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The Scientific World Journal
Volume 2013 (2013), Article ID 915437, 11 pages
http://dx.doi.org/10.1155/2013/915437
Research Article

An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order

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

Received 27 August 2013; Accepted 19 September 2013

Academic Editors: A. Atangana, S. C. O. Noutchie, and A. Secer

Copyright © 2013 Ricardo Almeida and Delfim F. M. Torres. 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

We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type.