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Inverse Problem for a Two-Dimensional Anomalous Diffusion Equation with a Fractional Derivative of the Riemann–Liouville Type

Author

Listed:
  • Rafał Brociek

    (Department of Mathematics Applications and Methods for Artificial Intelligence, Faculty of Applied Mathematics, Silesian University of Technology, 44-100 Gliwice, Poland)

  • Agata Wajda

    (Department of Technologies and Installations for Waste Management, Faculty of Energy and Environmental Engineering, Silesian University of Technology, 44-100 Gliwice, Poland)

  • Damian Słota

    (Department of Mathematics Applications and Methods for Artificial Intelligence, Faculty of Applied Mathematics, Silesian University of Technology, 44-100 Gliwice, Poland)

Abstract

The article presents a method for solving the inverse problem of a two-dimensional anomalous diffusion equation with a Riemann–Liouville fractional-order derivative. In the first part of the present study, the authors present a numerical solution of the direct problem. For this purpose, a differential scheme was developed based on the alternating direction implicit method. The presented method was accompanied by examples illustrating its accuracy. The second part of the study concerned the inverse problem of recreating the model parameters, including the orders of the fractional derivative, in the anomalous diffusion equation. Equations of this type can be used to describe, inter alia, the heat conductivity in porous materials. The ant colony optimization algorithm was used to solve this problem. The authors investigated the impact of the distribution of measurement points, the use of different mesh sizes, and the input data errors on the obtained results.

Suggested Citation

  • Rafał Brociek & Agata Wajda & Damian Słota, 2021. "Inverse Problem for a Two-Dimensional Anomalous Diffusion Equation with a Fractional Derivative of the Riemann–Liouville Type," Energies, MDPI, vol. 14(11), pages 1-17, May.
    • Handle: RePEc:gam:jeners:v:14:y:2021:i:11:p:3082-:d:562304
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    References listed on IDEAS

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    1. Socha, Krzysztof & Dorigo, Marco, 2008. "Ant colony optimization for continuous domains," European Journal of Operational Research, Elsevier, vol. 185(3), pages 1155-1173, March.
    2. Songshu Liu & Lixin Feng, 2020. "An Inverse Problem for a Two-Dimensional Time-Fractional Sideways Heat Equation," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-13, March.
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