Predictor-corrector strategies for single- and multi-term fractional
differential equations
In E. A. Lipitakis (ed.): Proceedings of the 5th Hellenic-European
Conference on Computer Mathematics and its Applications.
LEA Press, Athens (2002), 117-122.
[Zbl. Math. 1028.65081]
2000 Mathematics Subject Classification: Primary 65L05, 65L06;
Secondary 26A33, 65R20.
Key words: Fractional differential equation, multi-term equation,
predictor-corrector method
Abstract
We investigate strategies for the numerical solution of the initial
value problem
y(an)(x) =
f(x, y(x), y(a1)(x), ...,
y(an-1)(x)),
y(k)(0) = y0(k)
(k=0,1,..., b -1),
where b is the smallest integer greater than or equal to
an and
0 < a1 < a2 < ... < an.
Here y(aj) denotes the derivative
of order aj > 0
(aj not necessarily an integer) in the sense of Caputo.
We begin our investigations with the simplest case, n=1. Here we
propose a predictor-corrector (more precisely, PECE) method that can
be interpreted in the spirit of the classical Adams-Bashforth-Moulton
schemes for first-order equations. Specifically we analyse the
discretization error of this approach under various assumptions on the
given data.
On the basis of these results we then look at the general problem for
arbitrary n. Here we first generalize the classical approach
for higher order differential equations and investigate the properties
of the resulting scheme.
To overcome the disadvantages of this first approach, we alternatively
suggest to construct an approximate solution in
two steps. First we replace the given equation by a different
n-term fractional differential equation without changing the initial
conditions. This second differential equation is constructed in such a
way that its solution z does not differ significantly from the
solution y of the original problem, and that
its structure allows a conversion into an equivalent
system of one-term fractional differential equations of order a
whose dimension is comparatively small.
Then we solve the resulting system numerically with our predictor-corrector
algorithm. In view of the properties of the system that are guaranteed
by our construction, we find that the computational effort remains
reasonably small without sacrificing too much precision.
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