Optimality Conditions for Multidimensional Variational Problems Involving the Caputo-Type Fractional Derivative

  • Barbara Łupińska
  • Tatiana Odzijewicz
  • Ewa SchmeidelEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 230)


We study multidimensional variational problems, where the Lagrange function depends on the partial Caputo–Katugampola fractional derivatives, generalizing the Caputo and the Caputo–Hadamard fractional derivatives. We present sufficient and necessary conditions which determine the extremizers of a functional.


Fractional calculus Multidimensional variational calculus Caputo-type fractional derivative 

AMS Subject classification

26A33 34A08 34K28 



Research supported by the University of Białystok grant BST–137/2015 (B. Łupińska), and by the Warsaw School of Economics grant KAE/S15/35/15 (T. Odzijewicz).


  1. 1.
    Almeida, R.: Variational problems involving a Caputo-type fractional derivative. J. Optim. Theory Appl. 2, 1–19 (2016)Google Scholar
  2. 2.
    Almeida, R.: A Gronwall inequality for a general Caputo fractional operator. Math. Inequal. Appl. (in press)Google Scholar
  3. 3.
    Almeida, R., Torres, D.F.M.: Calculus of variations with fractional derivatives and fractional integrals. Appl. Math. Lett. 22(12), 1816–1820 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Almeida, R., Malinowska, A.B., Torres, D.F.M.: A fractional calculus of variations for multiple integrals with application to vibrating string. J. Math. Phys. 51(3), 033503 (2010), 12 ppMathSciNetCrossRefGoogle Scholar
  5. 5.
    Almeida, R., Malinowska, A.B., Odzijewicz, T.: Fractional differential equations with dependence on the Caputo–Katugampola derivative. J. Comput. Nonlinear Dyn. (in press)Google Scholar
  6. 6.
    Katugampola, U.N.: New approach to a generalized fractional integral. Appl. Math. Comput. 218(3), 860–865 (2011)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Katugampola, U.N.: A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl. 6(4), 1–15 (2014)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)CrossRefGoogle Scholar
  9. 9.
    Klimek, M.: On Solutions of Linear Fractional Differential Equations of a Variational Type. The Publishing Office of Czestochowa University of Technology, Czestochowa (2009)Google Scholar
  10. 10.
    Lánczos, C.: The Variational Principles of Mechanics. Mathematical Expositions, vol. 4, 4th edn. University of Toronto Press, Toronto (1970)Google Scholar
  11. 11.
    Łupińska, B., Odzijewicz, T., Schmeidel, E.: Some properties of generalized fractional integrals and derivatives. In: Proceedings of the International Conference of Numerical Analysis and Applied Mathematics 2016 (ICNAAM-2016) Book Series: AIP Conference Proceedings, vol. 1863 (publ. by) American Institute of Physics (2017), pp. 1–4 (article identifier 140010)Google Scholar
  12. 12.
    Malinowska, A.B., Torres, D.F.M.: Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. Comput. Math. Appl. 59(9), 3110–3116 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Malinowska, A.B., Odzijewicz, T., Torres, D.F.M.: Advanced Methods in the Fractional Calculus of Variations. Springer Briefs in Applied Sciences and Technology. Springer International Publishing, New York (2015)CrossRefGoogle Scholar
  14. 14.
    Odzijewicz, T.: Generalized fractional isoperimetric problem of several variables. Discret. Contin. Dyn. Syst. B 19(8), 2617–2629 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Odzijewicz, T., Malinowska, A.B., Torres, D.F.M.: Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal. 75(3), 1507–1515 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Odzijewicz, T., Malinowska, A.B., Torres, D.F.M.: Fractional calculus of variations of several independent variables. Eur. Phys. J. 222(8), 1813–1826 (2013)Google Scholar
  17. 17.
    Odzijewicz, T., Malinowska, A.B., Torres, D.F.M.: Noether’s theorem for fractional variational problems of variable order. Cent. Eur. J. Phys. 11(6), 691–701 (2013)Google Scholar
  18. 18.
    Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53(2), 1890–1899 (1996)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55(3), part B, 3581–3592 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Barbara Łupińska
    • 1
  • Tatiana Odzijewicz
    • 1
    • 2
  • Ewa Schmeidel
    • 1
    Email author
  1. 1.University of BialystokBiałystokPoland
  2. 2.Warsaw School of EconomicsWarsawPoland

Personalised recommendations