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

Department of Applied Mathematics I, School of Forest Engineering, Universidade de Vigo, Campus Universitario da Xunqueira, S/N, 36005 Pontevedra, Spain
Mathematics 2020, 8(7), 1093; https://doi.org/10.3390/math8071093
Received: 31 May 2020 / Revised: 30 June 2020 / Accepted: 2 July 2020 / Published: 4 July 2020
(This article belongs to the Special Issue Fractional Integrals and Derivatives: “True” versus “False”)
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. View Full-Text
Keywords: fractional differential equations; initial values; existence; uniqueness fractional differential equations; initial values; existence; uniqueness
MDPI and ACS Style

Cao Labora, D. Fractional Integral Equations Tell Us How to Impose Initial Values in Fractional Differential Equations. Mathematics 2020, 8, 1093. https://doi.org/10.3390/math8071093

AMA Style

Cao Labora D. Fractional Integral Equations Tell Us How to Impose Initial Values in Fractional Differential Equations. Mathematics. 2020; 8(7):1093. https://doi.org/10.3390/math8071093

Chicago/Turabian Style

Cao Labora, Daniel. 2020. "Fractional Integral Equations Tell Us How to Impose Initial Values in Fractional Differential Equations" Mathematics 8, no. 7: 1093. https://doi.org/10.3390/math8071093

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