Fractional Variational Calculus for Non-differentiable Functions



The fractional calculus of variations is a subject under strong research. Different definitions for fractional derivatives and integrals are used, depending on the purpose under study. The fractional operators in this paper are defined in the sense of Jumarie. This allows us to work with functions which are non-differentiable. We present necessary and sufficient optimality conditions for fractional problems of the calculus of variations with a Lagrangian density depending on the free end-points.


Fractional Derivative Fractional Operator Fractional Calculus Lagrangian Density Fractional Differential Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work is partially supported by the Portuguese Foundation for Science and Technology (FCT) through the Systems and Control Group of the R&D Unit CIDMA, and partially by BUT Grant S/WI/2/2011. The author is grateful to Delfim F. M. Torres for inspiring discussions and useful comments.


  1. 1.
    Agrawal OP (2006) Fractional variational calculus and the transversality conditions. J Phys A 39:10375–10384MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Almeida R, Torres DFM (2009) Calculus of variations with fractional derivatives and fractional integrals. Appl Math Lett 22:1816–1820MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Almeida R, Torres DFM (2009) Hölderian variational problems subject to integral constraints. J Math Anal Appl 359:674–681MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Almeida R, Torres DFM (2011) Fractional variational calculus for nondifferentiable functions. Comput Math Appl 61:3097–3104MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Almeida R, Malinowska AB, Torres DFM (2010) A fractional calculus of variations for multiple integrals. Application to vibrating string. J Math Phys 51:033503Google Scholar
  6. 6.
    Atanacković TM, Konjik S, Pilipović S (2008) Variational problems with fractional derivatives: Euler–Lagrange equations. J Phys A 41:095201MathSciNetCrossRefGoogle Scholar
  7. 7.
    Baleanu D (2008) Fractional constrained systems and caputo derivatives. J Comput Nonlinear Dynam 3:199–206MathSciNetCrossRefGoogle Scholar
  8. 8.
    Baleanu D, Golmankhaneh AK, Golmankhaneh AK, Baleanu, M.C. (2009) Fractional electromagnetic equations using fractional forms. Int J Theor Phy. 48:3114–3123MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Baleanu D, Guvenc ZB, Machado JAT (2010) New Trends in Nanotechnology and Fractional Calculus Applications, Springer Science Business MediaGoogle Scholar
  10. 10.
    Carpinteri A, Mainardi F (1997) Fractals and fractional calculus in continuum mechanics. Springer, ViennaMATHGoogle Scholar
  11. 11.
    Cresson J (2007) Fractional embedding of differential operators and Lagrangian systems. J Math Phys 48:033504MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cruz PAF, Torres DFM, Zinober ASI (2010) A non-classical class of variational problems. Int J Math Model Numerical Optimisation 1:227–236MATHCrossRefGoogle Scholar
  13. 13.
    Debnath L (2003) Recent applications of fractional calculus to science and engineering. Int J Math Math Sci 54:3413–3442MathSciNetCrossRefGoogle Scholar
  14. 14.
    El-Nabulsi RA, Torres DFM (2007) Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann–Liouville derivatives of order (α, β). Math Meth Appl Sci 30:1931–1939MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    El-Nabulsi RA, Torres DFM (2008) Fractional actionlike variational problems. J Math Phys 49:053521MathSciNetCrossRefGoogle Scholar
  16. 16.
    Frederico GSF, Torres DFM (2007) A formulation of Noether’s theorem for fractional problems of the calculus of variations. J Math Anal Appl 334:834–846MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Frederico GSF, Torres, DFM (2008) Fractional conservation laws in optimal control theory. Nonlinear Dynam 53:215–222MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Hilfer R (2000) Applications of fractional calculus in physics. World Scientific Publishing, River EdgeMATHCrossRefGoogle Scholar
  19. 19.
    Jumarie G (2005) On the representation of fractional Brownian motion as an integral with respect to (dt)a. Appl Math Lett 18:739–748MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Jumarie G (2007) Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function. J Appl Math Comput 23:215–228MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Jumarie G (2009) Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions. Appl Math Lett 22:378–385MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Jumarie G (2010) Analysis of the equilibrium positions of nonlinear dynamical systems in the presence of coarse-graining disturbance in space, J Appl Math Comput 32:329–351MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamMATHGoogle Scholar
  24. 24.
    Klimek M (2002) Lagrangean and Hamiltonian fractional sequential mechanics. Czechoslovak J Phys 52:1247–1253MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Kolwankar KM, Gangal AD (1997) Holder exponents of irregular signals and local fractional derivatives. Pramana J Phys 48:49–68CrossRefGoogle Scholar
  26. 26.
    Machado JAT, Silva MF, Barbosa RS, Jesus IS, Reis CM, Marcos MG, Galhano AF (2010) Some applications of fractional calculus in engineering. Mathematical Problems in Engineering, doi:10.1155/2010/639801Google Scholar
  27. 27.
    Malinowska AB, Sidi Ammi MR, Torres DFM (2010) Composition functionals in fractional calculus of variations, Commun Frac Calc 1:32–40Google Scholar
  28. 28.
    Malinowska AB, Torres DFM (2010) Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. Comput Math Appl 59:3110–3116MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Malinowska AB, Torres DFM (2010) Natural Boundary Conditions in the Calculus of Variations. Math Meth Appl Sc. 33:1712–1722MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Malinowska AB, Torres DFM (2011) Fractional calculus of variations for a combined Caputo derivative. Fract Calc Appl Anal 14(4), in pressGoogle Scholar
  31. 31.
    Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New YorkMATHGoogle Scholar
  32. 32.
    Odzijewicz T, Torres DFM 2010 Calculus of variations with fractional and classical derivatives. In: Podlubny I, Vinagre Jara BM, Chen YQ, Feliu Batlle V, Tejado Balsera I (ed) Proceedings of FDA’10, The 4th IFAC Workshop on Fractional Differentiation and its Applications, Badajoz, Spain, p. 5, 18–20, Article no. FDA10-076Google Scholar
  33. 33.
    Ortigueira MD, Machado JAT (2006) Fractional calculus applications in signals and systems. Signal Process 86:2503–2504MATHCrossRefGoogle Scholar
  34. 34.
    Podlubny I (1999) Fractional differential equations. Academic Press, San Diego, CA, USAMATHGoogle Scholar
  35. 35.
    Rabei EM, Ababneh BS (2008) Hamilton-Jacobi fractional mechanics. J Math Anal Appl 344:799–805MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Riewe F (1996) Nonconservative Lagrangian and Hamiltonian mechanics. Phys Rev E 53:1890–1899MathSciNetCrossRefGoogle Scholar
  37. 37.
    Ross B (1975) Fractional calculus and its applications, Springer, BerlinMATHCrossRefGoogle Scholar
  38. 38.
    Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives. Translated from the 1987 Russian original, Gordon and Breach, YverdonGoogle Scholar
  39. 39.
    Tarasov VE (2008) Fractional vector calculus and fractional Maxwell’s equations. Ann Phy 323:2756–2778MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    van Brunt B (2004) The calculus of variations. Springer, New YorkMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Faculty of Computer ScienceBiałystok University of TechnologyBiałystokPoland

Personalised recommendations