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Russia Ufa
Section Mathematics
Title Approximation of ordinary fractional differential equations by differential equations with a small parameter
Author(-s) Lukashchuk S.Yu.a
Affiliations Ufa State Aviation Technical Universitya
Abstract An approach to approximation of ordinary fractional differential equations by integer-order differential equations is proposed. It is assumed that the order of fractional differentiation is close to integer. Perturbation expansions for the Riemann-Liouville and Caputo fractional derivatives are derived in terms of a suitable small parameter extracted from the order of fractional differentiation. The first-order term of these expansions is represented by series depending on integer-order derivatives of all integer orders. The expansions obtained permit one to approximate ordinary fractional differential equations, involving such types of fractional derivatives, by integer-order differential equations with a small parameter. It is proved that, for fractional differential equations belonging to a certain class, corresponding approximate equations contain only a finite number of integer-order derivatives. Approximate solutions to such equations can be obtained using well-known perturbation techniques. The proposed approach is illustrated by several examples.
Keywords ordinary fractional differential equation, small parameter, approximation, approximate solution
UDC 517.928
MSC 34A08, 34E10
DOI 10.20537/vm170403
Received 21 August 2017
Language Russian
Citation Lukashchuk S.Yu. Approximation of ordinary fractional differential equations by differential equations with a small parameter, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 4, pp. 515-531.
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