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I am trying to read this paper and have a certain doubt. On page 8 the author comments that

First of all, there is a fundamental problem associated with all fractional differential operators (not only the Caputo version that we look at here): In contrast to differential operators of integer order, fractional derivatives are not local operators. As a matter of fact, this property is highly desirable from the physical point of view because it allows us to model phenomena with memory effects. But when it comes to numerical work we find that this non-locality leads to a significantly higher computational effort: The arithmetic complexity of our algorithm with step size h is $O(h^{-2})$, whereas a comparable algorithm for an integer-order initial value problem (thus involving a local operator) would only give rise to a $O(h^{-1})$ complexity. A number of ways have been suggested to overcome this difficulty. The first idea seems to have been the fixed memory principle of Podlubny[31].

What I understand from this is that since fractional differential operators have two terminal points within which they are evaluated so the derivative is not evaluated at a point. In that sense it is not local. The fixed memory principle in this book is not clear to me. I only found a reference to a short memory principle and even that wasn't clear. Can someone please explain this principle?

Thanks.

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up vote 2 down vote accepted

Things are less involved if you use the Grunwald-Letnikov derivative. You can fix a criterion to "cut" the summation. Besides this derivative does not have the problems of the others RL or C. I used the short memory principle in a paper about the two-sided derivative. I can send it to you. Send me a mail.

mdo@fct.unl.pt

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