Nonlinear Dynamics

, Volume 29, Issue 1–4, pp 3–22 | Cite as

A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

  • Kai Diethelm
  • Neville J. Ford
  • Alan D. Freed

Abstract

We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too.

fractional differential equation Caputo derivative numerical solution predictor-corrector method Adams method 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Butzer, P. L. and Westphal, U., 'An introduction to fractional calculus,' in Applications of Fractional Calculus in Physics, R. Hilfer (ed.), World Scientific, Singapore, 2000, pp. 1–85.Google Scholar
  2. 2.
    Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, CA, 1999.Google Scholar
  3. 3.
    Samko, S. G., Kilbas, A. A., and Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.Google Scholar
  4. 4.
    Caputo, M., 'Linear models of dissipation whose Q is almost frequency independent, II', The Geophysical Journal of the Royal Astronomical Society 13, 1967, 529–539.Google Scholar
  5. 5.
    Diethelm, K. and Ford, N. J., 'Analysis of fractional differential equations', Journal of Mathematical Analysis and Applications 265, 2002, 229–248.Google Scholar
  6. 6.
    Lorenzo, C. F. and Hartley, T. T., 'Initialized fractional calculus', International Journal of Applied Mathematics 3, 2000, 249–265.Google Scholar
  7. 7.
    Gorenflo, R., 'Fractional calculus: Some numerical methods', in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi (eds.), Springer-Verlag, Wien, 1997, pp. 277–290.Google Scholar
  8. 8.
    Gorenflo, R. and Mainardi, F., 'Fractional calculus: Integral and differential equations of fractional order', in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi (eds.), Springer-Verlag, Wien, 1997, pp. 223–276.Google Scholar
  9. 9.
    Diethelm, K. and Ford, N. J., 'Numerical solution of the Bagley-Torvik equation', BIT, to appear.Google Scholar
  10. 10.
    Diethelm, K. and Freed, A. D., '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, F. Keil, W. Mackens, H. Voß, and J. Werther (eds.), Springer-Verlag, Heidelberg, 1999, pp. 217–224.Google Scholar
  11. 11.
    Benson, D. A., 'The fractional advection-dispersion equation: Development and application', Ph.D. Thesis, University of Nevada Reno, 1998.Google Scholar
  12. 12.
    Blank, L., 'Numerical treatment of differential equations of fractional order', Numerical Analysis Report 287, Manchester Centre for Computational Mathematics, Manchester, 1996.Google Scholar
  13. 13.
    Chern, J.-T., 'Finite element modeling of viscoelastic materials on the theory of fractional calculus', Ph.D. Thesis, Pennsylvania State University, 1993.Google Scholar
  14. 14.
    Diethelm, K., 'An algorithm for the numerical solution of differential equations of fractional order', Electronic Transactions on Numerical Analysis 5, 1997, 1–6.Google Scholar
  15. 15.
    Diethelm, K. and Luchko, Y., 'Numerical solution of linear multi-term initial value problems of fractional order', Journal of Computational Analysis and Applications, to appear.Google Scholar
  16. 16.
    Diethelm, K. and Walz, G., 'Numerical solution of fractional order differential equations by extrapolation', Numerical Algorithms 16, 1997, 231–253.Google Scholar
  17. 17.
    Enelund, M., Fenander, Å., and Olsson, P., 'Fractional integral formulation of constitutive equations of viscoelasticity', AIAA Journal 35, 1997, 1356–1362.Google Scholar
  18. 18.
    Enelund, M. and Josefson, B. L., 'Time-domain finite element analysis of viscoelastic structures with fractional derivative constitutive relations', AIAA Journal 35, 1997, 1630–1637.Google Scholar
  19. 19.
    Enelund, M. and Lesieutre, G. A., 'Time domain modeling of damping using anelastic displacement fields and fractional calculus', International Journal of Solids and Structures 36, 1999, 4447–4472.Google Scholar
  20. 20.
    Enelund, M. and Olsson, P., 'Damping described by fading memory-Analysis and application to fractional derivative models', International Journal of Solids and Structures 36, 1998, 939–970.Google Scholar
