# The Fractional Calculus of Variations

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

## Abstract

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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© 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