Numerical solution of fractional order differential equations by extrapolation

Numerical Algorithms

Volumn 16, Issue 3-4, 1997, Pages 231-253

  Diethelm, Kai ^a   Walz, Guido ^b  

^a UNIVERSITY OF HILDESHEIM   (Germany)

^b UNIVERSITY OF MANNHEIM   (Germany)

Author keywords

Asymptotic expansion; Extrapolation; Fractional order derivative; Fractional order differential equation; Quadrature; Trapezoidal formula

Indexed keywords

EID: 0043044718     PISSN: 10171398     EISSN: None     Source Type: Journal    
DOI: 10.1023/a:1019147432240     Document Type: Article

References (21)

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