An efficient adaptive moving grid algorithm for time-fractional integrodifferential equations governing viscoelastic nanofluid dynamics

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Abstract

An adaptive moving grid method is developed to solve the time-fractional integrodifferential governing equations of viscoelastic nanofluid. The momentum equation is derived based on a dual-parameter fractional Maxwell constitutive relation, and the energy equation employs a generalized Cattaneo heat conduction relation. To improve solution accuracy, a monitor function based on the equidistribution principle is constructed, and an adaptive mesh redistribution strategy is developed in the spatial domain. The temporal fractional-order operators are approximated by the L1 algorithm and the weighted-shifted Grünwald difference method. Numerical experiments demonstrate that the adaptive grid achieves 77.6–88.4% higher accuracy compared to uniform grids at the same grid scale, along with enhanced stability in convergence. Parametric analysis indicates that increasing the fractional-order derivative in the energy equation results in a thickening of both the velocity and thermal boundary layers. Furthermore, the dual-fractional Maxwell model exhibits a thicker velocity boundary layer than its classical single-parameter counterpart. The proposed method offers an efficient and robust approach for simulating complex viscoelastic nanofluid systems with memory effects and multi-field coupling.

Keywords

adaptive moving grid / fractional integrodifferential equations / viscoelastic nanofluid / fractional Maxwell model / generalized Cattaneo heat conduction relation

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References

[1] R. P. Chhabra, Non-Newtonian fluids: An introduction, in Rheology of Complex Fluids (eds. J. Krishnan, A. Deshpande, P. Kumar), Springer, New York, (2010), 3-34. https://doi.org/10.1007/978-1-4419-6494-6

[2] J. Azaiez, G. M. Homsy, Linear stability of free shear flow of viscoelastic liquids, J. Fluid Mech., 268(1994), 37-69. https://doi.org/10.1017/S0022112094001254

[3] D. Tripathi, S. K. Pandey, S. Das, Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel, Appl. Math. Comput., 215(2010), 3645-3654. https://doi.org/10.1016/j.amc.2009.11.002

[4] H. Qi, M. Xu, Some unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative, Appl. Math. Model., 33(2009), 4184-4191. https://doi.org/10.1016/j.apm.2009.03.002

[5] D. Song, T. Jiang, Study on the constitutive equation with fractional derivative for the viscoelastic fluids-modified Jeffreys model and its application, Rheol. Acta, 37(1998), 512-517. https://doi.org/10.1007/s003970050138

[6] F. Irgens, Rheology and Non-Newtonian Fluids , Springer, New York, 2014. https://doi.org/10.1007/978-3-319-01053-3

[7] J. Zhao, L. Zheng, X. Zhang, F. Liu, Unsteady natural convection boundary layer heat transfer of fractional Maxwell viscoelastic fluid over a vertical plate, Int. J. Heat Mass Transfer., 97(2016), 760-766. https://doi.org/10.1016/j.ijheatmasstransfer.2016.02.059

[8] L. Liu, L. Feng, Q. Xu, L. Zheng, F. Liu, Flow and heat transfer of generalized Maxwell fluid over a moving plate with distributed order time fractional constitutive models, Int. Commun. Heat Mass Transfer., 116(2020), 104679. https://doi.org/10.1016/j.icheatmasstransfer.2020.104679

[9] L. Feng, F. Liu, I. Turner, V. V. Anh, Magnetohydrodynamics flow and heat transfer of novel generalized Kelvin-Voigt viscoelastic nanofluids over a moving plate, Phys. Fluids, 36(2024), 063109. https://doi.org/10.1063/5.0213855

[10] D. Qaiser, Z. Zheng, M. R. Khan, Numerical assessment of mixed convection flow of Walters-B nanofluid over a stretching surface with Newtonian heating and mass transfer, Therm. Sci. Eng. Prog., 22(2021), 100801. https://doi.org/10.1016/j.tsep.2020.100801

[11] M. Aleem, M. I. Asjad, A. Shaheen, I. Khan, MHD Influence on different water based nanofluids (TiO₂ , Al₂O₃ , CuO) in porous medium with chemical reaction and newtonian heating, Chaos Solitons Fractals, 130(2020), 109437. https://doi.org/10.1016/j.chaos.2019.109437

[12] Z. Mao, L. Feng, I. Turner, A. Xiao, F. Liu, Transient free convective flow of viscoelastic nanofluids governed by fractional integrodifferential equations under Newtonian heating and thermal radiation, Chin. J. Phys., 93(2025), 584-600. https://doi.org/10.1016/j.cjph.2024.12.025

