zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Remarks on fractional derivatives. (English) Zbl 1125.26009
The authors give a historical sketch of fractional calculus, which is readily available in details in the references given. On the other hand, they should also have mentioned the monograph “The fractional calculus. Theory and applications of differentiation and integration to arbitrary order” (1974; Zbl 0292.26011) by {\it K. B. Oldham} and {\it J. Spanier}, which happens to be the maiden text made available to researchers of this area of research. Moreover, this book contains an excellent chronological bibliography on fractional calculus by {\it B. Ross} (pp. 3--15). The authors study in the present paper some properties of fractional derivatives, which is claimed to be interesting and new (not found elsewhere) by the authors. Grünwald-Letnikov fractional derivative, Riemann-Liouville fractional derivative and Caputo derivative are studied here. The Riemann-Liouville and the Caputo derivatives are compared and, further the sequential property of Caputo derivative is derived and simultaneously authors compare these two (mentioned above) derivatives with the classical derivative. The authors also give a sketch map, which illustrates consistency of fractional derivatives or integrals, which appears to be useful for some problems which require geometrical interpretation.

MSC:
26A33Fractional derivatives and integrals (real functions)
WorldCat.org
Full Text: DOI
References:
[1] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations. (1993) · Zbl 0789.26002
[2] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008
[3] Butzer, P. L.; Westphal, U.: An introduction to fractional calculus. (2000) · Zbl 0987.26005
[4] Diethelm, K.; Ford, N. J.; Freed, A. D.; Luchko, Yu.: Algorithms for the fractional calculus: a selection of numerical methods. Comput. methods appl. Mech. eng. 194, 743-773 (2005) · Zbl 1119.65352
[5] W.H. Deng, C.P. Li, Stability analysis of nonlinear fractional differential equations (submitted for publication).
[6] Zhou, T. S.; Li, C. P.: Synchronization in fractional-order differential systems. Physica D 212, 111-125 (2005) · Zbl 1094.34034
[7] D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Engineering in Systems and Application Multiconference, IMACS, IEEE-SMC, Lille, France, vol. 2, 1996, pp. 963 -- 968.
[8] Heymans, N.; Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann -- Liouville fractional derivatives. Rheol. acta 45, 765-772 (2006)
[9] Dzhrbashyan, M. M.; Nersesyan, A. B.: Criteria of expansibility of functions in Dirichlet series. Izv. akad. Nauk. arm. SSR, ser. Fiz. mat. 11, No. 5, 85-106 (1958) · Zbl 0086.05701
[10] Dzhrbashyan, M. M.; Nersesyan, A. B.: On the use of some integro-differential operators. Dokl. akad. Nauk. SSSR 121, No. 2, 210-213 (1958) · Zbl 0095.08504
[11] Dzhrbashyan, M. M.; Nersesyan, A. B.: Expansion in some biorthogonal systems and boundary-value problems for differential equations of fractional order. Trudy mosk. Mat. ob. 10, 89-179 (1961)
[12] Dzhrbashyan, M. M.: Integral transforms and representations of functions in the complex domain. (1966) · Zbl 0148.30002
[13] Dzhrbashyan, M. M.; Nersesyan, A. B.: Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv. akad. Nauk. arm. SSR 3, No. 1, 3-29 (1968)
[14] W.H. Deng, C.P. Li, Q. Guo, Analysis of fractional differential equations with multi-order (submitted for publication).
[15] Sun, H. H.; Abdelwahab, A. A.; Onaral, B.: Linear approximation of transfer function with a pole of fractional order. IEEE trans. Automat. contr. 29, 441-444 (1984) · Zbl 0532.93025
[16] Diethelm, K.; Ford, N. J.: Numerical solution of the bagley -- torvik equation. Bit 42, 490-507 (2002) · Zbl 1035.65067
[17] Diethelm, K.; Ford, N. J.; Freed, A. D.: A predictor -- corrector approach for the numerical solution of fractional differential equations. Nonlinear dyn. 29, 3-22 (2002) · Zbl 1009.65049
[18] Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations. J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003
[19] Diethelm, K.; Ford, N. J.: Multi-order fractional differential equations and their numerical solution. Appl. math. Comput. 154, 621-640 (2004) · Zbl 1060.65070
[20] Daftardar-Gejji, V.; Babakhani, A.: Analysis of a system of fractional differential equations. J. math. Anal. appl. 293, 511-522 (2004) · Zbl 1058.34002
[21] Edwards, J. T.; Ford, N. J.; Simpson, A. C.: The numerical solution of linear multi-term fractional differential equations: systems of equations. J. math. Anal. appl. 148, 401-418 (2002) · Zbl 1019.65048
[22] El-Raheem, Zaki F. A.: Modification of the application of a contraction mapping method on a class of fractional differential equation. Appl. math. Comput. 137, 371-374 (2003) · Zbl 1034.34070
[23] Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, solitons & fractals 7, 1461-1477 (1996) · Zbl 1080.26505
[24] Matignon, D.: Observer-based controllers for fractional differential equations. Conference on decision and control, 4967-4972 (1997)
[25] Bonnet, C.; Partington, J. R.: Coprime factorizations and stability of fractional differential systems. Sys. contr. Lett. 41, 167-174 (2000) · Zbl 0985.93048
[26] Li, C. P.; Peng, G. J.: Chaos in Chen’s system with a fractional order. Chaos, solitons & fractals 22, 443-450 (2004) · Zbl 1060.37026
[27] Deng, W. H.; Li, C. P.: Synchronization of chaotic fractional Chen system. J. phys. Soc. jpn. 74, 1645-1648 (2005) · Zbl 1080.34537
[28] Deng, W. H.; Li, C. P.: Chaos synchronization of the fractional Lü system. Physica A 353, 61-72 (2005)
[29] Li, C. P.; Deng, W. H.; Xu, D.: Chaos synchronization of the Chua system with a fractional order. Physica A 360, 171-185 (2006)
[30] Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. calculus appl. Anal. 5, 367-386 (2002) · Zbl 1042.26003