×

zbMATH — the first resource for mathematics

Analysis and numerical methods for fractional differential equations with delay. (English) Zbl 1291.65214
Summary: We consider fractional differential equations with delay. We focus on linear equations. We summarise existence and uniqueness theory based on the method of steps and we give a theorem on the propagation of derivative discontinuities. We discuss the dependence of the solution on the parameters of the equation and conclude with a numerical treatment and examples based on the adaptation of a fractional backward difference method to the delay case.

MSC:
65L05 Numerical methods for initial value problems
34A08 Fractional ordinary differential equations and fractional differential inclusions
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
34K06 Linear functional-differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Lakshmikantham, V., Theory of fractional functional differential equations, J. Nonlinear Sci., 69, 3337-3343, (2008) · Zbl 1162.34344
[2] Liao, C.; Ye, H., Existence of positive solutions of nonlinear fractional delay differential equations, Positivity, 13, 601-609, (2009) · Zbl 1177.34081
[3] Ye, H.; Ding, Y.; Gao, J., The existence of a positive solution of \(D^\alpha [x(t) - x(0)] = x(t) f(t, x t)\), Positivity, 11, 341-350, (2007) · Zbl 1121.34064
[4] Chen, Y.; Moore, K. L., Analytical stability bound for a class of delayed fractional-order dynamic systems, Nonlinear Dynam., 29, 191-200, (2002) · Zbl 1020.34064
[5] Lazarević, Mihailo P.; Spasić, Aleksandar M., Finite-time stability analysis of fractional order time-delay systems: gronwall’s approach, Math. Comput. Modelling, 49, 475-481, (2009) · Zbl 1165.34408
[6] Krol, K., Asymptotic properties of fractional delay differential equations, Appl. Math. Comput., 218, 5, 1515-1532, (2011) · Zbl 1239.34095
[7] Deng, W.; Li, C.; Lu, J., Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dynam., 48, 409-416, (2007) · Zbl 1185.34115
[8] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003
[9] Bellman, R.; Cooke, K. L., Differential-difference equations, (1963), New York, London Academic Press · Zbl 0105.06402
[10] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, (2006) · Zbl 1092.45003
[11] Bellen, A.; Zennaro, M., Numerical methods for delay differential equations, (2003), Oxford Science Publications, Clarendon Press Oxford · Zbl 0749.65042
[12] Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248, (2002) · Zbl 1014.34003
[13] Diethelm, K., (The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Series Lecture Notes in Mathematics, vol. 2004, (2010), Springer)
[14] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal., 5, 1-6, (1997) · Zbl 0890.65071
[15] Ford, N. J.; Connolly, J. A., Comparison of numerical methods for fractional differential equations, Commun. Pure Appl. Anal., 5, 2, 289-307, (2006) · Zbl 1133.65115
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.