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AIMS Mathematics

2024, Volume 9, Issue 6: 15246-15262. doi: 10.3934/math.2024740
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Efficient spectral collocation method for nonlinear systems of fractional pantograph delay differential equations

  • 1.
    Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), P.O. Box-65892, Riyadh 11566, Saudi Arabia
  • 2.
    Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
  • 3.
    Art and Science Faculty, Department of Mathematics, Siirt University, 56100 Siirt, Turkey
  • 4.
    Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon
  • 5.
    Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg 620002, Russia
  • Received: 09 March 2024 Revised: 22 April 2024 Accepted: 23 April 2024 Published: 28 April 2024

  • MSC : 26A33, 33D45, 65M70

  • Caputo-Hadamard-type fractional calculus involves the logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenges in numerical approximations. In this paper, we construct and analyze a spectral collocation approach using mapped Jacobi functions as basis functions and construct an efficient algorithm to solve systems of fractional pantograph delay differential equations involving Caputo-Hadamard fractional derivatives. What we study is the error estimates of the derived method. In addition, we tabulate numerical results to support our theoretical analysis.

    Citation: M. A. Zaky, M. Babatin, M. Hammad, A. Akgül, A. S. Hendy. Efficient spectral collocation method for nonlinear systems of fractional pantograph delay differential equations[J]. AIMS Mathematics, 2024, 9(6): 15246-15262. doi: 10.3934/math.2024740

    Related Papers:

  • Caputo-Hadamard-type fractional calculus involves the logarithmic function of an arbitrary exponent as its convolutional kernel, which causes challenges in numerical approximations. In this paper, we construct and analyze a spectral collocation approach using mapped Jacobi functions as basis functions and construct an efficient algorithm to solve systems of fractional pantograph delay differential equations involving Caputo-Hadamard fractional derivatives. What we study is the error estimates of the derived method. In addition, we tabulate numerical results to support our theoretical analysis.


