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2025, Volume 33, Issue 9: 5518-5535. doi: 10.3934/era.2025247
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Linear barycentric rational collocation method for solving nonlinear time-fractional Cable equation

  • School of Science, Shandong Jianzhu University, Jinan 250101, China
  • Published: 16 September 2025

  • The time-fractional Cable (TFC) equation constitutes a significant advancement of the classical Cable equation within the context of fractional calculus. In contrast to conventional numerical techniques such as finite difference and finite element methods, the barycentric interpolation method exhibits superior capability in handling the nonlocal properties inherent in fractional operators. This study thoroughly examines the principles and attributes of the barycentric interpolation method, integrating them with the unique aspects of the TFC equation to develop an appropriate matrix formulation. This transformation facilitates the conversion of the TFC equation into a solvable algebraic system. The proposed algorithm's accuracy and convergence rate are meticulously evaluated. Numerical experiments are performed to underscore the benefits of this approach in solving the TFC equation, thereby confirming the efficacy of the error analysis and convergence assessment.

    Citation: Bo Liu, Di Liang. Linear barycentric rational collocation method for solving nonlinear time-fractional Cable equation[J]. Electronic Research Archive, 2025, 33(9): 5518-5535. doi: 10.3934/era.2025247

    Related Papers:

  • The time-fractional Cable (TFC) equation constitutes a significant advancement of the classical Cable equation within the context of fractional calculus. In contrast to conventional numerical techniques such as finite difference and finite element methods, the barycentric interpolation method exhibits superior capability in handling the nonlocal properties inherent in fractional operators. This study thoroughly examines the principles and attributes of the barycentric interpolation method, integrating them with the unique aspects of the TFC equation to develop an appropriate matrix formulation. This transformation facilitates the conversion of the TFC equation into a solvable algebraic system. The proposed algorithm's accuracy and convergence rate are meticulously evaluated. Numerical experiments are performed to underscore the benefits of this approach in solving the TFC equation, thereby confirming the efficacy of the error analysis and convergence assessment.



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    [1] N. R. Bayramov, J. K. Kraus, On the stable solution of transient convection-diffusion equations, J. Comput. Appl. Math., 280 (2015), 275–293. https://doi.org/10.1016/j.cam.2014.12.001 doi: 10.1016/j.cam.2014.12.001
    [2] T. Liu, H. Zhang, X. Yang, The ADI compact difference scheme for three-dimensional integro-partial differential equation with three weakly singular kernels, J. Appl. Math. Comput., 71 (2025), 3861–3889. https://doi.org/10.1007/s12190-025-02386-3 doi: 10.1007/s12190-025-02386-3
    [3] J. Zhang, X. Yang, S. Wang, The ADI difference and extrapolation scheme for high-dimensional variable coefficient evolution equations, Electron. Res. Arch., 33 (2025), 3305–3327. https://doi.org/10.3934/era.2025146 \newpage doi: 10.3934/era.2025146
    [4] Z. Zhang, X. Yang, S. Wang, The alternating direction implicit difference scheme and extrapolation method for a class of three-dimensional hyperbolic equations with constant coefficients, Electron. Res. Arch., 33 (2025), 3348–3377. https://doi.org/10.3934/era.2025148 doi: 10.3934/era.2025148
    [5] X. Yang, Z. Zhang, Superconvergence analysis of a robust orthogonal Gauss collocation method for 2D fourth-order subdiffusion equations, J. Sci. Comput., 100 (2024), 62. https://doi.org/10.1007/s10915-024-02616-z doi: 10.1007/s10915-024-02616-z
