×

zbMATH — the first resource for mathematics

On the fractional Adams method. (English) Zbl 1189.65142
Summary: The generalized Adams-Bashforth-Moulton method, often simply called “the fractional Adams method”, is a useful numerical algorithm for solving a fractional ordinary differential equation: \(D^{\alpha}_*y(t)=f(t,y(t))\), \(y^{(k)}(0)=y_0^{(k)}\), \(k=0,1,\dots,n-1\), where \(\alpha >0\), \(n=\lceil \alpha\rceil\) is the first integer not less than \(\alpha \), and \(D^{\alpha}_*y(t)\) is the \(\alpha\)th-order fractional derivative of \(y(t)\) in the Caputo sense. Although error analyses for this fractional Adams method have been given for (a) \(0<\alpha\), \(D^{\alpha}_*y(t)\in C^2[0,T]\), (b) \(\alpha >1\), \(C^{1+\lceil\alpha\rceil}[0,T]\), (c) \(0<\alpha <1\), \(y\in C^2[0,T]\), (d) \(\alpha >1, f\in C^{3}(G)\), there are still some unsolved problems- (i) the error estimates for \(\alpha\in (0,1)\), \(f\in C^{3}(G)\), (ii) the error estimates for \(\alpha\in (0,1)\), \(f\in C^{2}(G)\), (iii) the solution \(y(t)\) having some special forms. In this paper, we mainly study the error analyses of the fractional Adams method for the fractional ordinary differential equations for the three cases (i)-(iii). Numerical simulations are also included which are in line with the theoretical analysis.

MSC:
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations and fractional differential inclusions
44A15 Special integral transforms (Legendre, Hilbert, etc.)
45J05 Integro-ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York, renewed 2006 · Zbl 0428.26004
[2] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley-Interscience (John Wiley & Sons) New York · Zbl 0789.26002
[3] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[4] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier New York · Zbl 1092.45003
[5] 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
[6] Zhou, T.S.; Li, C.P., Synchronization in fractional-order differential systems, Physica D, 212, 111-125, (2005) · Zbl 1094.34034
[7] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003
[8] 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
[9] Lubich, C., Runge – kutta theory for Volterra and Abel integral equations of the second kind, Math. comput., 41, 87-102, (1983) · Zbl 0538.65091
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.