Fractional differences and integration by parts. (English) Zbl 1225.39008
Inspired by the results of F. M. Atici and P. W. Eloe [Int. J. Difference Equ. 2, No. 2, 165–176 (2007), Proc. Am. Math. Soc. 137, No. 3, 981–989 (2009; Zbl 1166.39005)] and K. S. Miller and B. Ross [An introduction to the fractional calculus and fractional differential equations. New York: John Wiley & Sons (1993; Zbl 0789.26002)], the authors introduce right fractional sum and difference operators. Based upon the provided theory, a by-part formula is given analogous to the one in usual fractional calculus. Towards the end of the paper, the obtained results are implemented to derive Euler-Lagrange equations for a discrete variational problem in fractional calculus.
Reviewer: Murat Adivar (Izmir)
MSC:
39A12 | Discrete version of topics in analysis |
26A33 | Fractional derivatives and integrals |
39A10 | Additive difference equations |
49J15 | Existence theories for optimal control problems involving ordinary differential equations |
49M25 | Discrete approximations in optimal control |
39A70 | Difference operators |