Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 653939, 28 pages
Research Article

Solving Linear Coupled Fractional Differential Equations by Direct Operational Method and Some Applications

1Faculty of Engineering, Multimedia University, Selangor Darul Ehsan, 63100 Cyberjaya, Malaysia
2Department of Chemistry, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
3School of Information Science & Technology, East China Normal University, No. 500, Dong-Chuan Road, Shanghai 200241, China
4College of Computer Science, Zhejiang University of Technology, Hangzhou 310023, China

Received 20 July 2011; Accepted 7 September 2011

Academic Editor: Carlo Cattani

Copyright © 2012 S. C. Lim et al. 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.


A new direct operational inversion method is introduced for solving coupled linear systems of ordinary fractional differential equations. The solutions so-obtained can be expressed explicitly in terms of multivariate Mittag-Leffler functions. In the case where the multiorders are multiples of a common real positive number, the solutions can be reduced to linear combinations of Mittag-Leffler functions of a single variable. The solutions can be shown to be asymptotically oscillatory under certain conditions. This technique is illustrated in detail by two concrete examples, namely, the coupled harmonic oscillator and the fractional Wien bridge circuit. Stability conditions and simulations of the corresponding solutions are given.