Advertisement

Fractional Euler–Lagrange Differential Equations via Caputo Derivatives

  • Ricardo AlmeidaEmail author
  • Agnieszka B. Malinowska
  • Delfim F. M. Torres
Chapter

Abstract

We review some of the recent results of the fractional variational calculus. Necessary optimality conditions of Euler–Lagrange type for functionals with a Lagrangian containing left and right Caputo derivatives are given. Several problems are considered: with fixed or free boundary conditions, and in presence of integral constraints that also depend on Caputo derivatives.

Notes

Acknowledgements

Work supported by FEDER funds through COMPETE — Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Competitividade”) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveir) and the Portuguese Foundation for Science and Technology (“FCT — Fundação para a Ciência e a Tecnologia”), within project PEst-C/MAT/Ul4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690. Agnieszka Malinowska is also supported by Białystok University of Technology grant S/WI/2/2011.

References

  1. 1.
    Agrawal OP (2007) Generalized Euler-Lagrange equations and transversality conditions for fvps in terms of the Caputo derivative. J Vib Contr 13(9–10):1217–1237zbMATHCrossRefGoogle Scholar
  2. 2.
    Almeida R, Malinowska AB Generalized transversality conditions in fractional calculus of variations. (Submitted)Google Scholar
  3. 3.
    Almeida R, Malinowska AB, Torres DFM (2010) A fractional calculus of variations for multiple integrals with application to vibrating string. J Math Phys 51(3):033503, 12 ppGoogle Scholar
  4. 4.
    Almeida R, Torres DFM (2009) Hölderian variational problems subject to integral constraints. J Math Anal Appl 359(2):674–681MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Almeida R, Torres DFM (2009) Isoperimetric problems on time scales with nabla derivatives. J Vib Contr 15(6):951–958MathSciNetCrossRefGoogle Scholar
  6. 6.
    Almeida R, Torres DFM (2009) Calculus of variations with fractional derivatives and fractional integrals. Appl Math Lett 22(12):1816–1820MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Almeida R, Torres DFM (2010) Leitmann’s direct method for fractional optimization problems. Appl Math Comput 217(3):956–962MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Almeida R, Torres DFM (2011) Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Comm Nonlinear Sci Numer Simult 16(3):1490–1500MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Atanacković TM, Konjik S, Pilipović S (2007) Variational problems with fractional derivatives: Euler-Lagrange equations. J Phys A 41(9):095201, 12 ppGoogle Scholar
  10. 10.
    Baleanu D (2008) Fractional constrained systems and caputo derivatives. J Comput Nonlinear Dynam 3(2):021102MathSciNetCrossRefGoogle Scholar
  11. 11.
    Baleanu D (2008) New applications of fractional variational principles. Rep Math Phys 61(2):199–206MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Baleanu D, Güvenç ZB, Tenreiro Machado JA (2010) New trends in nanotechnology and fractional calculus applications. Springer, New YorkzbMATHCrossRefGoogle Scholar
  13. 13.
    Bastos NRO, Ferreira RAC, Torres DFM (2011) Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin Dyn Syst 29(2):417–437MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bastos NRO, Ferreira RAC, Torres DFM (2011) Discrete-time fractional variational problems. Signal Process 91(3):513–524zbMATHCrossRefGoogle Scholar
  15. 15.
    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 Methods Appl Sci 30(15):1931–1939MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    El-Nabulsi RA, Torres DFM (2008) Fractional actionlike variational problems. J Math Phys 49(5):053521, 7 ppGoogle Scholar
  17. 17.
    Frederico GSF, Torres DFM (2007) A formulation of Noether’s theorem for fractional problems of the calculus of variations. J Math Anal Appl 334(2):834–846MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Frederico GSF, Torres DFM (2008) Fractional conservation laws in optimal control theory. Nonlinear Dynam 53(3):215–222MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Frederico GSF, Torres DFM (2010) Fractional Noether’s theorem in the Riesz-Caputo sense. Appl Math Comput 217(3):1023–1033MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamzbMATHGoogle Scholar
  21. 21.
    Klimek M (2009) On solutions of linear fractional differential equations of a variational type. Czestochowa Series Monographs 172, Czestochowa University of Technology, CzestochowaGoogle Scholar
  22. 22.
    Magin R, Ortigueira MD, Podlubny I, Trujillo J (2011) On the fractional signals and systems. Signal Process 91(3):350–371zbMATHCrossRefGoogle Scholar
  23. 23.
    Malinowska AB, Torres DFM (2010) Natural boundary conditions in the calculus of variations. Math Methods Appl Sci 33(14):1712–1722MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Malinowska AB, Torres DFM (2010) Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. Comput Math Appl 59(9):3110–3116MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Malinowska AB, Torres DFM (2010) Leitmann’s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales. Appl Math Comput 217(3):1158–1162MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Malinowska AB, Torres DFM (2010) The Hahn quantum variational calculus. J Optim Theory Appl 147(3):419–442MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Mozyrska D, Torres DFM (2010) Minimal modified energy control for fractional linear control systems with the Caputo derivative. Carpathian J Math 26(2):210–221MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mozyrska D, Torres DFM (2011) Modified optimal energy and initial memory of fractional continuous-time linear systems. Signal Process 91(3):379–385zbMATHCrossRefGoogle Scholar
  29. 29.
    Odzijewicz T, Malinowska AB, Torres DFM (2011) Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal (in press) DOI: 10.1016/j.na.2011.01.010Google Scholar
  30. 30.
    Podlubny I (1999) Fractional differential equations. Academic Press, San Diego, CAzbMATHGoogle Scholar
  31. 31.
    Riewe F (1997) Mechanics with fractional derivatives. Phys Rev E 55(3):3581–3592MathSciNetCrossRefGoogle Scholar
  32. 32.
    Torres DFM, Leitmann G (2008) Contrasting two transformation-based methods for obtaining absolute extrema. J Optim Theory Appl 137(1):53–59MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    van Brunt B (2004) The calculus of variations. Universitext, Springer, New YorkzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Ricardo Almeida
    • 1
    Email author
  • Agnieszka B. Malinowska
    • 2
  • Delfim F. M. Torres
    • 1
  1. 1.Center for Research and Development in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal
  2. 2.Faculty of Computer ScienceBiałystok University of TechnologyBiałystokPoland

Personalised recommendations