Michael Holm

The University of Nebraska-Lincoln,USA

ABSTRACT

We introduce fractional sum and difference operators, study their behavior and develop a complete theory governing their compositions. This theory is then applied to solve a general, fractional initial value problem.

Keywords: Discrete Fractional Calculus, Fractional Sum, Fractional Difference, Composition Rule, Fractional Initial Value Problem.

Mathematics Subject Classification: 34

RESUMEN

Introducimos operadores de suma y diferencia fraccionaria, estudiamos su comportamiento y desarrollamos una teoria completa que rige sus composiciones. Aplicamos esta teoría para resolver un problema fraccionado de valor inicial

References

[1] FM Atici and PW Eloe, A Transform Method in Discrete Fractional Calculus, International Journal of Difference Equations, 2 2007 165-176.         [ Links ]

[2] FM Atici and PW Eloe, Initial Value Problems in Discrete Fractional Calculus, Proc. Amer. Math. Soc., 137 2009 981-989.         [ Links ]

[3] FM Atici and PW Eloe, Discrete Fractional Calculus with the Nabla Operator, Electronic Journal of Qualitative Theory of Differential Equations, Spec. Ed. I 2009 1-12.B Kuttner, On Differences of Fractional Order, Proc. London Math. Soc., 7 1957 453-466.         [ Links ]

[4] JB Diaz and TJ Olser, Differences of Fractional Order, Mathematics of Computation, 28 1974.         [ Links ]

[5] Henry L Gray and N Zhang, On a New Definition of the Fractional Difference, Mathematics of Computation, 50 1988 513-529.         [ Links ]

[6] W Kelley and A Peterson, Difference Equations: An Introduction with Application, Second Edition, Academic Press, New York, New York, 2000.         [ Links ]

[7] KS Miller and B Ross, Fractional Difference Calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and their Applications, Nihon University, Koriyama, Japan, 1988 139-152.         [ Links ]

[8] K Oldham and J Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover Publications, Inc., Mineola, New York, 2002.         [ Links ]

[9] I Podlubny, Fractional Differential Equations, Academic Press, New York, New York, 1999.         [ Links ]

Received: August 2010. Revised: September 2010.