Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 64, 2015 - Issue 6: Optimization in the natural sciences
255
Views
17
CrossRef citations to date
0
Altmetric
Articles

Optimality conditions for fractional variational problems with dependence on a combined Caputo derivative of variable order

, &
Pages 1381-1391 | Received 05 Feb 2014, Accepted 08 Jan 2015, Published online: 16 Feb 2015

References

  • Malinowska AB, Torres DFM. Introduction to the fractional calculus of variations. London: Imperial College Press; 2012.
  • Almeida R, Pooseh S, Torres DFM. Computational methods in the fractional calculus of variations. London: Imperial College Press; 2015.
  • Almeida R, Torres DFM. Calculus of variations with fractional derivatives and fractional integrals. Appl. Math. Lett. 2009;22:1816–1820.
  • Atanacković TM, Konjik S, Pilipović S. Variational problems with fractional derivatives: Euler–Lagrange equations. J. Phys. A. 2008;41:095201. 12 p.
  • Baleanu D. New applications of fractional variational principles. Rep. Math. Phys. 2008;61:199–206.
  • Baleanu D, Golmankhaneh AK, Nigmatullin R, Golmankhaneh AK. Fractional Newtonian mechanics. Cent. Eur. J. Phys. 2010;8:120–125.
  • Cresson J. Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 2007;48:033504. 34 p.
  • Rabei EM, Nawafleh KI, Hijjawi RS, Muslih SI, Baleanu D. The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 2007;327:891–897.
  • Tarasov VE. Fractional variations for dynamical systems: Hamilton and Lagrange approaches. J. Phys. A. 2006;39:8409–8425.
  • Pooseh S, Almeida R, Torres DFM. Approximation of fractional integrals by means of derivatives. Comput. Math. Appl. 2012;64:3090–3100.
  • Pooseh S, Almeida R, Torres DFM. Discrete direct methods in the fractional calculus of variations. Comput. Math. Appl. 2013;66:668–676.
  • Lotfi A, Yousefi SA. A numerical technique for solving a class of fractional variational problems. J. Comput. Appl. Math. 2013;237:633–643.
  • Blaszczyk T, Ciesielski M. Numerical solution of fractional Sturm–Liouville equation in integral form. Fract. Calc. Appl. Anal. 2014;17:307–320.
  • Xu Y, Agrawal OP. Models and numerical solutions of generalized oscillator equations. J. Vib. Acoust. 2014;136:050903. 7 p.
  • Sumelka W, Blaszczyk T. Fractional continua for linear elasticity. Arch. Mech. 2014;66:147–172.
  • Almeida R, Malinowska AB. Generalized transversality conditions in fractional calculus of variations. Commun. Nonlinear Sci. Numer. Simul. 2013;18:443–452.
  • Chiang AC. Elements of dynamic optimization. Singapore: McGraw-Hill; 1992.
  • Malinowska AB, Torres DFM. Fractional calculus of variations for a combined Caputo derivative. Fract. Calc. Appl. Anal. 2011;14:523–537.
  • Malinowska AB, Torres DFM. Multiobjective fractional variational calculus in terms of a combined Caputo derivative. Appl. Math. Comput. 2012;218:5099–5111.
  • Odzijewicz T, Malinowska AB, Torres DFM. Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal. 2012;75:1507–1515.
  • Almeida R, Torres DFM. An expansion formula with higher-order derivatives for fractional operators of variable order. Sci. World J. 2013;2013:915437. 11 p.
  • Odzijewicz T, Malinowska AB, Torres DFM. Fractional variational calculus of variable order. In: Almeida A, Castro LF, Speck FO, editors. Advances in harmonic analysis and operator theory. Vol. 229, Operator theory advances and applications. Basel: Birkhäuser/Springer Basel AG; 2013. p. 291–301.
  • Samko SG. Fractional integration and differentiation of variable order. Anal. Math. 1995;21:213–236.
  • Fu Z-J, Chen W, Yang H-T. Boundary particle method for Laplace transformed time fractional diffusion equations. J. Comput. Phys. 2013;235:52–66.
  • Sun H, Chen W, Li C, Chen Y. Finite difference schemes for variable-order time fractional diffusion equation. Int. J. Bifur. Chaos Appl. Sci. Eng. 2012;22:1250085. 16 p.
  • Sun H, Hu S, Chen Y, Chen W, Yu Z. A dynamic-order fractional dynamic system. Chinese Phys. Lett. 2013;30:046601. 4 p.
  • Caputo M. Linear model of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astr. Soc. 1967;13:529–539.
  • Riewe F. Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E. 1996;53:1890–1899.
  • Riewe F. Mechanics with fractional derivatives. Phys. Rev. E. 1997;55:3581–3592.
  • Coimbra CFM. Mechanics with variable-order differential operators. Ann. Phys. 2003;12:692–703.
  • Coimbra CFM, Soon CM, Kobayashi MH. The variable viscoelasticity operator. Annalen der Physik. 2005;14:378–389.
  • Ramirez LES, Coimbra CFM. On the variable order dynamics of the nonlinear wake caused by a sedimenting particle. Phys. D. 2011;240:1111–1118.
  • Sheng H, Sun HG, Coopmans C, Chen YQ, Bohannan GW. A physical experimental study of variable-order fractional integrator and differentiator. Eur. Phys. J. 2011;193:93–104.
  • Sun HG, Chen W, Chen YQ. Variable order fractional differential operators in anomalous diffusion modeling. Phys. A. 2009;388:4586–4592.
  • Odzijewicz T, Malinowska AB, Torres DFM. Noether’s theorem for fractional variational problems of variable order. Cent. Eur. J. Phys. 2013;11:691–701.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.