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International Journal of Differential Equations
Volume 2012, Article ID 521750, 38 pages
http://dx.doi.org/10.1155/2012/521750
Review Article

Generalized Multiparameters Fractional Variational Calculus

Mechanical Engineering and Energy Processes, Southern Illinois University, Carbondale, IL 62901, USA

Received 2 April 2012; Accepted 8 August 2012

Academic Editor: Fawang Liu

Copyright © 2012 Om Prakash Agrawal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper builds upon our recent paper on generalized fractional variational calculus (FVC). Here, we briefly review some of the fractional derivatives (FDs) that we considered in the past to develop FVC. We first introduce new one parameter generalized fractional derivatives (GFDs) which depend on two functions, and show that many of the one-parameter FDs considered in the past are special cases of the proposed GFDs. We develop several parts of FVC in terms of one parameter GFDs. We point out how many other parts could be developed using the properties of the one-parameter GFDs. Subsequently, we introduce two new two- and three-parameter GFDs. We introduce some of their properties, and discuss how they can be used to develop FVC. In addition, we indicate how these formulations could be used in various fields, and how the generalizations presented here can be further extended.