IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v13y2025i16p2544-d1720683.html
   My bibliography  Save this article

A Spectral Approach to Variable-Order Fractional Differential Equations: Improved Operational Matrices for Fractional Jacobi Functions

Author

Listed:
  • Hany M. Ahmed

    (Department of Mathematics, Faculty of Technology and Education, Helwan University, Cairo 11281, Egypt)

  • Mohammad Izadi

    (Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman 76169-14111, Iran)

  • Carlo Cattani

    (Engineering School (DEIM), University of Tuscia, 01100 Viterbo, Italy)

Abstract

The current paper presents a novel numerical technique to handle variable-order multiterm fractional differential equations (VO-MTFDEs) supplemented with initial conditions (ICs) by introducing generalized fractional Jacobi functions (GFJFs). These GFJFs satisfy the associated ICs. A crucial part of this approach is using the spectral collocation method (SCM) and building operational matrices (OMs) for both integer-order and variable-order fractional derivatives in the context of GFJFs. These lead to efficient and accurate computations. The suggested algorithm’s convergence and error analysis are proved. The feasibility of the suggested procedure is confirmed via five numerical test examples.

Suggested Citation

  • Hany M. Ahmed & Mohammad Izadi & Carlo Cattani, 2025. "A Spectral Approach to Variable-Order Fractional Differential Equations: Improved Operational Matrices for Fractional Jacobi Functions," Mathematics, MDPI, vol. 13(16), pages 1-18, August.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:16:p:2544-:d:1720683
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/13/16/2544/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/13/16/2544/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Omar Abu Arqub & Ahlem Ben Rabah & Shaher Momani, 2023. "A spline construction scheme for numerically solving fractional Bagley–Torvik and Painlevé models correlating initial value problems concerning the Caputo–Fabrizio derivative approach," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 34(09), pages 1-22, September.
    2. M. R. Balooch Shahriyar & F. Ismail & S. Aghabeigi & A. Ahmadian & S. Salahshour, 2013. "An Eigenvalue-Eigenvector Method for Solving a System of Fractional Differential Equations with Uncertainty," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-11, July.
    3. Fukang Yin & Junqiang Song & Yongwen Wu & Lilun Zhang, 2013. "Numerical Solution of the Fractional Partial Differential Equations by the Two‐Dimensional Fractional‐Order Legendre Functions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    4. Jianping Liu & Xia Li & Limeng Wu, 2016. "An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation," Mathematical Problems in Engineering, Hindawi, vol. 2016, pages 1-10, June.
    5. Mahmoud Abd El-Hady & Mohamed El-Gamel & Homan Emadifar & Atallah El-shenawy, 2025. "Analysis of RL electric circuits modeled by fractional Riccati IVP via Jacobi-Broyden Newton algorithm," PLOS ONE, Public Library of Science, vol. 20(1), pages 1-28, January.
    6. Fukang Yin & Junqiang Song & Yongwen Wu & Lilun Zhang, 2013. "Numerical Solution of the Fractional Partial Differential Equations by the Two-Dimensional Fractional-Order Legendre Functions," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-13, November.
    7. Singh, Harendra & Srivastava, H.M., 2019. "Jacobi collocation method for the approximate solution of some fractional-order Riccati differential equations with variable coefficients," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 1130-1149.
    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. Singh, Harendra, 2020. "Analysis for fractional dynamics of Ebola virus model," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    2. Izadi, Mohammad & Srivastava, H.M., 2021. "Numerical approximations to the nonlinear fractional-order Logistic population model with fractional-order Bessel and Legendre bases," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    3. Hassan, Shahzaib Ahmed & Raja, Muhammad Junaid Ali Asif & Chang, Chuan-Yu & Shu, Chi-Min & Shoaib, Muhammad & Kiani, Adiqa Kausar & Raja, Muhammad Asif Zahoor, 2024. "Nonlinear chaotic Lorenz-Lü-Chen fractional order dynamics: A novel machine learning expedition with deep autoregressive exogenous neural networks," Chaos, Solitons & Fractals, Elsevier, vol. 189(P1).
    4. İbrahim Avcı & Nazim I. Mahmudov, 2020. "Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration," Mathematics, MDPI, vol. 8(1), pages 1-24, January.
    5. Singh, Harendra, 2021. "Analysis of drug treatment of the fractional HIV infection model of CD4+ T-cells," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    6. Mukhtar, Roshana & Chang, Chuan-Yu & Raja, Muhammad Asif Zahoor & Chaudhary, Naveed Ishtiaq & Shu, Chi-Min, 2024. "Novel nonlinear fractional order Parkinson's disease model for brain electrical activity rhythms: Intelligent adaptive Bayesian networks," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    7. Singh, Harendra & Baleanu, Dumitru & Singh, Jagdev & Dutta, Hemen, 2021. "Computational study of fractional order smoking model," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    8. Muhammad Imran Asjad & Saif Ur Rehman & Ali Ahmadian & Soheil Salahshour & Mehdi Salimi, 2021. "First Solution of Fractional Bioconvection with Power Law Kernel for a Vertical Surface," Mathematics, MDPI, vol. 9(12), pages 1-18, June.
    9. Haifa Bin Jebreen & Ioannis Dassios, 2022. "A Biorthogonal Hermite Cubic Spline Galerkin Method for Solving Fractional Riccati Equation," Mathematics, MDPI, vol. 10(9), pages 1-14, April.
    10. Srivastava, H.M. & Saad, Khaled M. & Khader, M.M., 2020. "An efficient spectral collocation method for the dynamic simulation of the fractional epidemiological model of the Ebola virus," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    11. Saad, Khaled M. & Srivastava, H.M. & Gómez-Aguilar, J.F., 2020. "A Fractional Quadratic autocatalysis associated with chemical clock reactions involving linear inhibition," Chaos, Solitons & Fractals, Elsevier, vol. 132(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:gam:jmathe:v:13:y:2025:i:16:p:2544-:d:1720683. 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.