IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i7p1134-d383217.html
   My bibliography  Save this article

A Numerical Method for a System of Fractional Differential-Algebraic Equations Based on Sliding Mode Control

Author

Listed:
  • Yongpeng Tai

    (College of Automobile and Traffic Engineering, Nanjing Forestry University, Nanjing 210037, China)

  • Ning Chen

    (College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China)

  • Lijin Wang

    (College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China)

  • Zaiyong Feng

    (Department of Mathematics Teaching, Nanjing Institute of Railway Technology, Nanjing 210031, China)

  • Jun Xu

    (College of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, China)

Abstract

Fractional calculus is widely used in engineering fields. In complex mechanical systems, multi-body dynamics can be modelled by fractional differential-algebraic equations when considering the fractional constitutive relations of some materials. In recent years, there have been a few works about the numerical method of the fractional differential-algebraic equations. However, most of the methods cannot be directly applied in the equations of dynamic systems. This paper presents a numerical algorithm of fractional differential-algebraic equations based on the theory of sliding mode control and the fractional calculus definition of Grünwald–Letnikov. The algebraic equation is considered as the sliding mode surface. The validity of the present method is verified by comparing with an example with exact solutions. The accuracy and efficiency of the present method are studied. It is found that the present method has very high accuracy and low time consumption. The effect of violation corrections on the accuracy is investigated for different time steps.

Suggested Citation

  • Yongpeng Tai & Ning Chen & Lijin Wang & Zaiyong Feng & Jun Xu, 2020. "A Numerical Method for a System of Fractional Differential-Algebraic Equations Based on Sliding Mode Control," Mathematics, MDPI, vol. 8(7), pages 1-13, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:7:p:1134-:d:383217
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/7/1134/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/7/1134/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Hongliang Liu & Yayun Fu & Bailing Li, 2017. "Discrete Waveform Relaxation Method for Linear Fractional Delay Differential-Algebraic Equations," Discrete Dynamics in Nature and Society, Hindawi, vol. 2017, pages 1-9, June.
    2. Giona, Massimiliano & Cerbelli, Stefano & Roman, H.Eduardo, 1992. "Fractional diffusion equation and relaxation in complex viscoelastic materials," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 191(1), pages 449-453.
    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. Wei, T. & Li, Y.S., 2018. "Identifying a diffusion coefficient in a time-fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 151(C), pages 77-95.
    2. Zeid, Samaneh Soradi, 2019. "Approximation methods for solving fractional equations," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 171-193.
    3. Yan, Xiong-bin & Zhang, Zheng-qiang & Wei, Ting, 2022. "Simultaneous inversion of a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    4. Li, Zhiyuan & Liu, Yikan & Yamamoto, Masahiro, 2015. "Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 381-397.
    5. Vikram Singh & Dwijendra N. Pandey, 2020. "Exact Controllability of Multi-Term Time-Fractional Differential System with Sequencing Techniques," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(1), pages 105-120, March.
    6. Sun, Yuting & Hu, Cheng & Yu, Juan & Shi, Tingting, 2023. "Synchronization of fractional-order reaction-diffusion neural networks via mixed boundary control," Applied Mathematics and Computation, Elsevier, vol. 450(C).
    7. Jürgen Geiser & Eulalia Martínez & Jose L. Hueso, 2020. "Serial and Parallel Iterative Splitting Methods: Algorithms and Applications to Fractional Convection-Diffusion Equations," Mathematics, MDPI, vol. 8(11), pages 1-42, November.

    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:jmathe:v:8:y:2020:i:7:p:1134-:d:383217. 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.