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    [1] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equation, Amsterdam: Elsevier, 2006.
    [2] F. Mainardi, Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models, London: Imperial College Press, 2010.
    [3] C. J. Li, H. X. Zhang, X. H. Yang, A new nonlinear compact difference scheme for a fourth-order nonlinear Burgers type equation with a weakly singular kernel, J. Appl. Math. Comput., 2024, 1–33. https://doi.org/10.1007/s12190-024-02039-x
    [4] L. J. Qiao, W. L. Qiu, M. A. Zaky, A. S. Hendy, Theta-type convolution quadrature OSC method for nonlocal evolution equations arising in heat conduction with memory, Fract. Calc. Appl. Anal., 2024, 1–26. https://doi.org/10.1007/s13540-024-00265-5
    [5] X. Y. Peng, W. L. Qiu, A. S. Hendy, M. A. Zaky, Temporal second-order fast finite difference/compact difference schemes for time-fractional generalized burgers' equations, J. Sci. Comput., 99 (2024), 52. https://doi.org/10.1007/s10915-024-02514-4 doi: 10.1007/s10915-024-02514-4
    [6] H. Chen, M. A. Zaky, X. C. Zheng, A. S. Hendy, W. L. Qiu, Spatial two-grid compact difference method for nonlinear Volterra integro-differential equation with Abel kernel, Numer. Algor., 2024, 1–42. https://doi.org/10.1007/s11075-024-01811-1
    [7] H. Chen, W. L. Qiu, M. A. Zaky, A. S. Hendy, A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with weakly singular kernel, Calcolo, 60 (2023), 13. https://doi.org/10.1007/s10092-023-00508-6 doi: 10.1007/s10092-023-00508-6
    [8] J. Hadamard, Essai sur l'étude des fonctions données par leur développement de Taylor, J. Math. Pure. Appl., 8 (1892), 101–186.
    [9] B. Ahmad, A. Alsaedi, S. K. Ntouyas, J. Tariboon, Hadamard-type fractional differential equations, inclusions and inequalities, Cham: Springer, 2017. https://doi.org/10.1007/978-3-319-52141-1
    [10] A. A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191–1204.
    [11] L. Ma, C. P. Li, On Hadamard fractional calculus, Fractals, 25 (2017), 1750033. https://doi.org/10.1142/S0218348X17500335 doi: 10.1142/S0218348X17500335
    [12] M. Gohar, C. P. Li, C. T. Yin, On Caputo-Hadamard fractional differential equations, Int. J. Comput. Math., 97 (2020), 1459–1483. https://doi.org/10.1080/00207160.2019.1626012 doi: 10.1080/00207160.2019.1626012
    [13] G. T. Wang, K. Pei, Y. Q. Chen, Stability analysis of nonlinear Hadamard fractional differential system, J. Franklin Inst., 356 (2019), 6538–6546. https://doi.org/10.1016/j.jfranklin.2018.12.033 doi: 10.1016/j.jfranklin.2018.12.033
    [14] H. Belbali, M. Benbachir, S. Etemad, C. Park, S. Rezapour, Existence theory and generalized Mittag-Leffler stability for a nonlinear Caputo-Hadamard FIVP via the Lyapunov method, AIMS Math., 7 (2022), 14419–14433. https://doi.org/10.3934/math.2022794 doi: 10.3934/math.2022794
    [15] S. Aljoudi, B. Ahmad, J. J. Nieto, A. Alsaedi, A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions, Chaos Soliton Fract., 91 (2016), 39–46. https://doi.org/10.1016/j.chaos.2016.05.005 doi: 10.1016/j.chaos.2016.05.005
    [16] S. Dhaniya, A. Kumar, A. Khan, T. Abdeljawad, M. A. Alqudah, Existence results of Langevin equations with Caputo-Hadamard fractional operator, J. Math., 2023 (2023), 1–12. https://doi.org/10.1155/2023/2288477 doi: 10.1155/2023/2288477
    [17] M. T. Beyene, M. D. Firdi, T. T. Dufera, Analysis of Caputo-Hadamard fractional neutral delay differential equations involving Hadamard integral and unbounded delays: existence and uniqueness, Research Math., 11 (2024), 2321669. https://doi.org/10.1080/27684830.2024.2321669 doi: 10.1080/27684830.2024.2321669
    [18] B. B. He, H. C. Zhou, C. H. Kou, Stability analysis of Hadamard and Caputo-Hadamard fractional nonlinear systems without and with delay, Fract. Calc. Appl. Anal., 25 (2022), 2420–2445. https://doi.org/10.1007/s13540-022-00106-3 doi: 10.1007/s13540-022-00106-3
    [19] M. Gohar, C. P. Li, Z. Q. Li, Finite difference methods for Caputo-Hadamard fractional differential equations, Mediterr. J. Math., 17 (2020), 194. https://doi.org/10.1007/s00009-020-01605-4 doi: 10.1007/s00009-020-01605-4
    [20] C. P. Li, Z. Q. Li, Z. Wang, Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation, J. Sci. Comput., 85 (2020), 1–27. https://doi.org/10.1007/s10915-020-01353-3 doi: 10.1007/s10915-020-01353-3
    [21] E. Y. Fan, C. P. Li, Z. Q. Li, Numerical approaches to Caputo-Hadamard fractional derivatives with applications to long-term integration of fractional differential systems, Commun. Nonlinear Sci. Numer. Simul., 106 (2022), 106096. https://doi.org/10.1016/j.cnsns.2021.106096 doi: 10.1016/j.cnsns.2021.106096
    [22] G. Istafa, M. Rehman, Numerical solutions of Hadamard fractional differential equations by generalized Legendre functions, Math. Methods Appl. Sci., 46 (2023), 6821–6842. https://doi.org/10.1002/mma.8942 doi: 10.1002/mma.8942
    [23] M. A. Zaky, A. S. Hendy, D. Suragan, Logarithmic Jacobi collocation method for Caputo-Hadamard fractional differential equations, Appl. Numer. Math., 181 (2022), 326–346. https://doi.org/10.1016/j.apnum.2022.06.013 doi: 10.1016/j.apnum.2022.06.013
    [24] P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl., 269 (2002), 387–400. https://doi.org/10.1016/S0022-247X(02)00049-5 doi: 10.1016/S0022-247X(02)00049-5
    [25] X. Yang, H. Zhang, The uniform $l^{1}$ long-time behavior of time discretization for time-fractional partial differential equations with nonsmooth data, Appl. Math. Lett., 124 (2022), 107644. https://doi.org/10.1016/j.aml.2021.107644 doi: 10.1016/j.aml.2021.107644
    [26] X. H. Yang, H. X. Zhang, Q. Zhang, G. W. Yuan, Z. Q. Sheng, The finite volume scheme preserving maximum principle for two-dimensional time-fractional Fokker-Planck equations on distorted meshes, Appl. Math. Lett., 97 (2019), 99–106. https://doi.org/10.1016/j.aml.2019.05.030 doi: 10.1016/j.aml.2019.05.030
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