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Licensed Unlicensed Requires Authentication Published by De Gruyter December 19, 2019

Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation

  • Matthias Hinze EMAIL logo , André Schmidt and Remco I. Leine


In this paper, we propose a novel approach for the numerical solution of fractional-order ordinary differential equations. The method is based on the infinite state representation of the Caputo fractional differential operator, in which the entire history of the state of the system is considered for correct initialization. The infinite state representation contains an improper integral with respect to frequency, expressing the history dependence of the fractional derivative. The integral generally has a weakly singular kernel, which may lead to problems in numerical computations. A reformulation of the integral generates a kernel that decays to zero at both ends of the integration interval leading to better convergence properties of the related numerical scheme. We compare our method to other schemes by considering several benchmark problems.

MSC 2010: 34A08; 34K28; 65L03


This work is supported by the Federal Ministry of Education and Research of Germany under Grant No. 01IS17096.


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Received: 2019-05-06
Published Online: 2019-12-19
Published in Print: 2019-10-25

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