Numerical Algorithms

, Volume 36, Issue 1, pp 31–52 | Cite as

Detailed Error Analysis for a Fractional Adams Method

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


We investigate a method for the numerical solution of the nonlinear fractional differential equation D*αy(t)=f(t,y(t)), equipped with initial conditions y(k)(0)=y0(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.

fractional differential equation Caputo derivative Adams–Bashforth–Moulton method 


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Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Kai Diethelm
    • 1
  • Neville J. Ford
    • 2
  • Alan D. Freed
    • 3
  1. 1.Institut Computational MathematicsTechnische Universität BraunschweigBraunschweigGermany
  2. 2.Department of MathematicsChester CollegeChesterUK
  3. 3.Polymers Branch, MS 49-3NASA's John H. Glenn Research Center at Lewis FieldClevelandUSA

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