Section
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Mathematics
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Title
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Approximation of ordinary fractional differential equations by differential equations with a small parameter
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Author(-s)
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Lukashchuk S.Yu.a
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Affiliations
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Ufa State Aviation Technical Universitya
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Abstract
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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.
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Keywords
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ordinary fractional differential equation, small parameter, approximation, approximate solution
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UDC
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517.928
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MSC
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34A08, 34E10
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DOI
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10.20537/vm170403
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Received
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21 August 2017
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Language
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Russian
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Citation
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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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References
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