    [6] X. Yang, Z. Zhang, Analysis of a new NFV scheme preserving DMP for two-dimensional sub-diffusion equation on distorted meshes, J. Sci. Comput., 99 (2024), 80. https://doi.org/10.1007/s10915-024-02511-7 doi: 10.1007/s10915-024-02511-7
    [7] Y. Shi, X. Yang, The pointwise error estimate of a new energy-preserving nonlinear difference method for supergeneralized viscous Burgers' equation, Comput. Appl. Math., 44 (2025), 257. https://doi.org/10.1007/s40314-025-03222-x doi: 10.1007/s40314-025-03222-x
    [8] J. Wang, X. Jiang, H. Zhang, A BDF3 and new nonlinear fourth-order difference scheme for the generalized viscous Burgers' equation, Appl. Math. Lett., 151 (2024), 109002. https://doi.org/10.1016/j.aml.2024.109002 doi: 10.1016/j.aml.2024.109002
    [9] Y. Wang, Y. Liu, H. Li, J. Wang, Finite element method combined with second-order time discrete scheme for nonlinear fractional Cable equation, Eur. Phys. J. Plus, 131 (2016), 61. https://doi.org/10.1140/epjp/i2016-16061-3 doi: 10.1140/epjp/i2016-16061-3
    [10] F. Zeng, C. Li, A new Crank-Nicolson finite element method for the time-fractional subdiffusion equation, Appl. Numer. Math., 121 (2017), 82–95. https://doi.org/10.1016/j.apnum.2017.06.011 doi: 10.1016/j.apnum.2017.06.011
    [11] Y. Liu, Y. Du, H. Li, J. Wang, A two-grid finite element approximation for a nonlinear time-fractional Cable equation, Nonlinear Dyn., 85 (2016), 2535–2548. https://doi.org/10.1007/s11071-016-2843-9 doi: 10.1007/s11071-016-2843-9
    [12] J. Shen, T. Tang, L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, Heidelberg, Germany, 2011. https://doi.org/10.1007/978-3-540-71041-7
    [13] A. G. Atta, Two spectral Gegenbauer methods for solving linear and nonlinear time fractional Cable problems, Int. J. Mod. Phys. C, 35 (2024), 1–22. https://doi.org/10.1142/S0129183124500700 doi: 10.1142/S0129183124500700
    [14] H. Liu, S. Lü, Galerkin spectral method for nonlinear time fractional Cable equation with smooth and nonsmooth solutions, Appl. Math. Comput., 350 (2019), 32–47. https://doi.org/10.1016/j.amc.2018.12.072 doi: 10.1016/j.amc.2018.12.072
    [15] X. Yang, X. Jiang, H. Zhang, A time-space spectral tau method for the time fractional Cable equation and its inverse problem, Appl. Numer. Math., 130 (2018), 95–111. https://doi.org/10.1016/j.apnum.2018.03.016 doi: 10.1016/j.apnum.2018.03.016
    [16] H. Liu, S. Lü, T. Jiang, Analysis of Legendre pseudospectral approximations for nonlinear time fractional diffusion-wave equations, Int. J. Comput. Math., 98 (2021), 1769–1791. https://doi.org/10.1080/00207160.2020.1846731 doi: 10.1080/00207160.2020.1846731
    [17] Y. Chen, X. Lin, M. Zhang, Y. Huang, Stability and convergence of L1-Galerkin spectral methods for the nonlinear time fractional Cable equation, East Asian J. Appl. Math., 13 (2023), 22–46. https://doi.org/10.4208/eajam.020521.140522 doi: 10.4208/eajam.020521.140522
    [18] S. Kumar, D. Baleanu, Numerical solution of two-dimensional time fractional Cable equation with Mittag-Leffler kernel, Math. Methods Appl. Sci., 43 (2020), 8348–8362. https://doi.org/10.1002/mma.6491 doi: 10.1002/mma.6491
    [19] M. A. Arefin, U. Sadiya, M. Inc, M. H. Uddin, Adequate soliton solutions to the space-time fractional telegraph equation and modified third-order KdV equation through a reliable technique, Opt. Quantum Electron., 54 (2022), 309. https://doi.org/10.1007/s11082-022-03640-9 doi: 10.1007/s11082-022-03640-9
    [20] N. H. Sweilam, S. M. AL-Mekhlafi, A novel numerical method for solving the 2-D time fractional Cable equation, Eur. Phys. J. Plus, 134 (2019), 323. https://doi.org/10.1140/epjp/i2019-12730-y doi: 10.1140/epjp/i2019-12730-y
    [21] Y. Liu, Z. Yu, H. Li, F. Liu, J. Wang, Time two-mesh algorithm combined with finite element method for time fractional water wave model, Int. J. Heat Mass Transfer, 120 (2018), 1132–1145. https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.118 doi: 10.1016/j.ijheatmasstransfer.2017.12.118
