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)
Analytic study on linear systems of fractional differential equations. (English) Zbl 1189.34017
Summary: An analytic study on linear systems of fractional differential equations with constant coefficients is presented. We briefly describe the issues of existence, uniqueness and stability of the solutions for two classes of linear fractional differential systems. This paper deals with systems of differential equations of fractional order, where the orders are equal to real number or rational numbers between zero and one. Exact solutions for initial value problems of linear fractional differential systems are analytically derived. Existence and uniqueness results are proved for two classes. The presented results are illustrated by analyzing some examples to demonstrate the effectiveness of the presented analytical approaches.

34A08Fractional differential equations
26A33Fractional derivatives and integrals (real functions)
34A30Linear ODE and systems, general
34D20Stability of ODE
45J05Integro-ordinary differential equations
Full Text: DOI
[1] Bagley, R. L.; Torvik, P. L.: On the fractional calculus models of viscoelastic behaviour, J. rheol. 30, 133-155 (1986) · Zbl 0613.73034 · doi:10.1122/1.549887
[2] Gaul, L.; Klein, P.; Kempfle, S.: Damping description involving fractional operators, Mech. syst. Signal process. 5, 81-88 (1991)
[3] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993) · Zbl 0789.26002
[4] Samko, G.; Kilbas, A.; Marichev, O.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[5] Mainardi, F.: On the initial value problem for the fractional diffusion-wave equation, Waves and stability in continuous media (Bologna 1993), 246-251 (1994)
[6] Metzler, F.; Schick, W.; Kilian, H. G.; Nonnenmacher, T. F.: Relaxation in filled polymers: A fractional calculus approach, J. chem. Phys. 103, 7180-7186 (1995)
[7] Rossikhin, Y.; Shitikova, M.: Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. mech. Rev. 50, 15-67 (1997) · Zbl 0901.73030
[8] Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics, 291-348 (1997) · Zbl 0917.73004
[9] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[10] Metzler, R.; Klafter, J.: The random walks guide to anomalous diffusion: A fractional dynamic approach, Phys. rep. 339, No. 1, 1-77 (2000) · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[11] , Applications of fractional calculus in physics (2000) · Zbl 0998.26002
[12] Matsuzaki, T.; Nakagawa, M.: A chaos neuron model with fractional differential equation, J. phys. Soc. Japan 72, 2678-2684 (2003)
[13] Magin, R. L.: Fractional calculus in bioengineering, Crit. rev. Biomed. eng. 32, 1-104 (2004)
[14] Zaslavsky, G. M.: Hamiltonian chaos and fractional dynamics, (2005) · Zbl 1083.37002
[15] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, (2006) · Zbl 1092.45003
[16] Bonilla, B.; Rivero, M.; Rodríguez-Germá, L.; Trujillo, J. J.: Fractional differential equations as alternative models to nonlinear differential equations, Appl. math. Comput. 187, No. 1, 79-88 (2007) · Zbl 1120.34323 · doi:10.1016/j.amc.2006.08.105
[17] Podlubny, I.: The Laplace transform method for linear differential equations of fractional order, (1994)
[18] Schneider, W.; Wyss, W.: Fractional diffusion and wave equations, J. math. Phys. 30, 134-144 (1989) · Zbl 0692.45004 · doi:10.1063/1.528578
[19] Mainardi, F.; Luchko, Y.; Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation, Fract. calc. Appl. anal. 4, 153-192 (2001) · Zbl 1054.35156
[20] Diethelm, K.; Ford, N.; Freed, A.: A predictor--corrector approach for the numerical solution of fractional differential equations, Nonlinear dynam. 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[21] Diethelm, K.; Ford, N.; Freed, A.: Detailed error analysis for a fractional Adams method, Numer. algorithms 36, 31-52 (2004) · Zbl 1055.65098 · doi:10.1023/B:NUMA.0000027736.85078.be
