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Fractional Dynamics and Control pp 109–118Cite as

Fractional Euler–Lagrange Differential Equations via Caputo Derivatives

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Abstract

We review some of the recent results of the fractional variational calculus. Necessary optimality conditions of Euler–Lagrange type for functionals with a Lagrangian containing left and right Caputo derivatives are given. Several problems are considered: with fixed or free boundary conditions, and in presence of integral constraints that also depend on Caputo derivatives.

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Notes

  1. 1.

    Along the work we use round brackets for the arguments of functions, and square brackets for the arguments of operators. By definition, an operator receives a function and returns another function.

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Acknowledgements

Work supported by FEDER funds through COMPETE — Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Competitividade”) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveir) and the Portuguese Foundation for Science and Technology (“FCT — Fundação para a Ciência e a Tecnologia”), within project PEst-C/MAT/Ul4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690. Agnieszka Malinowska is also supported by Białystok University of Technology grant S/WI/2/2011.

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Authors and Affiliations

  1. Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    Ricardo Almeida & Delfim F. M. Torres

  2. Faculty of Computer Science, Białystok University of Technology, 15-351, Białystok, Poland

    Agnieszka B. Malinowska

Authors
  1. Ricardo Almeida
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  2. Agnieszka B. Malinowska
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  3. Delfim F. M. Torres
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Corresponding author

Correspondence to Ricardo Almeida .

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Editors and Affiliations

  1. Faculty of Art and Sciences, Mathematics and Computer Sciences, Cankaya University, Ankara, Turkey

    Dumitru Baleanu

  2. Institute of Engineering, Department of Electrical Engineering, Polytechnic Institute of Porto, Rua Dr António Bernadino Almeida 431, Porto, 4200-072, Portugal

    José António Tenreiro Machado

  3. Edwardsville, School of Engineering, Southern Illinois University, Room: EB2064, Edwardsville, 62026-1805, USA

    Albert C. J. Luo

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Almeida, R., Malinowska, A.B., Torres, D.F.M. (2012). Fractional Euler–Lagrange Differential Equations via Caputo Derivatives. In: Baleanu, D., Machado, J., Luo, A. (eds) Fractional Dynamics and Control. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0457-6_9

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  • DOI: https://doi.org/10.1007/978-1-4614-0457-6_9

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  • Online ISBN: 978-1-4614-0457-6

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