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Fractional Integral Equations Tell Us How to Impose Initial Values in Fractional Differential Equations

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  • Daniel Cao Labora

    (Department of Applied Mathematics I, School of Forest Engineering, Universidade de Vigo, Campus Universitario da Xunqueira, S/N, 36005 Pontevedra, Spain)

Abstract

One major question in Fractional Calculus is to better understand the role of the initial values in fractional differential equations. In this sense, there is no consensus about what is the reasonable fractional abstraction of the idea of “initial value problem”. This work provides an answer to this question. The techniques that are used involve known results concerning Volterra integral equations, and the spaces of summable fractional differentiability introduced by Samko et al. In a few words, we study the natural consequences in fractional differential equations of the already existing results involving existence and uniqueness for their integral analogues, in terms of the Riemann–Liouville fractional integral. In particular, we show that a fractional differential equation of a certain order with Riemann–Liouville derivatives demands, in principle, less initial values than the ceiling of the order to have a uniquely determined solution, in contrast to a widely extended opinion. We compute explicitly the amount of necessary initial values and the orders of differentiability where these conditions need to be imposed.

Suggested Citation

  • Daniel Cao Labora, 2020. "Fractional Integral Equations Tell Us How to Impose Initial Values in Fractional Differential Equations," Mathematics, MDPI, vol. 8(7), pages 1-17, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:7:p:1093-:d:380238
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    References listed on IDEAS

    as
    1. Rudolf Hilfer & Yuri Luchko, 2019. "Desiderata for Fractional Derivatives and Integrals," Mathematics, MDPI, vol. 7(2), pages 1-5, February.
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