Trends, Directions for Further Research, and Some Open Problems of Fractional Calculus

Diethelm, Kai; Kiryakova, Virginia; Luchko, Yuri; Machado, J. A. Tenreiro; Tarasov, Vasily E.

doi:10.48550/arxiv.2108.04241

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2021

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“…For the numerical solution of such intial value problems, many methods have been proposed; cf., e.g., [2,11,12,17,21]. A common feature of a large part of these methods is that, in order to correctly handle the non-locality of the fractional differential operators, they require a relatively large amount of time and/or computer memory [14].…”

Section: Introductionmentioning

confidence: 99%

Fast Solution Methods for Fractional Differential Equations in the Modeling of Viscoelastic Materials

Diethelm¹

2021

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Fractional order models have proven to be a very useful tool for the modeling of the mechanical behaviour of viscoelastic materials. Traditional numerical solution methods exhibit various undesired properties due to the non-locality of the fractional differential operators, in particular regarding the high computational complexity and the high memory requirements. The infinite state representation is an approach on which one can base numerical methods that overcome these obstacles. Such algorithms contain a number of parameters that influence the final result in nontrivial ways. Based on numerical experiments, we initiate a study leading to good choices of these parameters.

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