case. The fractional generalization of $Fa\grave{a}$ di Bruno formula is used in order to obtain the concrete expression of the fractional Lagrangians which differs from a given fractional Lagrangian by adding a fractional derivative. The fractional Euler-Lagrange and Hamilton equations corresponding to the obtained fractional Lagrangians are investigated and two examples are analyzed in details.
In this paper, first a class of fractional differential equations are obtained by using the fractional variational principles. We find a fractional Lagrangian $L(x(t)$, where $_a^cD_t^αx(t))$ and $0<α< 1$, such that the following is the corresponding Euler-Lagrange % \begin{equation}_tD_b^α(_a^cD_t^α) x(t)+ b(t,x(t))(_a^cD_t^αx(t))+f(t,x(t))=0. \end{equation} % At last, exact solutions for some Euler-Lagrange equations are presented. In particular, we consider the following equations % \begin{equation}_tD_b^α(_a^cD_t^αx(t))=λx(t), (λ\in R) \end{equation} % \begin{equation}_tD_b^α(_a^cD_t^αx(t))+g(t)_a^cD_t^αx(t)=f(t), \end{equation} where g(t) and f(t) are suitable functions.
Wentzel, Kramers, 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 details.
In this paper we develop a fractional Hamilton-Jacobi formulation for discrete systems in terms of fractional Caputo derivatives. The fractional action function is obtained and the solutions of the equations of motion are recovered. An example is studied in details.
The Hamilton-Jacobi analysis is applied to the dynamics of the scalar fluctuations about the Friedmann-Robertson-Walker (FRW). The gauge conditions are found from the consistency conditions. The physical degrees of freedom of the model are obtain by symplectic projector method. The role of the linearly dependent Hamiltonians and the gauge variables in Hamilton-Jacobi formalism is discussed.
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.
The recently modified Faddeev-Jackiw formalism for systems having one chain of four levels of only second-class constraints is applied to the non-trivial a=1 bosonized chiral Schwinger model in (1+1) dimensions as well as to one mechanical system. The sets of obtained constraints are in agreement with Dirac's canonical formulation.
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.
Systems invariant under the reparametrization of time were treated as constrained systems within Hamilton-Jacobi formalism. After imposing the integrability conditions the time-dependent Schrödinger equation was obtained. Three examples are investigated in details.
Metafluid dynamics was investigated within Hamilton-Jacobi formalism and the existence of the hidden gauge symmetry was analyzed. The obtained results are in agreement with those of Faddeev-Jackiw approach.
Hamilton-Jacobi formalism is used to study 2D-gravity and its SL(2, R) hidden symmetry. If the contribution of the surface term is considered the obtained results coincide with those given by the Dirac and Faddeev-Jackiw approaches.
The O(3) non-linear sigma model is investigated using multi Hamilton-Jacobi formalism. The integrability conditions are investigated and the results are in agreement with those obtained by Dirac's method. By choosing an adequate extension of phase space we describe the transformed system by a set of three Hamilton-Jacobi equations and calculate the corresponding action.
In this paper a non-relativistic particle moving on a hypersurface in a curved space and the multidimensional rotator are investigated using the Hamilton-Jacobi formalism. The equivalence with the Dirac Hamiltonian formalism is demonstrated in both Cartesian and curvilinear coordinates. The energy spectrum of the multidimensional rotator is equal to that of a pure Laplace-Beltrami operator with no additional constant arising from the curvature of the sphere.
The solution of the Lax tensor equations in the case $L_{αβγ}=-L_{βαγ}$ was analyzed. The Lax tensors on the dual metrics were investigated. We classified all two dimensional metrics having the symmetric Lax tensor $L_{αβγ}$. The Lax tensors of the flat space, Rindler system and its dual were found.
In this paper the symmetries of the dual manifold were investigated. We found the conditions when the manifold and its dual admit the same Killing vectors and Killing-Yano tensors. In the case of an Einstein's metric $g_{μν}$ the corresponding equations for its dual were found. The examples of Kerr-Newman geometry and the separable coordinates in 1+1 dimensions were analyzed in details.