Dual identities in fractional difference calculus within Riemann
- PDF / 235,981 Bytes
- 16 Pages / 595.28 x 793.7 pts Page_size
- 39 Downloads / 149 Views
RESEARCH
Open Access
Dual identities in fractional difference calculus within Riemann Thabet Abdeljawad* *
Correspondence: [email protected] Department of Mathematics and Computer Science, Çankaya University, Ankara, 06530, Turkey
Abstract We investigate two types of dual identities for Riemann fractional sums and differences. The first type relates nabla- and delta-type fractional sums and differences. The second type represented by the Q-operator relates left and right fractional sums and differences. These dual identities insist that in the definition of right fractional differences, we have to use both nabla and delta operators. The solution representation for a higher-order Riemann fractional difference equation is obtained as well. Keywords: right (left) delta and nabla fractional sums; right (left) delta and nabla Riemann; Q-operator; dual identity
1 Introduction During the last two decades, due to its widespread applications in different fields of science and engineering, fractional calculus has attracted the attention of many researchers [–]. Starting from the idea of discretizing the Cauchy integral formula, Miller and Ross [] and Gray and Zhang [] obtained discrete versions of left-type fractional integrals and derivatives, called fractional sums and differences. Fifteen years later, several authors started to deal with discrete fractional calculus [–], benefiting from the theory of time scales originated by Hilger in (see []). In this article, we summarize some of the results mentioned in the above references and add more in the right-type and higher order fractional cases. Throughout the article, we almost agree with the previously presented definitions except for the definition of right fractional difference. We will figure out that these definitions seem to be more convenient than the previously presented ones by proving some dual identities. These identities fall into two kinds. The first kind relates nabla-type fractional differences and sums to delta ones. The second kind represented by the Q-operator relates left and right fractional sums and differences. This setting enables us to get identities resembling better the ordinary fractional case. Along with the previously mentioned points, we are able to fit a reasonable nabla integration by parts formula which remains in accordance with the one obtained in [] but different from those obtained in [] and []. The obtained dual identities are also used to obtain a delta integration by parts formula from the nabla one. The solution representation for the higher-order Riemann fractional difference equation is obtained as well and thus the result in [] is generalized. We will see that the higherorder Riemann fractional difference initial value problem of non-integer order needs only © 2013 Abdeljawad; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provid
Data Loading...
