zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On Riemann and Caputo fractional differences. (English) Zbl 1228.26008
Summary: We define left and right Caputo fractional sums and differences, study some of their properties and then relate them to Riemann-Liouville ones studied before. Also, the discrete version of the $Q$-operator is used to relate the left and right Caputo fractional differences. A Caputo fractional difference equation is solved. The solution proposes discrete versions of Mittag-Leffler functions.

26A33Fractional derivatives and integrals (real functions)
34A08Fractional differential equations
Full Text: DOI arXiv
[1] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives -- theory and applications, (1993) · Zbl 0818.26003
[2] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[3] Kilbas, A.; Srivastava, M. H.; Trujillo, J. J.: Theory and application of fractional differential equations, North holland mathematics studies 204 (2006) · Zbl 1092.45003
[4] Zaslavsky, G. M.: Hamiltonian chaos and fractional dynamics, (2005) · Zbl 1083.37002
[5] Magin, R. L.: Fractional calculus in bioengineering, (2006)
[6] Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, solitons and fractals 7, No. 9 (1996) · Zbl 1080.26505 · doi:10.1016/0960-0779(95)00125-5
[7] Kilbas, A. A.; Rivero, M.; Trujillo, J. J.: Existence and uniqueness theorems for differential equations of fractional order in weighted spaces of continuous functions, Fractional calculus applied analysis 6, No. 4, 363-400 (2003) · Zbl 1085.34002
[8] Silva, M. F.; Machado, J. A. T.; Lopes, A. M.: Modelling and simulation of artificial locomotion systems, Robotica 23, 595-606 (2005)
[9] Agrawal, O. P.; Baleanu, D.: A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems, Journal of vibration and control 13, No. 9--10, 1269-1281 (2007) · Zbl 1182.70047 · doi:10.1177/1077546307077467
[10] Scalas, E.: Mixtures of compound Poisson processes as models of tick-by-tick financial data, Chaos, solitons and fractals 34, No. 1, 33-40 (2007) · Zbl 1142.60392 · doi:10.1016/j.chaos.2007.01.047
[11] Diethelm, K.; Ford, N. J.; Freed, A. D.; Luchko, Y.: Algorithms for the fractional calculus: a selection of numerical methods, Computer methods in applied mechanics and engineering 194, No. 6--8, 743-773 (2005) · Zbl 1119.65352 · doi:10.1016/j.cma.2004.06.006
[12] Atanackovic, T. M.; Stankovic, B.: On a class of differential equations with left and right fractional derivatives, Zeitschrift fur angewandte Mathematik und mechanik 87, No. 7, 537-546 (2007) · Zbl 1131.34003 · doi:10.1002/zamm.200710335
[13] Abdeljawad (Maraaba), T.; Baleanu, D.; Jarad, F.: Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, Journal of mathematical physics 49, 083507 (2008) · Zbl 1152.81550 · doi:10.1063/1.2970709
[14] Maraaba (Abdeljawad), T.; Jarad, F.; Baleanu, D.: On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Science in China series A: mathematics 51, No. 10, 1775-1786 (2008) · Zbl 1179.26024 · doi:10.1007/s11425-008-0068-1
[15] Baleanu, D.; Trujillo, J. J.: On exact solutions of a class of fractional Euler--Lagrange equations, Nonlinear dynamics 52, No. 4, 331-335 (2008) · Zbl 1170.70328 · doi:10.1007/s11071-007-9281-7
[16] K.S. Miller, B. Ross, Fractional difference calculus, in: Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, 1989, pp. 139--152. · Zbl 0693.39002
[17] Atıcı, F. M.; Eloe, P. W.: A transform method in discrete fractional calculus, International journal of difference equations 2, No. 2, 165-176 (2007)
[18] Atıcı, F. M.; Eloe, P. W.: Initial value problems in discrete fractional calculus, Proceedings of the American mathematical society 137, 981-989 (2009) · Zbl 1166.39005 · doi:10.1090/S0002-9939-08-09626-3
[19] Abdeljawad, T.; Baleanu, D.: Fractional differences and integration by parts, Journal of computational analysis and applications 13, No. 3, 574-582 (2011) · Zbl 1225.39008
[20] Kilbas, Samko G.; Marichev, A. A.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[21] Bastos, Nuno R. O.; Ferreira, Rui A. C.; Torres, Delfim F. M.: Discrete-time fractional variational problems, Signal processing 91, No. 3, 513-524 (2011) · Zbl 1203.94022 · doi:10.1016/j.sigpro.2010.05.001
[22] Atıcı, Ferhan M.; Eloe, Paul W.: Discrete fractional calculus with the nabla operator, Electronic journal of qualitative theory of differential equations, No. 3, 1-12 (2009) · Zbl 1189.39004 · emis:journals/EJQTDE/sped1/103.pdf