Advanced Methods in the Fractional Calculus of Variations, 2015 SpringerBriefs in Applied Sciences and Technology Series

This brief presents a general unifying perspective on the fractional calculus. It brings together results of several recent approaches in generalizing the least action principle and the Euler?Lagrange equations to include fractional derivatives.

The dependence of Lagrangians on generalized fractional operators as well as on classical derivatives is considered along with still more general problems in which integer-order integrals are replaced by fractional integrals. General theorems are obtained for several types of variational problems for which recent results developed in the literature can be obtained as special cases. In particular, the authors offer necessary optimality conditions of Euler?Lagrange type for the fundamental and isoperimetric problems, transversality conditions, and Noether symmetry theorems. The existence of solutions is demonstrated under Tonelli type conditions. The results are used to prove the existence of eigenvalues and corresponding orthogonal eigenfunctions of fractional Sturm?Liouville problems.

Advanced Methods in the Fractional Calculus of Variations is a self-contained text which will be useful for graduate students wishing to learn about fractional-order systems. The detailed explanations will interest researchers with backgrounds in applied mathematics, control and optimization as well as in certain areas of physics and engineering.

1. Introduction.- 2. Fractional Calculus.- 3. Fractional Calculus of Variations.- 4. Standard Methods in Fractional Variational Calculus.- 5. Direct Methods in Fractional Calculus of Variations.- 6. Application to the Sturm-Liouville Problem.- 7. Conclusion.- Appendix - Two Convergence Lemmas.- Index.

Agnieszka B. Malinowska is an Assistant Professor in the Bialystok University of Technology, Poland, she is affiliated with this university since 1995. From 2008 to 2011 she was a Senior Researcher at the University of Aveiro, Portugal. She obtained her M.Sc. in Mathematics from the Warsaw University, Ph.D. and Habilitation  in Technical Sciences (Automation and Robotics) from the Systems Research Institute of the Polish Academy of Sciences. Her research interests include topics in the areas of Multi objective Optimization, Calculus of Variations, Time Scale Theory and Fractional Calculus. She is co-author of a book about the Fractional Calculus of Variations with Imperial College Press and World Scientific Publishing, 2012, a book on Quantum Variational Calculus with Springer, 2014, and author and co-author of 38 papers in international journals ranked by ISI Web of Science.

Tatiana Odzijewicz is a Post-doc Researcher at the Center for Research and Development in Mathematicsand Applications (CIDMA), University of Aveiro, Portugal. She received M.Sc. from University of Bialystok, Poland (2009) and PhD in Mathematics and Applications from University of Aveiro, Portugal (2013). Her research is mainly in fractional calculus, generalized operators, the calculus of variations and optimal control. Tatiana Odzijewicz has been recipient, in 2012, of the celebrated Grunvald-Letnikov Award.

Delfim F. M. Torres was born August 16, 1971, in Nampula, Mozambique. He is currently an Associate Professor with Habilitation at the Department of Mathematics, University of Aveiro (UA), Portugal; Coordinator of the group on Mathematical Theory of Systems and Control of the Center for Research and Development in Mathematics and Applications (CIDMA); Director of The FCT Doctoral Program in Mathematics and Applications (FCT MAP-PDMA) of the Universities of Minho, Aveiro, Porto and UBI, Portugal; Editor-in-Chief of the Int. J. Appl. Math. Stat. (IJAMAS) since 2008; Editor-in-