  21. 21.
    Lubich, C., 'Runge-Kutta theory for Volterra and Abel integral equations of the second kind', Mathematics of Computation 41, 1983, 87–102.Google Scholar
  22. 22.
    Lubich, C., 'Fractional linear multistep methods for Abel-Volterra integral equations of the second kind', Mathematics of Computation 45, 1985, 463–469.Google Scholar
  23. 23.
    Lubich, C., 'Discretized fractional calculus', SIAM Journal on Mathematical Analysis 17, 1986, 704–719.Google Scholar
  24. 24.
    Ruge, P. and Wagner, N., 'Time-domain solutions for vibration systems with fading memory', in Proceedings of the European Conference on Computational Mechanics 1999, W. Wunderlich (ed.), CD-ROM, Lehrstuhl für Statik, Technische Universität München, 1999 (http://www.isd.uni-stuttgart.de/~nwagner/eccm99.ps).Google Scholar
  25. 25.
    Shokooh, A. and Suarez, L. E., 'A comparison of numerical methods applied to a fractional derivative model of damping materials', Journal of Vibration and Control 5, 1999, 331–354.Google Scholar
  26. 26.
    Yuan, L. and Agrawal, O. P., 'A numerical scheme for dynamic systems containing fractional derivatives', in Proceedings of the 1998 ASME Design Engineering Technical Conferences, Atlanta, GA, September 13–16, ASME International, New York, 1998, CD-ROM publication (http://heera.engr.siu.edu/mech/faculty/agrawal/mech5857.pdf).Google Scholar
  27. 27.
    Hairer, E., Nørsett, S. P., and Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd revised edition, Springer-Verlag, Berlin, 1993.Google Scholar
  28. 28.
    Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, 1991.Google Scholar
  29. 29.
    Diethelm, K. and Freed, A. D., 'The FracPECE subroutine for the numerical solution of differential equations of fractional order', in Forschung und wissenschaftliches Rechnen 1998, S. Heinzel and T. Plesser (eds.), Gesellschaft für wissenschaftliche Datenverarbeitung, Göttingen, 1999, pp. 57–71.Google Scholar
  30. 30.
    Diethelm, K., Ford, N. J., and Freed, A. D., 'Detailed error analysis for a fractional Adams method', Berichte der Mathematischen Institut der TU Braunschweig O2/02, Braunschweig, submitted for publication (http://www.tu-bs.de/~diethelm/publications/adams.ps).Google Scholar
  31. 31.
    Ford, N. J. and Simpson, A. C., 'The numerical solution of fractional differential equations: Speed versus accuracy', Numerical Algorithms 26, 2001, 333–346.Google Scholar
  32. 32.
    de Hoog, F. and Weiss, R., 'Asymptotic expansions for product integration', Mathematics of Computation 27, 1973, 295–306.Google Scholar
  33. 33.
    Walz, G., Asymptotics and Extrapolation, Akademie-Verlag, Berlin, 1996.Google Scholar
  34. 34.
    Torvik, P. J. and Bagley, R. L., 'On the appearance of the fractional derivative in the behavior of real materials', Journal of Applied Mechanics 51, 1984, 294–298.Google Scholar
  35. 35.
    Koeller, R. C., 'Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics', Acta Mechanica 58, 1986, 251–264.Google Scholar
  36. 36.
    Diethelm, K. and Ford, N. J., 'Numerical solution of linear and non-linear fractional differential equations involving fractional derivatives of several orders', Numerical Analysis Report 379, Manchester Centre for Computational Mathematics, Manchester, 2001, submitted for publication (http://www.maths.man. ac.uk/~nareports/narep379.ps.gz).Google Scholar
  37. 37.
    Bagley, R. L. and Calico, R. A., 'Fractional order state equations for the control of viscoelastically damped structures', Journal of Guidance, Control, and Dynamics 14, 1991, 304–311.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Kai Diethelm
    • 1
  • Neville J. Ford
    • 2
  • Alan D. Freed
    • 3
  1. 1.Institut für Angewandte MathematikTechnische Universität BraunschweigBraunschweigGermany
  2. 2.Department of MathematicsChester CollegeChesterUnited Kingdom
  3. 3.Polymers BranchNASA's John H. Glenn Research Center at Lewis FieldClevelandU.S.A

Personalised recommendations