zbMATH — the first resource for mathematics

Explicit methods for fractional differential equations and their stability properties. (English) Zbl 1169.65121
The authors investigate a class of multistep methods for fractional differential equations and study the stability. A formula for the region of stability of the methods under investigation is obtained. The stability of some existing explicit methods is also studied. The authors derive new methods of the first and second order with interval of stability and provide numerical examples to illustrate the methods discussed.

65R20 Numerical methods for integral equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
26A33 Fractional derivatives and integrals
45J05 Integro-ordinary differential equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
Full Text: DOI
[1] Cuesta, E.; Palencia, C., A fractional trapezoidal rule for integro-differential equations of fractional order in Banach spaces, Appl. numer. math., 45, 2-3, 139-159, (2003) · Zbl 1023.65151
[2] 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
[3] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 2, 229-248, (2002) · Zbl 1014.34003
[4] Diethelm, K.; Ford, J.M.; Ford, N.J.; Weilbeer, M., Pitfalls in fast numerical solvers for fractional differential equations, J. comput. appl. math., 186, 2, 482-503, (2006) · Zbl 1078.65550
[5] Diethelm, K.; Weilbeer, M., A numerical approach for joulin’s model of a point source initiated flame, Fract. calc. appl. anal., 7, 2, 191-212, (2004) · Zbl 1094.80006
[6] Dubois, F.; Mengué, S., Mixed collocation for fractional differential equations, Numer. algorithms, 34, 2-4, 303-311, (2003) · Zbl 1038.65059
[7] Elaydi, S., An introduction to difference equations, () · Zbl 0855.39003
[8] Ford, N.J.; Connolly, J.A., Comparison of numerical methods for fractional differential equations, Commun. pure appl. anal., 5, 2, 289-306, (2006) · Zbl 1133.65115
[9] Galeone, L.; Garrappa, R., On multistep methods for differential equations of fractional order, Mediterr. J. math., 3, 3-4, 565-580, (2006) · Zbl 1167.65399
[10] L. Galeone, R. Garrappa, Second order multistep methods for fractional differential equations, Technical Report 20/2007, Department of Mathematics, University of Bari, 2007 · Zbl 1167.65399
[11] Gorenflo, R.; Abdel-Rehim, E.A., Discrete models of time-fractional diffusion in a potential well, Fract. calc. appl. anal., 8, 2, 173-200, (2005) · Zbl 1129.26002
[12] Joulin, G., Point – source initiation of Lean spherical flames of light reactants: an asymptotic theory, Combustion sci. technol., 43, 99-113, (1985)
[13] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Science B.V. Amsterdam · Zbl 1092.45003
[14] Lubich, C., Discretized fractional calculus, SIAM J. math. anal., 17, 3, 704-719, (1986) · Zbl 0624.65015
[15] Lubich, C., A stability analysis of convolution quadratures for abel – volterra integral equations, IMA J. numer. anal., 6, 1, 87-101, (1986) · Zbl 0587.65090
[16] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York, London · Zbl 0428.26004
[17] Podlubny, I., Fractional differential equations, () · Zbl 0918.34010
[18] Yuste, S.B.; Acedo, L., An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. numer. anal., 42, 5, 1862-1874, (2005) · Zbl 1119.65379
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.