The Fractional Calculus of Variations

  • Ricardo Almeida
  • Dina Tavares
  • Delfim F. M. Torres
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)


In this chapter, we consider general fractional problems of the calculus of variations, where the Lagrangian depends on a combined Caputo fractional derivative of variable fractional order \(^CD_\gamma ^{{\alpha (\cdot ,\cdot )},{\beta (\cdot ,\cdot )}}\) given as a combination of the left and the right Caputo fractional derivatives of orders, respectively, \({\alpha (\cdot ,\cdot )}\) and \({\beta (\cdot ,\cdot )}\). More specifically, here we study some problems of the calculus of variations with integrands depending on the independent variable t, an arbitrary function x and a fractional derivative \(^CD_\gamma ^{{\alpha (\cdot ,\cdot )},{\beta (\cdot ,\cdot )}}x\).


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Copyright information

© The Author(s), under exclusive license to Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Ricardo Almeida
    • 1
  • Dina Tavares
    • 2
  • Delfim F. M. Torres
    • 1
  1. 1.Department of MathematicsUniversity of AveiroAveiroPortugal
  2. 2.Polytechnic Institute of LeiriaLeiriaPortugal

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