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Novel operational matrices-based method for solving fractional-order delay differential equations via shifted Gegenbauer polynomials

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  • Usman, M.
  • Hamid, M.
  • Zubair, T.
  • Haq, R.U.
  • Wang, W.
  • Liu, M.B.

Abstract

Accurate solutions of nonlinear multi-dimensional delay problems of fractional-order arising in mathematical physics and engineering recently have been found to be a challenging task for the research community. This paper witnesses that an efficient fully spectral operational matrices-based scheme is developed and successfully applied for stable solutions of time-fractional delay differential equations (DDEs). Monomials are introduced in order to proposed the novel operational matrices for fractional-order integration Iν and derivative Dν by means of shifted Gegenbauer polynomials. Some ordinary and partial delay differential equations of fractional-order are considered to show reliability, efficiency and appropriateness of the proposed method. In order to approximate the delay term in DDEs a novel delay operational matrix Θba is introduced with the help of shifted Gegenbauer polynomials. The proposed algorithm transform the problem understudy into a system of algebraic equations which are easier to tackle. Analytical solutions of the mentioned problem are effectively obtained, and an inclusive comparative study is reported which reveals that the proposed computational scheme is effective, accurate and well-matched to investigate the solutions of aforementioned problems. Error bound analysis is enclosed in our investigation to reveal the consistency and support the mathematical formulation of the algorithm. This proposed scheme can be extended to explore the solution of more dervisfy problem of physical nature in complex geometry.

Suggested Citation

  • Usman, M. & Hamid, M. & Zubair, T. & Haq, R.U. & Wang, W. & Liu, M.B., 2020. "Novel operational matrices-based method for solving fractional-order delay differential equations via shifted Gegenbauer polynomials," Applied Mathematics and Computation, Elsevier, vol. 372(C).
    • Handle: RePEc:eee:apmaco:v:372:y:2020:i:c:s0096300319309774
    DOI: 10.1016/j.amc.2019.124985
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    References listed on IDEAS

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    1. Saeed, Umer & Rehman, Mujeeb ur & Iqbal, Muhammad Asad, 2015. "Modified Chebyshev wavelet methods for fractional delay-type equations," Applied Mathematics and Computation, Elsevier, vol. 264(C), pages 431-442.
    2. F. Mohammadi & M.M. Hosseini & Syed Tauseef Mohyud-Din, 2011. "Legendre wavelet Galerkin method for solving ordinary differential equations with non-analytic solution," International Journal of Systems Science, Taylor & Francis Journals, vol. 42(4), pages 579-585.
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    Cited by:

    1. Usman, Muhammad & Hamid, Muhammad & Liu, Moubin, 2021. "Novel operational matrices-based finite difference/spectral algorithm for a class of time-fractional Burger equation in multidimensions," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    2. Zubair, Tamour & Lu, Tiao & Usman, Muhammad, 2021. "Higher dimensional semi-relativistic time-fractional Vlasov-Maxwell code for numerical simulation based on linear polarization and 2D Landau damping instability," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    3. Bukhari, Ayaz Hussain & Raja, Muhammad Asif Zahoor & Rafiq, Naila & Shoaib, Muhammad & Kiani, Adiqa Kausar & Shu, Chi-Min, 2022. "Design of intelligent computing networks for nonlinear chaotic fractional Rossler system," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    4. Dehestani, H. & Ordokhani, Y. & Razzaghi, M., 2020. "Application of fractional Gegenbauer functions in variable-order fractional delay-type equations with non-singular kernel derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    5. Jaradat, Imad & Alquran, Marwan & Sulaiman, Tukur A. & Yusuf, Abdullahi, 2022. "Analytic simulation of the synergy of spatial-temporal memory indices with proportional time delay," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    6. Usman, Muhammad & Hamid, Muhammad & Khan, Zafar Hayat & Haq, Rizwan Ul, 2021. "Neuronal dynamics and electrophysiology fractional model: A modified wavelet approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 570(C).

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