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Initial value problems in discrete fractional calculus
Open accessAbstract
This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations. For this purpose we first carefully develop the commutativity properties of the fractional sum and the fractional difference operators. Then a $\nu$-th ($0 < \nu \leq 1$) order fractional difference equation is defined. A nonlinear problem with an initial condition is solved and the corresponding linear problem with constant coefficients is solved as an example. Further, the half-order linear problem with constant coefficients is solved with a method of undetermined coefficients and with a transform method.
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Cites
Fractional Differential Equations
(2025) · MDPI eBooks · doi:10.3390/books978-3-7258-4742-6 · Cited by 20,278
An Introduction to the Fractional Calculus and Fractional Differential Equations
Miller, K. S., & Ross, B. (1993) · Cited by 8,193
Fractional Integrals and Derivatives, Theory and Applications
Samko, S. G., Kilbas, A. A., & Marichev, O. I. (1987) · CERN Document Server (European Organization for Nuclear Research) · Cited by 7,769
(1999) · Mathematics in Science and Engineering · doi:10.1016/s0076-5392(99)x8001-5 · Cited by 4,936
Dynamic Equations on Time Scales
Bohner, M., & Peterson, A. (2001) · Birkhäuser Boston eBooks · doi:10.1007/978-1-4612-0201-1 · Cited by 2,934
Matlab, An Introduction With Applications
Gilat, A. (2003) · CERN Document Server (European Organization for Nuclear Research) · Cited by 557
Cited by
On Riemann and Caputo fractional differences
Abdeljawad, T. (2011) · Computers & Mathematics with Applications · doi:10.1016/j.camwa.2011.03.036 · Cited by 733
Chaos in fractional-order discrete neural networks with application to image encryption
Chen, L., Yin, H., Huang, T., Yuan, L., Zheng, S., & Yin, L. (2020) · Neural Networks · doi:10.1016/j.neunet.2020.02.008 · Cited by 255
Metadata
- Authors
- Atici, F. M., & Eloe, P. W.
- Venue
- Proceedings of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Published
- 2008-09-10
- Volume
- 137
- Issue
- 03
- Pages
- 981-989
- ISSN
- 0002-9939 (ISSN-L)
- OpenAlex
- W1965801804
- Cited by
- 675
- References
- 6
- FWCI
- 25.49
- Type
- journal article
- Language
- en
- Open access
- hybrid
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Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
