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Abstract and Applied Analysis
Volume 2015, Article ID 952057, 9 pages
http://dx.doi.org/10.1155/2015/952057
Research Article

A New Grünwald-Letnikov Derivative Derived from a Second-Order Scheme

1School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa
2DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, South Africa

Received 25 May 2015; Accepted 16 August 2015

Academic Editor: Tiecheng Xia

Copyright © 2015 B. A. Jacobs. 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

A novel derivation of a second-order accurate Grünwald-Letnikov-type approximation to the fractional derivative of a function is presented. This scheme is shown to be second-order accurate under certain modifications to account for poor accuracy in approximating the asymptotic behavior near the lower limit of differentiation. Some example functions are chosen and numerical results are presented to illustrate the efficacy of this new method over some other popular choices for discretizing fractional derivatives.