Numerical Algorithms

, Volume 16, Issue 3–4, pp 231–253 | Cite as

Numerical solution of fractional order differential equations by extrapolation

  • Kai Diethelm
  • Guido Walz


We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published algorithm by Diethelm [6], possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no additional numerical costs. This is also illustrated by a number of numerical examples.

fractional order derivative fractional order differential equation quadrature extrapolation asymptotic expansion trapezoidal formula 26A33 41A55 65B05 65L05 65L06 65D30 


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  1. [1]
    H. Brass, Quadraturverfahren (Vandenhoeck andRuprecht, Göttingen, 1977).Google Scholar
  2. [2]
    C. Brezinski, A generalextrapolation algorithm, Numer. Math. 35 (1980) 175–187.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    C. Brezinski and M. Redivo-Zaglia, Extrapolation Methods, Theory and Practice (North-Holland, Amsterdam, 1992).Google Scholar
  4. [4]
    H. Brunner and P.J. van derHouwen, The Numerical Solution of Volterra Equations (North-Holland, Amsterdam, 1986).Google Scholar
  5. [5]
    K. Diethelm, Generalized compound quadrature formulae forfinite-part integrals, IMA Numer. Anal. 17 (1997) 479–493.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal. 5 (1997) 1–6.zbMATHMathSciNetGoogle Scholar
  7. [7]
    D. Elliott, An asymptotic analysis of two algorithms for certain Hadamard finite-part integrals, IMA J. Numer. Anal. 13 (1993) 445–462.zbMATHMathSciNetGoogle Scholar
  8. [8]
    W.B. Gragg, Repeated extrapolation to the limit in the numerical solution of ordinary differential equations, Thesis, University of California, Los Angeles (1964).Google Scholar
  9. [9]
    W.B. Gragg, On extrapolation algorithms for ordinary initial valueproblems, SIAM J. Numer. Anal. 2 (1965) 384–403.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    E. Hairer and Ch. Lubich, Asymptotic expansions of the global error of fixed-stepsize methods, Numer. Math. 45 (1984) 345–360.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (Springer, Berlin, 1987).Google Scholar
  12. [12]
    F. de Hoog and R. Weiss,Asymptotic expansions for product integration, Math. Comp. 27 (1973) 295–306.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    L. Lapidus and J.H. Seinfeld, Numerical Solution ofOrdinary Differential Equations (Academic Press, New York, 1971).Google Scholar
  14. [14]
    P. Linz, Analytical and Numerical Methods for Volterra Equations(SIAM, Philadelphia, PA, 1985).Google Scholar
  15. [15]
    Ch. Lubich, Discretizedfractional calculus, SIAM J. Math. Anal. 17 (1986) 704–719.zbMATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    J.N. Lyness, Finite-part integration and the Euler-MacLaurin expansion,in: Approximation and Computation, ed. R.V.M. Zahar, Internat. Ser. Numer. Math. 119 (Birkhäuser, Basel, 1994) pp. 397–407.Google Scholar
  17. [17]
    G. Meinardus and G. Merz, Praktische Mathematik II (Bibl. Institut, Mannheim, 1982).Google Scholar
  18. [18]
    K.B. Oldham and J. Spanier, The FractionalCalculus (Academic Press, New York, 1974).Google Scholar
  19. [19]
    W. Romberg,Vereinfachte numerische Integration, Det Kong. Norske Vid. Selskab Forhdl. 28 (1955) 30–36.zbMATHMathSciNetGoogle Scholar
  20. [20]
    A. Sard, Integral representations ofremainders, Duke Math. J. 15 (1948) 333–345.zbMATHMathSciNetCrossRefGoogle Scholar
  21. [21]
    G. Walz,Asymptotics and Extrapolation (Akademie-Verlag, Berlin, 1996).Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Kai Diethelm
    • 1
  • Guido Walz
    • 2
  1. 1.Mathematical InstituteUniversity of HildesheimHildesheimGermany
  2. 2.Department of Mathematics and Computer ScienceUniversity of MannheimMannheimGermany

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