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On the Atangana–Baleanu Derivative and Its Relation to the Fading Memory Concept: The Diffusion Equation Formulation

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Fractional Derivatives with Mittag-Leffler Kernel

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 194))

Abstract

The constructions of physically adequate forms of the diffusion equation with implementation of the Atangana–Baleanu derivative with Mittag-Leffler exponential kernel have been discussed. The specific form of the corresponding Atangana–Baleanu integral relates it directly to the fading memory concept, following the Boltzmann linear superposition principle with the standard Riemann-Liouville integral as the time-fading term. This approach relates the new fractional operators with non-singular kernel to the classical Riemann-Liouville integral. Using the concept of the fading memory and the specific form of the Atangana–Baleanu integral three forms of the diffusion equation have been investigated. The adequate definition of the flux to gradient relationship has been the main focus of the study resulting in two physically adequate formulations of the diffusion equation. The direct (formalistic) fractionalization of the classical diffusion equation results in physically inadequate relationships.

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Authors and Affiliations

  1. Department of Chemical Engineering, University of Chemical Technology and Metallurgy, 8 Kliment Ohridsky, Blvd., 1756, Sofia, Bulgaria

    Jordan Hristov

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  1. Jordan Hristov
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Correspondence to Jordan Hristov .

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Editors and Affiliations

  1. CONACYT-Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    José Francisco Gómez

  2. CONACYT-Instituto de Ingeniería, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Lizeth Torres

  3. Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    Ricardo Fabricio Escobar

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Hristov, J. (2019). On the Atangana–Baleanu Derivative and Its Relation to the Fading Memory Concept: The Diffusion Equation Formulation. In: Gómez, J., Torres, L., Escobar, R. (eds) Fractional Derivatives with Mittag-Leffler Kernel. Studies in Systems, Decision and Control, vol 194. Springer, Cham. https://doi.org/10.1007/978-3-030-11662-0_11

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  • DOI: https://doi.org/10.1007/978-3-030-11662-0_11

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  • Online ISBN: 978-3-030-11662-0

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