Skip to main content
SpringerLink
Log in

Detailed Error Analysis for a Fractional Adams Method

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C.T.H. Baker, The Numerical Treatment of Integral Equations (Clarendon Press, Oxford, 1977).

    Google Scholar 

  2. D.A. Benson, The fractional advection-dispersion equation: Development and application, Ph.D. thesis, University of Nevada Reno (1998).

  3. H. Brass, Quadraturverfahren (Vandenhoeck & Ruprecht, Göttingen, 1977).

    Google Scholar 

  4. 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.

    Google Scholar 

  5. M. Caputo, Linear models of dissipation whose Q is almost frequency independent — II, Geophys. J. Royal Astronom. Soc. 13 (1967) 529–539.

    Google Scholar 

  6. M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids, Rivista del Nuovo Cimento 1 (1971) 161–198.

    Google Scholar 

  7. M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure Appl. Geophys. 91 (1971) 134–147.

    Google Scholar 

  8. J.-T. Chern, Finite element modeling of viscoelastic materials on the theory of fractional calculus, Ph.D. thesis, Pennsylvania State University (1993).

  9. F. de Hoog and R. Weiss, Asymptotic expansions for product integration, Math. Comp. 27 (1973) 295–306.

    Google Scholar 

  10. K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Elec. Trans. Numer. Anal. 5 (1997) 1–6.

    Google Scholar 

  11. K. Diethelm and N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248.

    Google Scholar 

  12. K. Diethelm and N.J. Ford, Numerical solution of the Bagley—Torvik equation, BIT 42 (2002) 490–507.

    Google Scholar 

  13. 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.

    Google Scholar 

  14. 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.

    Google Scholar 

  15. L. Gaul, P. Klein and S. Kempfle, Damping description involving fractional operators, Mech. Systems Signal Processing 5 (1991) 81–88.

    Google Scholar 

  16. W.G. Glöckle and T.F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J. 68 (1995) 46–53.

    Google Scholar 

  17. R. Gorenflo, G. De Fabritiis and F. Mainardi, Discrete random walk models for symmetric Lévy-Feller diffusion processes, Phys. A 269 (1999) 79–89.

    Google Scholar 

  18. 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.

    Google Scholar 

  19. 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.

    Google Scholar 

  20. 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.

    Google Scholar 

  21. E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd revised ed. (Springer, Berlin, 1993).

    Google Scholar 

  22. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Springer, Berlin, 1991).

    Google Scholar 

  23. R. Hilfer, ed., Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000).

  24. C. Lubich, Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Math. Comp. 41 (1983) 87–102.

    Google Scholar 

  25. C. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comp. 45 (1985) 463–469.

    Google Scholar 

  26. 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.

    Google Scholar 

  27. 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.

    Google Scholar 

  28. 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.

    Google Scholar 

  29. 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.

    Google Scholar 

  30. 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.

    Google Scholar 

  31. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (2000) 1–77.

    Google Scholar 

  32. 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.

    Google Scholar 

  33. 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.

    Google Scholar 

  34. T.F. Nonnenmacher and R. Metzler, On the Riemann-Liouville fractional calculus and some recent applications, Fractals 3 (1995) 557–566.

    Google Scholar 

  35. K.B. Oldham and J. Spanier, The Fractional Calculus (Academic Press, New York, 1974).

    Google Scholar 

  36. W.E. Olmstead and R.A. Handelsman, Diffusion in a semi-infinite region with nonlinear surface dissipation, SIAM Rev. 18 (1976) 275–291.

    Google Scholar 

  37. I. Podlubny, Fractional-order systems and fractional-order controllers, Technical Report UEF-03-94, Slovak Acad. Sci. (1994).

  38. I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999).

    Google Scholar 

  39. 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.

    Google Scholar 

  40. S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, Yverdon, 1993).

    Google Scholar 

  41. E. Scalas, R. Gorenflo and F. Mainardi, Fractional calculus and continuous-time finance, Phys. A 284 (2000) 376–384.

    Google Scholar 

  42. 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).

  43. 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.

    Google Scholar 

  44. G. Walz, Asymptotics and Extrapolation (Akademie-Verlag, Berlin, 1996).

    Google Scholar 

  45. 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.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Computational Mathematics, Technische Universität Braunschweig, Pockelsstraße 14, 38106, Braunschweig, Germany

    Kai Diethelm

  2. Department of Mathematics, Chester College, Parkgate Road, Chester, CH1 4BJ, UK

    Neville J. Ford

  3. Polymers Branch, MS 49-3, NASA's John H. Glenn Research Center at Lewis Field, 21000 Brookpark Road, Cleveland, OH, 44135, USA

    Alan D. Freed

Authors
  1. Kai Diethelm
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Neville J. Ford
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alan D. Freed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kai Diethelm.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:NUMA.0000027736.85078.be

Search

Navigation