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Fractional Calculus and Applied Analysis

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Fractional calculus of variations for a combined Caputo derivative

1Faculty of Computer Science, Białystok University of Technology, 15-351, Białystok, Poland

2Center for Research and Development in Mathematics and Applications Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

© 2011 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Citation Information: Fractional Calculus and Applied Analysis. Volume 14, Issue 4, Pages 523–537, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: 10.2478/s13540-011-0032-6, September 2011

Publication History

Published Online:
2011-09-29

Abstract

We generalize the fractional Caputo derivative to the fractional derivative C D γα,β, which is a convex combination of the left Caputo fractional derivative of order α and the right Caputo fractional derivative of order β. The fractional variational problems under our consideration are formulated in terms of C D γα,β. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.

MSC: Primary 26A33; Secondary 49K05

Keywords: fractional derivatives; Caputo derivatives; fractional variational principles; Euler-Lagrange equations; isoperimetric constraints; transversality conditions

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