    [22] Z. Wang, On Caputo-Type Cable equation: Analysis and computation, CMES-Comput. Model. Eng. Sci., 123 (2020), 353–376. https://doi.org/10.32604/cmes.2020.08776 doi: 10.32604/cmes.2020.08776
    [23] H. R. Ghehsareh, M. S. Seidzadeh, S. K. Etesami, Numerical simulation of a generalized anomalous electro-diffusion process in nerve cells by a localized meshless approach in Pseudospectral mode, Int. J. Numer. Modell. Electron. Networks Devices Fields, 33 (2020), e2756. https://doi.org/10.1002/jnm.2756 doi: 10.1002/jnm.2756
    [24] W. Zou, Y. Tang, V. R. Hosseini, The numerical meshless approach for solving the 2D time nonlinear multi-term fractional Cable equation in complex geometries, Fractals, 30 (2022), 2240170. https://doi.org/10.1142/S0218348X22401703 doi: 10.1142/S0218348X22401703
    [25] Y. Liu, Y. Du, H. Li, F. Liu, Y. Wang, Some second-order $\theta$ schemes combined with finite element method for nonlinear fractional Cable equation, Numer. Algorithms, 80 (2019), 533–555. https://doi.org/10.1007/s11075-018-0496-0 doi: 10.1007/s11075-018-0496-0
    [26] J. P. Berrut, M. S. Floater, G. Klein, Convergence rates of derivatives of a family of barycentric rational interpolants, Appl. Numer. Math., 61 (2011), 989–1000. https://doi.org/10.1016/j.apnum.2011.05.001 doi: 10.1016/j.apnum.2011.05.001
    [27] J. P. Berrut, S. A. Hosseini, G. Klein, The linear barycentric rational quadrature method for Volterra integral equations, SIAM J. Sci. Comput., 36 (2014), A105–A123. https://doi.org/10.1137/120904020 doi: 10.1137/120904020
    [28] J. P. Berrut, G. Klein, Recent advances in linear barycentric rational interpolation, J. Comput. Appl. Math., 259 (2014), 95–107. https://doi.org/10.1016/j.cam.2013.03.044 doi: 10.1016/j.cam.2013.03.044
    [29] E. Cirillo, K. Hormann, On the Lebesgue constant of barycentric rational Hermite interpolants at equidistant nodes, J. Comput. Appl. Math., 349 (2019), 292–301. https://doi.org/10.1016/j.cam.2018.06.011 doi: 10.1016/j.cam.2018.06.011
    [30] M. S. Floater, K. Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Numer. Math., 107 (2007), 315–331. https://doi.org/10.1007/s00211-007-0093-y doi: 10.1007/s00211-007-0093-y
    [31] G. Klein, J. P. Berrut, Linear rational finite differences from derivatives of barycentric rational interpolants, SIAM J. Numer. Anal., 50 (2012), 643–656. https://doi.org/10.1137/110827156 doi: 10.1137/110827156
    [32] G. Klein, J. P. Berrut, Linear barycentric rational quadrature, BIT Numer. Math., 52 (2012), 407–424. https://doi.org/10.1007/s10543-011-0357-x doi: 10.1007/s10543-011-0357-x
    [33] A. Abdi, J. P. Berrut, S. A. Hosseini, The linear barycentric rational method for a class of delay Volterra integro-differential equations, J. Sci. Comput., 75 (2018), 1757–1775. https://doi.org/10.1007/s10915-017-0608-3 doi: 10.1007/s10915-017-0608-3
    [34] J. Pour-Mahmoud, M. Y. Rahimi-Ardabili, S. Shahmorad, Numerical solution of the system of Fredholm integro-differential equations by the tau method, Appl. Math. Comput., 168 (2005), 465–478. https://doi.org/10.1016/j.amc.2004.09.026 doi: 10.1016/j.amc.2004.09.026
    [35] S. M. Hosseini, S. Shahmorad, Numerical solution of a class of integro-differential equations by the tau method with an error estimation, Appl. Math. Comput., 136 (2003), 559–570. https://doi.org/10.1016/S0096-3003(02)00081-4 doi: 10.1016/S0096-3003(02)00081-4
    [36] S. Yousefi, M. Razzaghi, Legendre wavelets method for the nonlinear Volterra Fredholm integral equations, Math. Comput. Simul., 70 (2005), 1–8. https://doi.org/10.1016/j.matcom.2005.02.035 doi: 10.1016/j.matcom.2005.02.035
    [37] S. Yalcinbas, M. Sezer, H. H. Sorkun, Legendre polynomial solutions of high-order linear Fredholm integro-differential equations, Appl. Math. Comput., 210 (2009), 334–349. https://doi.org/10.1016/j.amc.2008.12.090 doi: 10.1016/j.amc.2008.12.090
    [38] S. Li, Z. Wang, High Precision Meshless barycentric Interpolation Collocation Method–Algorithmic Program and Engineering Application (in Chinese), Science Press, Beijing, China, 2012.