[22] Odibat, Z.; Momani, S.: An algorithm for the numerical solution of differential equations of fractional order, J. appl. Math. inform. 26, No. 1--2, 15-27 (2008) · Zbl 1133.65116
[23] Odibat, Z.; Momani, S.: Numerical methods for nonlinear partial differential equations of fractional order, Appl. math. Modelling. 32, No. 12, 28-39 (2008) · Zbl 1133.65116 · doi:10.1016/j.apm.2006.10.025
[24] Odibat, Z.; Momani, S.: Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos solitons fractals 36, No. 1, 167-174 (2008) · Zbl 1152.34311 · doi:10.1016/j.chaos.2006.06.041
[25] Cang, J.; Tan, Y.; Xu, H.; Liao, S. J.: Series solutions of non-linear Riccati differential equations with fractional order, Chaos solitons fractals 40, No. 1, 1-9 (2009) · Zbl 1197.34006 · doi:10.1016/j.chaos.2007.04.018
[26] Momani, S.; Odibat, Z.: A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor’s formula, J. comput. Appl. math. 220, No. 1--2, 85-95 (2008) · Zbl 1148.65099 · doi:10.1016/j.cam.2007.07.033
[27] Odibat, Z.; Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear sci. Numer. simul. 7, No. 1, 27-34 (2006) · Zbl 05675858
[28] Delbosco, D.; Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation, J. math. Anal. appl. 204, 609-625 (1996) · Zbl 0881.34005 · doi:10.1006/jmaa.1996.0456
[29] Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194
[30] Lakshmikantham, V.; Vatsala, A. S.: Basic theory of fractional differential equations, Nonlinear anal. TMA 69, No. 8, 2677-2682 (2008) · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[31] Daftardar-Gejji, V.: Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives, J. math. Anal. appl. 328, No. 2, 1026-1033 (2007) · Zbl 1115.34006 · doi:10.1016/j.jmaa.2006.06.007
[32] Bonilla, B.; Rivero, M.; Trujillo, J. J.: On systems of linear fractional differential equations with constant coefficients, Appl. math. Comput. 187, No. 1, 68-78 (2007) · Zbl 1121.34006 · doi:10.1016/j.amc.2006.08.104
[33] D. Matignon, Stability results of fractional differential equations with applications to control processing, in: Proceeding of IMACS, IEEE-SMC, Lille, France (1996) pp. 963--968
[34] Deng, W.; Li, C.; Lü, J.: Stability analysis of linear fractional differential system with multiple time delays, Nonlinear dynam. 48, 409-416 (2007) · Zbl 1185.34115 · doi:10.1007/s11071-006-9094-0
[35] Tavazoei, M. S.; Haeri, M.: A note on the stability of fractional order systems, Math. comput. Simulation. 79, No. 5, 1566-1576 (2009) · Zbl 1168.34036 · doi:10.1016/j.matcom.2008.07.003
[36] I. Petrás?, Stability of fractional-order systems with rational orders
[37] Momani, S.; Odibat, Z.: Numerical approach to differential equations of fractional order, J. comput. Appl. math. 207, No. 1, 96-110 (2007) · Zbl 1119.65127 · doi:10.1016/j.cam.2006.07.015
[38] Abdulaziz, O.; Noor, N.; M, I.: Further accuracy tests on Adomian decomposition method for chaotic systems, Chaos solitons fractals 36, No. 5, 1405-1411 (2008)
[39] Wang, Y.; Li, C.: Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle?, Phys. lett. A 363, 414-419 (2007)
[40] Grigorenko, I.; Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system, Phys. rev. Lett. 91, No. 3, 034101 (2003) · Zbl 1234.49040
[41] Deng, W. H.; Li, C. P.: Chaos synchronization of the fractional Lü system, Physica A 353, 61-72 (2005)
[42] Lu, J. G.; Chen, G.: A note on the fractional-order Chen system, Chaos solitons fractals 27, No. 3, 685-688 (2006) · Zbl 1101.37307 · doi:10.1016/j.chaos.2005.04.037
[43] Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, part II, J. roy. Astr. soc. 13, 529-539 (1967)
[44] Oldham, K. B.; Spanier, J.: The fractional calculus, (1974) · Zbl 0292.26011
[45] R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in: Carpinteri, Mainardi (Eds.), Fractals and fractional calculus, New York, 1997 · Zbl 0934.35008
[46] R. Gorenflo, F. Mainardi, Fractional oscillation and Mittag-Leffler functions, Fachbereich Mathematik and Informatic, A14/96, Freie Universitaet, Berlin, 1996, pp. 1--22 · Zbl 0916.34011