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The Scientific World Journal
Volume 2014, Article ID 402373, 21 pages
Research Article

Determination of Coefficients of High-Order Schemes for Riemann-Liouville Derivative

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 10 December 2013; Accepted 4 February 2014; Published 15 April 2014

Academic Editors: B. Carpentieri and C. Silva

Copyright © 2014 Rifang Wu 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.


Although there have existed some numerical algorithms for the fractional differential equations, developing high-order methods (i.e., with convergence order greater than or equal to 2) is just the beginning. Lubich has ever proposed the high-order schemes when he studied the fractional linear multistep methods, where he constructed the th order schemes for the th order Riemann-Liouville integral and th order Riemann-Liouville derivative. In this paper, we study such a problem and develop recursion formulas to compute these coefficients in the higher-order schemes. The coefficients of higher-order schemes are also obtained. We first find that these coefficients are oscillatory, which is similar to Runge’s phenomenon. So, they are not suitable for numerical calculations. Finally, several numerical examples are implemented to testify the efficiency of the numerical schemes for .