    [39] Z. Wang, S. Li, Barycentric Interpolation Collocation Method for Nonlinear Problems (in Chinese), National Defense Industry Press, 2015.
    [40] Z. Wang, Z. Xu, J. Li, Mixed barycentric interpolation collocation method of displacement-pressure for incompressible plane elastic problems (in Chinese), Chin. J. Appl. Mech., 35 (2018), 631–636+695.
    [41] Z. Wang, L. Zhang, Z. Xu, J. Li, Barycentric interpolation collocation method based on mixed displacement-stress formulation for solving plane elastic problems (in Chinese), Chin. J. Appl. Mech., 35 (2018), 304–308+451.
    [42] K. Maleknejad, N. Aghazadeh, Numerical solutions of Volterra integral equations of the second kind with convolution kernel by using Taylor-series expansion method, Appl. Math. Comput., 161 (2005), 915–922. https://doi.org/10.1016/j.amc.2003.12.075 doi: 10.1016/j.amc.2003.12.075
    [43] J. Li, J. Qu, High precision barycentric interpolation collocation method to solve EFK equation, Int. J. Comput. Math., 2025 (2025), 1–19. https://doi.org/10.1080/00207160.2025.2522993 doi: 10.1080/00207160.2025.2522993
    [44] Z. Li, J. Li, Numerical solutions for (2+1)-dimensional ZK-MEW equation using linear barycentric rational collocation method, Comput. Math. Appl., 193 (2025), 332–345. https://doi.org/10.1016/j.camwa.2025.06.021 doi: 10.1016/j.camwa.2025.06.021
    [45] C. Li, S. Hon, Multilevel tau preconditioners for symmetrized multilevel Toeplitz systems with applications to solving space fractional diffusion equations, SIAM J. Matrix Anal. Appl., 46 (2025), 487–508. https://doi.org/10.1137/24M1647096 doi: 10.1137/24M1647096
    [46] J. Pan, M. Ng, H. Wang, Fast preconditioned iterative methods for finite volume discretization of steady-state space-fractional diffusion equations, Numer. Algorithms, 74 (2017), 153–173. https://doi.org/10.1007/s11075-016-0143-6 doi: 10.1007/s11075-016-0143-6
    [47] M. Mazza, S. Serra-Capizzano, R. L. Sormani, Algebra preconditionings for 2D Riesz distributed-order space-fractional diffusion equations on convex domains, Numer. Linear Algebra Appl., 31 (2024), e2536. https://doi.org/10.1002/nla.2536 doi: 10.1002/nla.2536
    [48] M. Donatelli, M. Mazza, S. Serra-Capizzano, Spectral analysis and multigrid methods for finite volume approximations of space-fractional diffusion equations, SIAM J. Sci. Comput., 40 (2018), A4007–A4039. https://doi.org/10.1137/17M115164X doi: 10.1137/17M115164X
    [49] Z. Chen, H. Zhang, H. Chen, ADI compact difference scheme for the two-dimensional integro-differential equation with two fractional Riemann Liouville integral kernels, Fractal Fract., 8 (2024), 707. https://doi.org/10.3390/fractalfract8120707 doi: 10.3390/fractalfract8120707
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