Some Fractional Variational Problems Involving Caputo Derivatives

  • Amele Taieb UMAB Mostaganem
  • Zoubir Dahmani UMAB Mostaganem
Keywords: Caputo Derivative, Fractional Calculus of Variations, Isoperimetric Problems


In this paper, we study some fractional variational problems with functionals that involve some unknown functions and their Caputo derivatives. We also consider Caputo iso-perimetric problems. Generalized fractional Euler-Lagrange equations for the problems are presented. Furthermore, we study the optimality conditions for functionals depending on the unknown functions and the optimal time T. In addition, some examples are discussed.


Download data is not yet available.

Author Biographies

Amele Taieb, UMAB Mostaganem

LPAM, Faculty ST, UMAB Mostaganem, Algeria

Zoubir Dahmani, UMAB Mostaganem

LPAM, Faculty SEI, UMAB Mostaganem, Algeria


O.P. Agrawal, Formulation of Euler–Lagrange equations for fractional variational problems, J. Math. Anal. Appl, 272 (2002), 368–379. MathLAB Journal Vol 3 (2019)

O.P. Agrawal, Fractional variational calculus and the transversality conditions, J. Phys. A: Math. Gen, 39 (33) (2006), 10375-10384.

O.P. Agrawal, Generalized Euler-Lagrange equations and transversality conditions for FVPs in terms of the Caputo derivative, J. Vib. Control, 13 (9-10) (2007), 1217–1237.

O.P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (24) (2007), 6287–6303.

O.P. Agrawal, Generalized variational problems and Euler-Lagrange equations, Comput. Math. Appl, 59 (5) (2010), 1852-1864.

R. Almeida, Fractional variational problems with the Riesz–Caputo derivative, Appl. Math. Lett, 25 (2012), 142–148.

R. Almeida, R.A.C. Ferreira and D.F.M.Torres, Isoperimetric problems of the calculus of variations with fractional derivatives, Acta Mathematica Scientia, 32 (2012), 619-630.

R. Almeida, A.B. Malinowska and D.F.M. Torres, Fractional Euler-Lagrange differential equations via Caputo derivatives, Fractional Dynamics and Control, Springer New York, (2012), 109-118.

R. Almeida and D.F.M. Torres, Calculus of Variations with fractional derivatives and fractional integrals, Appl. Math. Lett, 22 (12,) (2009), 1816–1820.

R. Almeida and D.F.M. Torres, Hölderian variational problems subject to integral constraints, J.Math. Anal. Appl, 359 (2,) (2009), 674–681.

T. M. Atanacković, S. Konjik and S. Pilipović, Variational problems with fractional derivatives: Euler-Lagrange equations, J. Phys. A., 41 (9) (2008), 12 pp.

B.V. Brunt, The calculus of variations, Universitext, Springer, New York, (2004).

C. Carathéodory, Calculus of variations and partial differential equations of the First Order, Chelsea, (1982).

J. Gregory, Generalizing variational theory to include the indefinite integral higher derivatives and a variety of means as cost variables, Methods Appl. Anal, 15 (4) (2008), 427–435.

R. Hilfer, Applications of fractional calculus in physics, World Scientific, River Edge, New Jersey, (2000).

A.A. Kilbas and S.A. Marzan, Nonlinear differential equation with the Caputo fraction derivative in the space of continuously differentiable functions, Differ. Equ., 41 (1) (2005), 84-89.

A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier B.V., Amsterdam, The Netherlands, (2006).

J. Klafter, S.C. Lim and R. Metzler, Fractional dynamics, recent advances, World Scientific, Singapore, (2011).

A.B. Malinowska and D.F.M. Torres, Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative, Comput. Math. Appl., 59 (9) (2010), 3110–3116.

A.B. Malinowska and D.F.M. Torres, Introduction to the fractional calculus of variations, Imperial College Press, (2012). 23 MathLAB Journal Vol 3 (2019)

K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations,Wiley, New York, (1993).

T. Odzijewicz and D.F.M. Torres, Calculus of variations with classical and fractional derivatives, Mathematica Balkanica New Series, 26 (2012), Fasc. 1-2.

K.B. Oldham and J. Spanier, The fractional calculus, Academic Press, New York, (1974).

F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E 3., 53 (2) (1996), 1890–1899.

A. Taïeb and Z. Dahmani, A coupled system of nonlinear differential equations involving m nonlinear terms, Georjian Math. Journal, 23 (3) (2016), 447–458.

A. Taïeb and Z. Dahmani, Generalized Isoperimetric FVPs Via Caputo Approach, Acta Mathematica, Accepted 2019.

A. Taïeb, Stability of Singular Fractional Systems of Nonlinear Integro-Differential Equations, Lobachevskii Journal of Mathematics., 40 (2) (2019), 219–229.

R. Weinstock, Calculus of variations with applications to physics and engineering, Dover, (1974).

W. Yourgrau and S. Mandelstam, Variational principles in dynamics and quantum theory, W. B. Saunders Company, Philadelphia, (1968).

S.A. Yousefi, M. Dehghanb and A. Lotfi, Generalized Euler–Lagrange equations for fractional variational problems with free boundary conditions, Computers and Mathematics with Applications, 62 (2011), 987–995.

How to Cite
Taieb, A., & Dahmani, Z. (2019). Some Fractional Variational Problems Involving Caputo Derivatives. MathLAB Journal, 4, 11-24. Retrieved from
Research Articles