IDEAS home Printed from https://ideas.repec.org/a/gam/jeners/v14y2021i11p3082-d562304.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/1996-1073/14/11/3082/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/1996-1073/14/11/3082/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Bera, Sasadhar & Mukherjee, Indrajit, 2016. "A multistage and multiple response optimization approach for serial manufacturing system," European Journal of Operational Research, Elsevier, vol. 248(2), pages 444-452.
    2. Zhang, Zhe & Song, Xiaoling & Gong, Xue & Yin, Yong & Lev, Benjamin & Zhou, Xiaoyang, 2024. "Coordinated seru scheduling and distribution operation problems with DeJong’s learning effects," European Journal of Operational Research, Elsevier, vol. 313(2), pages 452-464.
    3. Anand Kumar & Manoj Thakur & Garima Mittal, 2018. "A new ants interaction scheme for continuous optimization problems," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 9(4), pages 784-801, August.
    4. Nikolaos Ploskas & Nikolaos V. Sahinidis, 2022. "Review and comparison of algorithms and software for mixed-integer derivative-free optimization," Journal of Global Optimization, Springer, vol. 82(3), pages 433-462, March.
    5. Ozgur Kisi & Armin Azad & Hamed Kashi & Amir Saeedian & Seyed Ali Asghar Hashemi & Salar Ghorbani, 2019. "Modeling Groundwater Quality Parameters Using Hybrid Neuro-Fuzzy Methods," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 33(2), pages 847-861, January.
    6. Bera, Sasadhar & Mukherjee, Indrajit, 2012. "An ellipsoidal distance-based search strategy of ants for nonlinear single and multiple response optimization problems," European Journal of Operational Research, Elsevier, vol. 223(2), pages 321-332.
    7. Amjad Hudaib & Mohammad Khanafseh & Ola Surakhi, 2018. "An Improved Version of K-medoid Algorithm using CRO," Modern Applied Science, Canadian Center of Science and Education, vol. 12(2), pages 116-116, February.
    8. Liao, Tianjun & Stützle, Thomas & Montes de Oca, Marco A. & Dorigo, Marco, 2014. "A unified ant colony optimization algorithm for continuous optimization," European Journal of Operational Research, Elsevier, vol. 234(3), pages 597-609.
    9. Eroğlu, Yunus & Seçkiner, Serap Ulusam, 2012. "Design of wind farm layout using ant colony algorithm," Renewable Energy, Elsevier, vol. 44(C), pages 53-62.
    10. Martin Schlüter & Matthias Gerdts, 2010. "The oracle penalty method," Journal of Global Optimization, Springer, vol. 47(2), pages 293-325, June.
    11. Ali Sardar Shahraki & Mohim Tash & Tommaso Caloiero & Ommolbanin Bazrafshan, 2024. "Optimal Allocation of Water Resources Using Agro-Economic Development and Colony Optimization Algorithm," Sustainability, MDPI, vol. 16(13), pages 1-18, July.
    12. Luo, Qifang & Yang, Xiao & Zhou, Yongquan, 2019. "Nature-inspired approach: An enhanced moth swarm algorithm for global optimization," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 159(C), pages 57-92.
    13. Tiago Maritan Ugulino Araújo & Lisieux Marie M. S. Andrade & Carlos Magno & Lucídio Anjos Formiga Cabral & Roberto Quirino Nascimento & Cláudio N. Meneses, 2016. "DC-GRASP: directing the search on continuous-GRASP," Journal of Heuristics, Springer, vol. 22(4), pages 365-382, August.
    14. Shao, Peng & Liang, Ying & Li, Guangquan & Li, Xing & Yang, Le, 2023. "Birefringence learning: A new global optimization technology model based on birefringence principle in application on artificial bee colony," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 470-486.
    15. Andres Quiros-Granados & JAvier Trejos-Zelaya, 2019. "Estimation of the yield curve for Costa Rica using combinatorial optimization metaheuristics applied to nonlinear regression," Papers 2001.00920, arXiv.org.
    16. Slawomir Blasiak, 2021. "Heat Transfer Analysis for Non-Contacting Mechanical Face Seals Using the Variable-Order Derivative Approach," Energies, MDPI, vol. 14(17), pages 1-13, September.
    17. Sulaiman, Mohd Herwan & Mustaffa, Zuriani, 2024. "Chiller energy prediction in commercial building: A metaheuristic-Enhanced deep learning approach," Energy, Elsevier, vol. 297(C).
    18. Jelić, Marko & Batić, Marko & Krstić, Aleksandra & Bottarelli, Michele & Mainardi, Elena, 2023. "Comparative analysis of metaheuristic optimization approaches for multisource heat pump operation," Renewable and Sustainable Energy Reviews, Elsevier, vol. 188(C).
    19. Qiang Yang & Xu Guo & Xu-Dong Gao & Dong-Dong Xu & Zhen-Yu Lu, 2022. "Differential Elite Learning Particle Swarm Optimization for Global Numerical Optimization," Mathematics, MDPI, vol. 10(8), pages 1-32, April.
    20. Nantiwat Pholdee & Sujin Bureerat, 2016. "Hybrid real-code ant colony optimisation for constrained mechanical design," International Journal of Systems Science, Taylor & Francis Journals, vol. 47(2), pages 474-491, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jeners:v:14:y:2021:i:11:p:3082-:d:562304. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.