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Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation. (English) Zbl 1437.65059

Summary: 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:

65L03 Numerical methods for functional-differential equations
34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
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