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Short memory principle and a predictor-corrector approach for fractional differential equations. (English) Zbl 1121.65128
In recent years, fractional differential equations have proven to be a very useful tool for the modeling of various phenomena in physics, engineering, and other sciences. This is, at least in part, due to the fact that they can be used in a very natural way to model processes with memory. However, the fact that memory has to be handled has the disadvantage that the computational complexity of numerical algorithms for fractional differential equations is much higher than for their classical (non-fractional) counterparts. An approach that can be used to modify various numerical methods in such a way that the complexity is reduced substantially without the loss of accuracy has been proposed by N. J. Ford and A. C. Simpson [Numer. Algorithms 26, No. 4, 333–346 (2001; Zbl 0976.65062)].
In the paper under review, the concept of Ford and Simpson [loc. cit.] is combined with additional ideas to improve the performance, in particular for equations of order $$\alpha \in (1,2)$$. The implementation of the combination of this principle with the predictor-corrector method of the reviewer, N. J. Ford and A. D. Freed [Numer. Algorithms 36, No. 1, 31–52 (2004; Zbl 1055.65098)] is described in detail.

##### MSC:
 65R20 Numerical methods for integral equations 26A33 Fractional derivatives and integrals 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 45J05 Integro-ordinary differential equations
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