Abstract
We investigate a method for the numerical solution of the nonlinear fractional differential equation D * α y(t)=f(t,y(t)), equipped with initial conditions y (k)(0)=y 0 (k), k=0,1,...,⌈α⌉−1. Here α may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardson's extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour.
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References
C.T.H. Baker, The Numerical Treatment of Integral Equations (Clarendon Press, Oxford, 1977).
D.A. Benson, The fractional advection-dispersion equation: Development and application, Ph.D. thesis, University of Nevada Reno (1998).
H. Brass, Quadraturverfahren (Vandenhoeck & Ruprecht, Göttingen, 1977).
H. Brunner, A survey of recent advances in the numerical treatment of Volterra integral and integrodifferential equations, J. Comput. Appl. Math. 8 (1982) 213–229.
M. Caputo, Linear models of dissipation whose Q is almost frequency independent — II, Geophys. J. Royal Astronom. Soc. 13 (1967) 529–539.
M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids, Rivista del Nuovo Cimento 1 (1971) 161–198.
M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys. 91 (1971) 134–147.
J.-T. Chern, Finite element modeling of viscoelastic materials on the theory of fractional calculus, Ph.D. thesis, Pennsylvania State University (1993).
F. de Hoog and R. Weiss, Asymptotic expansions for product integration, Math. Comp. 27 (1973) 295–306.
K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Elec. Trans. Numer. Anal. 5 (1997) 1–6.
K. Diethelm and N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248.
K. Diethelm and N.J. Ford, Numerical solution of the Bagley—Torvik equation, BIT 42 (2002) 490–507.
K. Diethelm and A.D. Freed, The FracPECE subroutine for the numerical solution of differential equations of fractional order, in: Forschung und wissenschaftliches Rechnen: Beiträge zum Heinz-Billing-Preis 1998, eds. S. Heinzel and T. Plesser (Gesellschaft für wissenschaftliche Datenverarbeitung, Göttingen, 1999) pp. 57–71.
K. Diethelm and A.D. Freed, On the solution of nonlinear fractional differential equations used in the modeling of viscoplasticity, in: Scientific Computing in Chemical Engineering II — Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties, eds. F. Keil, W. Mackens, H. Voß and J. Werther (Springer, Heidelberg, 1999) pp. 217–224.
L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators, Mech. Systems Signal Processing 5 (1991) 81–88.
W.G. Glöckle and T.F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J. 68 (1995) 46–53.
R. Gorenflo, G. De Fabritiis and F. Mainardi, Discrete random walk models for symmetric Lévy-Feller diffusion processes, Phys. A 269 (1999) 79–89.
R. Gorenflo, I. Loutchko and Y. Luchko, Computation of the Mittag-Leffler function Eα,β (z) and its derivatives, Fract. Calc. Appl. Anal. 5 (2002) 491–518. Erratum: Fract. Calc. Appl. Anal. 6 (2003) 111-112.
R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in: Fractals and Fractional Calculus in Continuum Mechanics, eds. A. Carpinteri and F. Mainardi (Springer, Wien, 1997) pp. 223–276.
R. Gorenflo and R. Rutman, On ultraslow and intermediate processes, in: Transform Methods and Special Functions, Sofia, 1994, eds. P. Rusev, I. Dimovski and V. Kiryakova (Science Culture Technology, Singapore, 1995) pp. 61–81.
E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd revised ed. (Springer, Berlin, 1993).
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Springer, Berlin, 1991).
R. Hilfer, ed., Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000).
C. Lubich, Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Math. Comp. 41 (1983) 87–102.
C. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comp. 45 (1985) 463–469.
F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics, in: Fractals and Fractional Calculus in Continuum Mechanics, eds. A. Carpinteri and F. Mainardi (Springer, Wien, 1997) pp. 291–348.
F. Mainardi, M. Raberto, R. Gorenflo and E. Scalas, Fractional calculus and continuous-time finance II: The waiting-time distribution, Phys. A 287 (2000) 468–481.
R.J. Marks, II and M.W. Hall, Differintegral interpolation from a bandlimited signal's samples, IEEE Trans. Acoust. Speech Signal Processing 29 (1981) 872–877.
D. Matignon and G. Montseny, eds., Fractional Differential Systems: Models, Methods, and Applications, ESAIM Proceedings, Vol. 5 (SMAI, Paris, 1998), also http://www.emath.fr/Maths/ Proc/Vol.5/index.htm.
R. Metzler, Generalized Chapman-Kolmogorov equation: A unifying approach to the description of anomalous transport in external fields, Phys. Rev. E 62 (2000) 6233–6245.
R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (2000) 1–77.
R. Metzler, W. Schick, H.-G. Kilian and T.F. Nonnenmacher, Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys. 103 (1995) 7180–7186.
R.K. Miller and A. Feldstein, Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM J. Math. Anal. 2 (1971) 242–258.
T.F. Nonnenmacher and R. Metzler, On the Riemann-Liouville fractional calculus and some recent applications, Fractals 3 (1995) 557–566.
K.B. Oldham and J. Spanier, The Fractional Calculus (Academic Press, New York, 1974).
W.E. Olmstead and R.A. Handelsman, Diffusion in a semi-infinite region with nonlinear surface dissipation, SIAM Rev. 18 (1976) 275–291.
I. Podlubny, Fractional-order systems and fractional-order controllers, Technical Report UEF-03-94, Slovak Acad. Sci. (1994).
I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999).
I. Podlubny, L. Dorcak and J. Misanek, Application of fractional-order derivatives to calculation of heat load intensity change in blast furnace walls, Trans. Tech. Univ. Košice 5 (1995) 137–144.
S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, Yverdon, 1993).
E. Scalas, R. Gorenflo and F. Mainardi, Fractional calculus and continuous-time finance, Phys. A 284 (2000) 376–384.
S. Shaw, M.K. Warby and J.R. Whiteman, A comparison of hereditary integral and internal variable approaches to numerical linear solid elasticity, Technical Report, Brunel University (1997).
P.J. Torvik and R.L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech. 51 (1984) 294–298.
G. Walz, Asymptotics and Extrapolation (Akademie-Verlag, Berlin, 1996).
S.C. Woon, Analytic continuation of operators. Applications: From number theory and group theory to quantum field and string theories, Rev. Math. Phys. 11 (1999) 463–501.
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Diethelm, K., Ford, N.J. & Freed, A.D. Detailed Error Analysis for a Fractional Adams Method. Numerical Algorithms 36, 31–52 (2004). https://doi.org/10.1023/B:NUMA.0000027736.85078.be
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DOI: https://doi.org/10.1023/B:NUMA.0000027736.85078.be