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Maximum and Minimum Principles for the Generalized Fractional Diffusion Problem with a Scale Function-Dependent Derivative

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Theory and Applications of Non-integer Order Systems

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 407))

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Abstract

In the paper, we prove the necessary condition for the extremum existence in terms of the generalized function-dependent fractional derivatives. By using these results we extend the maximum and minimum principles, known from the theory of differential equations and from diffusion problems with the Caputo derivative of constant or distributed order. We study the fractional diffusion problem, where time evolution is determined by the scale function-dependent Caputo derivative and show that the maximum or respectively minimum principle is valid, provided the source function is a non-positive or a non-negative one in the domain. As an application, we demonstrate how the sign of the classical solution is controlled by the initial and boundary conditions.

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Author information

Authors and Affiliations

  1. Institute of Mathematics, Czestochowa University of Technology, Dabrowskiego 69, 42-201, Czestochowa, Poland

    Małgorzata Klimek

  2. Faculty of Mechanical Engineering and Computer Science, Czestochowa University of Technology, Dabrowskiego 69, 42-201, Czestochowa, Poland

    Kalina Kamińska

Authors
  1. Małgorzata Klimek
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  2. Kalina Kamińska
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Corresponding author

Correspondence to Małgorzata Klimek .

Editor information

Editors and Affiliations

  1. Institute of Automatic Control, Silesian University of Technology Institute of Automatic Control, Gliwice, Poland

    Artur Babiarz

  2. Institute of Automatic Control, Silesian University of Technology Institute of Automatic Control, Gliwice, Poland

    Adam Czornik

  3. Institute of Automatic Control, Silesian University of Technology Institute of Automatic Control, Gliwice, Poland

    Jerzy Klamka

  4. Institute of Automatic Control, Silesian University of Technology Institute of Automatic Control, Gliwice, Poland

    Michał Niezabitowski

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Klimek, M., Kamińska, K. (2017). Maximum and Minimum Principles for the Generalized Fractional Diffusion Problem with a Scale Function-Dependent Derivative. In: Babiarz, A., Czornik, A., Klamka, J., Niezabitowski, M. (eds) Theory and Applications of Non-integer Order Systems. Lecture Notes in Electrical Engineering, vol 407. Springer, Cham. https://doi.org/10.1007/978-3-319-45474-0_19

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  • DOI: https://doi.org/10.1007/978-3-319-45474-0_19

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45473-3

  • Online ISBN: 978-3-319-45474-0

  • eBook Packages: EngineeringEngineering (R0)

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