The Variable-Order Fractional Calculus of Variations is devoted to the study of fractional operators with variable order and, in particular, variational problems involving variable-order operators. This brief presents a new numerical tool for the solution of differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one, an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. The contributors consider variational problems that may be subject to one or more constraints, where the functional depends on a combined Caputo derivative of variable fractional order. In particular, they establish necessary optimality conditions of Euler-Lagrange type. As the terminal point in the cost integral is free, as is the terminal state, transversality conditions are also obtained. The Variable-Order Fractional Calculus of Variations is a valuable source of information for researchers in mathematics, physics, engineering, control and optimization; it provides both analytical and numerical methods to deal with variational problems. It is also of interest to academics and postgraduates in these fields, as it solves multiple variational problems subject to one or more constraints in a single brief.

Auteur
Ricardo Almeida received his Bachelor and Master degrees in Mathematics from University of Porto, Portugal, andhis Ph.D. degree in Mathematics from University of Aveiro, Portugal. He is currently an Assistant Professor in theUniversity of Aveiro. His research interests include fractional calculus, calculus of variations and optimal controltheory. Ricardo Almeida has written 4 books and more than 60 publications in international journals.
Dina Tavares obtained her Master and PhD degrees in Mathematics from University of Aveiro. She is an Adjunct Professor at the School of Education and Social Sciences (ESECS) of Polytechnic Institute of Leiria (IPL), Portugal, since 2017. Her research interests include fractional calculus, calculus of variationsand mathematics education.
Delfim F.M. Torres is a Full Professor of Mathematics at the University of Aveiro since 2015 and Coordinator of theSystems and Control Group of CIDMA since 2010. He obtained PhD in Mathematics in 2002 and DSc (Habilitation)in Mathematics in 2011. Professor Torres has been awarded in 2015, 2016 and 2017 with the title of ISI Highly CitedResearcher. He has written more than 400 publications, including two books with Imperial College Press, in 2012and 2015, and two books with Springer, in 2014 and 2015. He is the Director of the FCT Doctoral Program ofExcellence in Mathematics and Applications of Universities of Minho, Aveiro, Porto and UBI since 2013. SixteenPhD students in mathematics (six of which woman) have successfully finished under his supervision.

Texte du rabat
**The Variable-Order Fractional Calculus of Variations is devoted to the study of fractional operators with variable order and, in particular, variational problems involving variable-order operators. This brief presents a new numerical tool for the solution of differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one, an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided.
The contributors consider variational problems that may be subject to one or more constraints, where the functional depends on a combined Caputo derivative of variable fractional order. In particular, they establish necessary optimality conditions of EulerLagrange type. As the terminal point in the cost integral is free, as is the terminal state, transversality conditions are also obtained.
The Variable-Order Fractional Calculus of Variations is a valuable source of information for researchers in mathematics, physics, engineering, control and optimization; it provides both analytical and numerical methods to deal with variational problems. It is also of interest to academics and postgraduates in these fields, as it solves multiple variational problems subject to one or more constraints in a single brief.

Résumé
**The Variable-Order Fractional Calculus of Variations is devoted to the study of fractional operators with variable order and, in particular, variational problems involving variable-order operators. This brief presents a new numerical tool for the solution of differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one, an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided.
The contributors consider variational problems that may be subject to one or more constraints, where the functional depends on a combined Caputo derivative of variable fractional order. In particular, they establish necessary optimality conditions of EulerLagrange type. As the terminal point in the cost integral is free, as is the terminal state, transversality conditions are also obtained.
The Variable-Order Fractional Calculus of Variations is a valuable source of information for researchers in mathematics, physics, engineering, control and optimization; it provides both analytical and numerical methods to deal with variational problems. It is also of interest to academics and postgraduates in these fields, as it solves multiple variational problems subject to one or more constraints in a single brief.

Contenu
Fractional Calculus.- The Calculus of Variations.- Expansion Formulas for Fractional Derivatives.- The Fractional Calculus of Variations.