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On the fractional-order logistic equation. (English) Zbl 1140.34302
The authors investigate the fractional-order logistic equation. They study the stability, existence, uniqueness and numerical approximation of a solution.

34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
26A33Fractional derivatives and integrals (real functions)
34D20Stability of ODE
Full Text: DOI
[1] Ahmed, E.; El-Sayed, A. M. A.; El-Mesiry, E. M.; El-Saka, H. A. A.: Numerical solution for the fractional replicator equation. Internat. J. Modern phys. C. 16, No. 7, 1-9 (2005) · Zbl 1080.65536
[2] Ahmed, E.; El-Sayed, A. M. A.; El-Saka, H. A. A.: On some Routh--Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems. Phys. lett. A 358, No. 1 (2006) · Zbl 1142.30303
[3] Ahmed, E.; El-Sayed, A. M. A.; El-Saka, H. A. A.: Equilibrium points, stability and numerical solutions of fractional-order predator--prey and rabies models. J. math. Anal. appl. 325, 542-553 (2007) · Zbl 1105.65122
[4] Diethelm, K.; Freed, A.: On the solution of nonlinear fractional order differential equations used in the modelling of viscoplasticity. Scientific computing in chemical engineering II--computational fluid dynamics, reaction engineering, and molecular properties, 217-224 (1999)
[5] Diethelm, K.; Freed, A.: The fracpece subroutine for the numerical solution of differential equations of fractional order. Forschung und wissenschaftliches rechnen 1998, 57-71 (1999)
[6] K. Diethelm, N.J. Ford, The numerical solution of linear and non-linear fractional differential equations involving fractional derivatives several of several orders, Numerical Analysis Report 379, Manchester Center for Numerical Computational Mathematics
[7] Diethelm, K.: Predictor--corrector strategies for single- and multi-term fractional differential equations. Proceedings of the 5th hellenic--European conference on computer mathematics and its applications, 117-122 (2002) · Zbl 1028.65081
[8] 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
[9] Diethelm, K.; Ford, N. J.; Freed, A. D.: Detailed error analysis for a fractional Adams method. Numer. algorithms 36, 31-52 (2004) · Zbl 1055.65098
[10] El-Mesiry, E. M.; El-Sayed, A. M. A.; El-Saka, H. A. A.: Numerical methods for multi-term fractional (arbitrary) orders differential equations. Appl. math. Comput. 160, No. 3, 683-699 (2005) · Zbl 1062.65073
[11] El-Sayed, A. M. A.: Fractional differential--difference equations. J. fract. Calc. 10, 101-106 (1996) · Zbl 0888.34060
[12] El-Sayed, A. M. A.: Nonlinear functional differential equations of arbitrary orders. Nonlinear anal. 33, No. 2, 181-186 (1998) · Zbl 0934.34055
[13] El-Sayed, A. M. A.; Gaafar, F. M.: Fractional order differential equations with memory and fractional-order relaxation--oscillation model. Pure math. Appl. 12 (2001) · Zbl 1006.34008
[14] El-Sayed, A. M. A.; El-Mesiry, E. M.; El-Saka, H. A. A.: Numerical solution for multi-term fractional (arbitrary) orders differential equations. Comput. appl. Math. 23, No. 1, 33-54 (2004) · Zbl 1213.34025
[15] El-Sayed, A. M. A.; Gaafar, F. M.; Hashem, H. H.: On the maximal and minimal solutions of arbitrary orders nonlinear functional integral and differential equations. Math. sci. Res. J. 8, No. 11, 336-348 (2004) · Zbl 1068.45008
[16] Gorenflo, R.; Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. Fractals and fractional calculus in continuum mechanics, 223-276 (1997)
[17] Matignon, D.: Stability results for fractional differential equations with applications to control processing. Computational engineering in system application 2, 963 (1996)
[18] Podlubny, I.; El-Sayed, A. M. A.: On two definitions of fractional calculus. (1996)
[19] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008