Journal of Applied Mathematics
Volume 2012 (2012), Article ID 193061, 21 pages
http://dx.doi.org/10.1155/2012/193061
Research Article

Analysis of a System for Linear Fractional Differential Equations

Fang Wang,1,2 Zhen-hai Liu,3 and Ping Wang4

1School of Mathematical Science and Computing Technology, Central South University, Hunan, Changsha 410075, China
2School of Mathematical Science and Computing Technology, Changsha University of Science and Technology, Hunan, Changsha 410076, China
3School of Mathematical Science and Computing Technology, Guangxi University for Nationalities, Guangxi, Nanning 530006, China
4Xiamen Institute of Technology, Huaqiao University, Fujian, Xiamen 361021, China

Received 3 March 2012; Revised 21 July 2012; Accepted 23 July 2012

Academic Editor: Chein-Shan Liu

Copyright © 2012 Fang Wang 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.

Abstract

The main purpose of this paper is to obtain the unique solution of the constant coefficient homogeneous linear fractional differential equations 𝐷 𝑞 𝑡 0 𝑋 ( 𝑡 ) = 𝑃 𝑋 ( 𝑡 ) , 𝑋 ( 𝑎 ) = 𝐵 and the constant coefficient nonhomogeneous linear fractional differential equations 𝐷 𝑞 𝑡 0 𝑋 ( 𝑡 ) = 𝑃 𝑋 ( 𝑡 ) + 𝐷 , 𝑋 ( 𝑎 ) = 𝐵 if 𝑃 is a diagonal matrix and 𝑋 ( 𝑡 ) ∈ 𝐶 1 − 𝑞 [ 𝑡 0 , 𝑇 ] × 𝐶 1 − 𝑞 [ 𝑡 0 , 𝑇 ] × ⋯ × 𝐶 1 − 𝑞 [ 𝑡 0 , 𝑇 ] and prove the existence and uniqueness of these two kinds of equations for any 𝑃 ∈ 𝐿 ( 𝑅 𝑚 ) and 𝑋 ( 𝑡 ) ∈ 𝐶 1 − 𝑞 [ 𝑡 0 , 𝑇 ] × 𝐶 1 − 𝑞 [ 𝑡 0 , 𝑇 ] × ⋯ × 𝐶 1 − 𝑞 [ 𝑡 0 , 𝑇 ] . Then we give two examples to demonstrate the main results.