Lagrangians with linear velocities within Riemann-Liouville fractional derivatives

By Dumitru Băleanu, Tansel Avkar. 2004

180 citations recorded by the source corpus.

Lagrangians linear in velocities were analyzed using the fractional calculus and the Euler-Lagrange equations were derived. Two examples were investigated in details, the explicit solutions of Euler-Lagrange equations were obtained and the recovery of the classical results was discussed.

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Citations in the corpus

A new method of finding the fractional Euler-Lagrange and Hamilton equations within Caputo fractional derivatives (2010, background): “During the last years, the fractional variational principles have been applied successfully in control theory [17–19] aswell as in physics [20–29].”
Fractional optimal control problem for variable-order differential systems (2017, unknown): “We also refer to ([15] -[23]) and references therein for more literature on optimal control of fractional evolution and diffusion equations.”
Fractional optimal control problems: a pseudo-state-space approach (2011, methodology): “The fractional Euler–Lagrange equation has recently been used by Baleanu and coworkers to model fractional Lagrangians with linear velocities (Baleanu and Avkar, 2004), fractional Hamiltonian formulation of systems with linear velocity (Muslih and Baleanu, 2005a), Hamiltonian equations for…”
A formulation of Noether's theorem for fractional problems of the calculus of variations (2007, unknown): “More recently, O. Agrawal proved a formulation for variational problems with right and left fractional derivatives in the Riemann-Liouville sense [1].”
A Central Difference Numerical Scheme for Fractional Optimal Control Problems (2008, methodology): “The fractional Euler–Lagrange equation has recently been used by Baleanu and coworkers to model fractional Lagrangian with linear velocities (Baleanu and Avkar, 2004), fractional metafluid dynamics (Baleanu, 2004), fractional Lagrangian and Hamiltonian formulations of discrete and continuous systems”
Variational problems with fractional derivatives: Euler–Lagrange equations (2008, methodology): “In the special case, that was treated previously (cf [1, 2, 7, 8, 11]), it was assumed that a = A.”
Generalized Euler—Lagrange Equations and Transversality Conditions for FVPs in terms of the Caputo Derivative (2007, methodology): “The fractional Euler–Lagrange equation has recently been used by Baleanu and coworkers to model fractional Lagrangians with linear velocities (Baleanu and Avkar, 2004), fractional metafluid dynamics (Baleanu, 2004), fractional Hamiltonian formulation of systems with linear velocity (Muslih and Balea”
Necessary optimality conditions for fractional action‐like integrals of variational calculus with Riemann–Liouville derivatives of order (α, β) (2007, unknown): “Avkar to investigate problems with Lagrangians which are linear in the velocities [2].”
A Hamiltonian Formulation and a Direct Numerical Scheme for Fractional Optimal Control Problems (2007, methodology): “The fractional Euler–Lagrange equation has recently been used by Baleanu and coworkers to model fractional Lagrangians with linear velocities (Baleanu and Avkar, 2004), fractional metafluid dynamics (Baleanu, 2004), fractional Lagrangian and Hamiltonian formulations of discrete and continuous system”
Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives (2005, methodology): “, (26) were obtained as [ 21 ] @aj(q i )”
Numerical solution for a class of fractional convection–diffusion equations using the flatlet oblique multiwavelets (2014, background): “During recent years fractional variational principles have been developed (Baleanu and Avkar, 2004; Baleanu, 2006; Baleanu and Muslih, 2005a,b; Muslih and Baleanu, 2005).”
Fractional actionlike variational problems (2008, methodology): “Different methods were used to obtain the fractional Euler-Lagrange equations and the corresponding Hamiltonian canonical equations [1, 3, 4, 5, 6, 14, 22, 23, 28, 29, 33].”
Fractional variational optimal control problems with delayed arguments (2010, background): “Fractional calculus is as old as the classical calculus, and during the last decades it has become a serious candidate in solving problems of complex systems which appear frequently in various branches of science and engineering [1–14].”
Monotone iterative method for a class of nonlinear fractional differential equations (2012, background): “The range of applications of the fractional calculus is very large at this moment and it is expected to increase in the nearest future (see for example [14],[11],[3],[5],[2],[1] and the references therein).”