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Cubo (Temuco)

On-line version ISSN 0719-0646

Cubo vol.13 no.3 Temuco Oct. 2011

http://dx.doi.org/10.4067/S0719-06462011000300009 

CUBO A Mathematical Journal Vol.13, No03, (153-184). October 2011

 

Sum and Difference Compositions in Discrete Fractional Calculus

 

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

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[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.

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

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

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

[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.

[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.

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


Received: August 2010. Revised: September 2010.

 

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