IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v201y2025ip1s0960077925011701.html

The analysis and computation of nabla Mittag–Leffler functions deduced from the frequency domain

Author

Listed:
  • Wei, Yiheng
  • Zhou, Shuaiyu
  • Xu, Qiang
  • Du, Feifei

Abstract

This paper makes a pioneering contribution to discrete-time fractional calculus by focusing on the nabla Mittag-Leffler function. Motivated by identified limitations in the existing time-domain definition, this work systematically develops a frequency-domain definition. Through rigorous mathematical analysis, a novel framework is established to bridge the gap between time- and frequency-domain representations, with particular emphasis on the carefully constructed initial value conditions. Subsequently, a comprehensive suite of analytic properties is derived, i.e., the numerical relationship and the dynamic behavior. To enable practical applications, three innovative computational schemes are proposed, each accompanied by thorough convergence analysis. The efficacy of our methods is demonstrated through three benchmark numerical examples. These advancements provide new tools for typical applications of nabla fractional order systems.

Suggested Citation

  • Wei, Yiheng & Zhou, Shuaiyu & Xu, Qiang & Du, Feifei, 2025. "The analysis and computation of nabla Mittag–Leffler functions deduced from the frequency domain," Chaos, Solitons & Fractals, Elsevier, vol. 201(P1).
    • Handle: RePEc:eee:chsofr:v:201:y:2025:i:p1:s0960077925011701
    DOI: 10.1016/j.chaos.2025.117157
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077925011701
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2025.117157?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Li, Hui & Kao, Yonggui & Li, Hong-Li, 2021. "Globally β-Mittag-Leffler stability and β-Mittag-Leffler convergence in Lagrange sense for impulsive fractional-order complex-valued neural networks," Chaos, Solitons & Fractals, Elsevier, vol. 148(C).
    2. Bao, Baizeng & Xu, Liguang, 2025. "Mittag-Leffler ultimate boundedness of fractional-order nonautonomous delay systems," Chaos, Solitons & Fractals, Elsevier, vol. 197(C).
    3. Mohammed, Pshtiwan Othman & Kürt, Cemaliye & Abdeljawad, Thabet, 2022. "Bivariate discrete Mittag-Leffler functions with associated discrete fractional operators," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    4. Suwan, Iyad & Abdeljawad, Thabet & Jarad, Fahd, 2018. "Monotonicity analysis for nabla h-discrete fractional Atangana–Baleanu differences," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 50-59.
    5. Andrew Ly & Pulin Gong, 2025. "Optimization on multifractal loss landscapes explains a diverse range of geometrical and dynamical properties of deep learning," Nature Communications, Nature, vol. 16(1), pages 1-14, December.
    6. Du, Feifei & Jia, Baoguo, 2020. "Finite time stability of fractional delay difference systems: A discrete delayed Mittag-Leffler matrix function approach," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    7. Birte Riechers & Amlan Das & Eric Dufresne & Peter M. Derlet & Robert Maaß, 2024. "Intermittent cluster dynamics and temporal fractional diffusion in a bulk metallic glass," Nature Communications, Nature, vol. 15(1), pages 1-9, December.
    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. Yang, Zhanwen & Li, Qi & Yao, Zichen, 2023. "A stability analysis for multi-term fractional delay differential equations with higher order," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).
    2. Maximilian Frey & Nico Neuber & Sascha Sebastian Riegler & Antoine Cornet & Yuriy Chushkin & Federico Zontone & Lucas Matthias Ruschel & Bastian Adam & Mehran Nabahat & Fan Yang & Jie Shen & Fabian We, 2025. "Liquid-like versus stress-driven dynamics in a metallic glass former observed by temperature scanning X-ray photon correlation spectroscopy," Nature Communications, Nature, vol. 16(1), pages 1-12, December.
    3. Abdeljawad, Thabet, 2019. "Fractional difference operators with discrete generalized Mittag–Leffler kernels," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 315-324.
    4. Diblík, Josef, 2026. "Solution of a discrete matrix equation with second-order difference and single delay," Applied Mathematics and Computation, Elsevier, vol. 508(C).
    5. Du, Feifei & Lu, Jun-Guo, 2021. "Explicit solutions and asymptotic behaviors of Caputo discrete fractional-order equations with variable coefficients," Chaos, Solitons & Fractals, Elsevier, vol. 153(P1).
    6. Almusawa, Musawa Yahya & Mohammed, Pshtiwan Othman, 2023. "Approximation of sequential fractional systems of Liouville–Caputo type by discrete delta difference operators," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    7. Aquino, Gerardo & Bologna, Mauro, 2026. "Noise-induced resonant acceleration of a charge in an intermittent magnetic field: An exact solution for ergodic and non-ergodic fluctuations," Chaos, Solitons & Fractals, Elsevier, vol. 203(C).
    8. Zeng, Junwei & Qian, Yongsheng & Wang, Wenhai & Xu, Dejie & Li, Haijun, 2023. "The impact of connected automated vehicles and platoons on the traffic safety and stability in complex heterogeneous traffic systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 629(C).
    9. Chen, Yuting & Li, Xiaoyan & Liu, Song, 2021. "Finite-time stability of ABC type fractional delay difference equations," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    10. Mouataz Billah Mesmouli & Loredana Florentina Iambor & Amir Abdel Menaem & Taher S. Hassan, 2024. "Existence Results and Finite-Time Stability of a Fractional ( p , q )-Integro-Difference System," Mathematics, MDPI, vol. 12(9), pages 1-12, May.
    11. Ivan Pavlenko & Marek Ochowiak & Praveen Agarwal & Radosław Olszewski & Bernard Michałek & Andżelika Krupińska, 2021. "Improvement of Mathematical Model for Sedimentation Process," Energies, MDPI, vol. 14(15), pages 1-12, July.
    12. Panda, Sumati Kumari & Abdeljawad, Thabet & Ravichandran, C., 2020. "A complex valued approach to the solutions of Riemann-Liouville integral, Atangana-Baleanu integral operator and non-linear Telegraph equation via fixed point method," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    13. Jianying Xiao & Yongtao Li, 2022. "Novel Synchronization Conditions for the Unified System of Multi-Dimension-Valued Neural Networks," Mathematics, MDPI, vol. 10(17), pages 1-24, August.
    14. Rashid, Saima & Sultana, Sobia & Hammouch, Zakia & Jarad, Fahd & Hamed, Y.S., 2021. "Novel aspects of discrete dynamical type inequalities within fractional operators having generalized ℏ-discrete Mittag-Leffler kernels and application," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    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:eee:chsofr:v:201:y:2025:i:p1:s0960077925011701. 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: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    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.