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)
Linear systems of fractional nabla difference equations. (English) Zbl 1218.39003
The authors consider a system of linear fractional nabla difference equations with constant coefficients of the form $$\nabla_0^\nu y(t)=Ay(t)+f(t),\quad t=1,2,\dots,$$ where $A$ denotes an $n\times n$ matrix with constant entries, $0<\nu<1$ and $\nabla_0^\nu$ is the Riemann-Liouville fractional difference operator. They construct the fundamental matrix for the homogeneous system and the causal Green’s function for the nonhomogeneous system. Applications to asymptotic behavior are given.

39A06Linear equations (difference equations)
39A12Discrete version of topics in analysis
26A33Fractional derivatives and integrals (real functions)
34A08Fractional differential equations
Full Text: DOI
[1] M. Abramowitz and I. Stegun, Handbook of mathematical functions, with formulas, graphs and mathematical tables , Sixth printing, with corrections, in National Bureau of Standards Applied Mathematics Series 55 National Bureau of Standards, Washington, D.C., 1967. · Zbl 0543.33001
[2] R.P. Agarwal, A propos d’une note de M. Pierre Humbert , C.R. Acad. Sci. 236 (1953), 2031-2032. · Zbl 0051.30801
[3] G. Anastassiou, Nabla discrete fractional calculus and nabla inequalities , Math. Comput. Modelling 51 (2010), 562-571. · Zbl 1190.26001 · doi:10.1016/j.mcm.2009.11.006
[4] G.E. Andrews, R. Askey and R. Roy, Special functions , in Encyclopedia of mathematics and its applications 71 , Cambridge University Press, Cambridge, 1999.
[5] F.M. At\icı and P.W. Eloe, Discrete fractional calculus with the nabla operator , Electron. J. Qual. Theory Differ. Equat. 2009 Special Edition I, No. 3, 12 pages (electronic). · Zbl 1189.39004 · emis:journals/EJQTDE/sped1/103.pdf · eudml:227223
[6] ---, A transform method in discrete fractional calculus , Inter. J. Difference Equations 2 (2007), 165-176.
[7] ---, Fractional $q$-calculus on a time scale , J. Nonlinear Math. Phys. 14 (2007), 333-344.
[8] F.M. At\icı and P.W. Eloe, Initial value problems in discrete fractional calculus , Proc. Amer. Math. Soc. 137 (2009), 981-989. · Zbl 1166.39005 · doi:10.1090/S0002-9939-08-09626-3
[9] F.M. At\icı and S. Şengül, Modeling with fractional difference equations , J. Math. Anal. Appl. 369 (2010), 1-9. · Zbl 1204.39004 · doi:10.1016/j.jmaa.2010.02.009
[10] B. Bonilla, M. Rivero and J.J. Trujillo, On systems of linear fractional differential equations with constant coefficients , Appl. Math. Comput. 187 (2007), 68-78. · Zbl 1121.34006 · doi:10.1016/j.amc.2006.08.104
[11] H.J. Haubold, A.M. Mathai and R.K. Saxena, Mittag-Leffler functions and their applications , arxiv.org/abs /09090230. · Zbl 1218.33021
[12] R. Magin, Fractional calculus in bioengineering , Begell House Publishers, Redding, CT, 2004.
[13] G.M. Mittag-Leffler, Sur la nouvelle fonction $E_\alpha(x)$ , C.R. Acad. Sci. Paris 137 (1903), 554-558. · Zbl 34.0435.01
[14] A. Nagai, Discrete Mittag-Leffler function and its applications , Publ. Res. Inst. Math. Sci. Kyoto Univ. 1302 (2003), 1-20.
[15] I. Podlubny, Fractional differential equations , Academic Press, New York, 1999. · Zbl 0924.34008
[16] G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integrals and derivatives: Theory and applications , Gordon and Breach, Yverdon, 1993. · Zbl 0818.26003
[17] H.M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel , Appl. Math. Comput. 211 (2009), 198-210. · Zbl 05562749 · doi:10.1016/j.amc.2009.01.055