[13] D. G. Prakasha, N. S. Malagi, P. Veeresha, B. C. Prasannakumara, An efficient computational technique for time-fractional Kaup-Kupershmidt equation, Numer. Methods Partial Differ. Equations, 37(2021), 1299-1316. https://doi.org/10.1002/num.22580

[14] P. Veeresha, D. G. Prakasha, N. Magesh, A. John Christopher, D. U. Sarwe, Solution for fractional potential KdV and Benjamin equations using the novel technique, J. Ocean Eng. Sci., 6(2021), 265-275. https://doi.org/10.1016/j.joes.2021.01.003

[15] C. V. D. Kumar, D. G. Prakasha, N. B. Turki, Exploring the dynamics of fractional-order nonlinear dispersive wave system through homotopy technique, Open Phys., 23(2025), 20250128. https://doi.org/10.1515/phys-2025-0128

[16] J. Xie, M. Wan, F. Zhao, J. Zhang, W. Shi, Dynamic perturbation analysis of fractional order differential quasiperiodic Mathieu equation, Chaos, 33(2023), 123118. https://doi.org/10.1063/5.0163991

[17] M. Shen, Z. Zhou, H. Chen, H. Fang, M. Zhang, A novel physically motivated Zener model for magnetohydrodynamics viscoelastic fluids with differential-integral fractional operator, Phys. Fluids, 37(2025), 073128. https://doi.org/10.1063/5.0275223

[18] Z. Mao, L. Feng, A. Xiao, F. Liu, I. Turner, Magnetohydrodynamic transient flow of dual-parameter fractional maxwell nanofluids past a vertical plate with generalised dual-phase-lagging heat conduction under ramped wall temperature conditions, Eng. Comput., 42(2026), 51. https://doi.org/10.1007/s00366-025-02267-0

[19] A. Mehri, M. S. Abdo, H. Bouhadjera, A. Dawood, K. Aldwoah, R. Egami, Finite element analysis of a multi-term nonlinear time-fractional convection-diffusion equation with Caputo-Fabrizio derivative, Boundary Value Probl., 2025(2025), 1-26. https://doi.org/10.1186/s13661-025-02112-9

[20] I. P. A. Papadopoulos, S. Olver, A sparse spectral method for fractional differential equations in one-spatial dimension, Adv. Comput. Math., 50(2024), 69. https://doi.org/10.1007/s10444-024-10164-1

[21] J. C. Kalita, A. K. Dass, D. C. Dalal, A transformation-free HOC scheme for steady convection-diffusion on non-uniform grids, Int. J. Numer. Methods Fluids, 44(2004), 33-53. https://doi.org/10.1002/fld.621

[22] H. Wu, Y. He, G. Tang, W. Tao, Lattice Boltzmann simulation of flow in porous media on non-uniform grids, Prog. Comput. Fluid Dyn., 5(2005), 97-103. https://doi.org/10.1504/PCFD.2005.005821

[23] J. Yang, Z. Xie, Z. Ji, H. Meng, Real-time heat transfer model based on variable non-uniform grid for dynamic control of continuous casting billets, ISIJ Int., 54(2014), 328-335. https://doi.org/10.2355/isijinternational.54.328

[24] R. Webster, Algebraic multigrid and incompressible fluid flow, Int. J. Numer. Methods Fluids, 53(2007), 669-690. https://doi.org/10.1002/fld.1297

[25] L. Liu, G. Long, Z. Cen, A robust adaptive grid method for a nonlinear singularly perturbed differential equation with integral boundary condition, Numer. Algorithms, 83(2020), 719-739. https://doi.org/10.1007/s11075-019-00700-2

[26] G. Long, L. Liu, Z. Huang, Richardson extrapolation method on an adaptive grid for singularly perturbed Volterra integro-differential equations, Numer. Funct. Anal. Optim., 42(2021), 739-757. https://doi.org/10.1080/01630563.2021.1928698

[27] A. Gupta, A. Kaushik, M. Sharma, A higher-order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction-diffusion problems with robin-boundary conditions, Numer. Methods Partial Differ. Equations, 39(2023), 1220-1250. https://doi.org/10.1002/num.22931

[28] L. Liu, Y. Liang, J. Zhang, X. Bao, A robust adaptive grid method for singularly perturbed Burger-Huxley equations, Electron. Res. Arch., 28(2020), 1439. https://doi.org/10.3934/era.2020076

[29] Z. Mao, D. Luo, A robust adaptive grid method for first-order nonlinear singularly perturbed Fredholm integro-differential equations, Networks Heterogen. Media, 18(2023), 1006-1023. https://doi.org/10.3934/nhm.2023044

[30] L. Liu, Y. Liang, X. Bao, H. Fang, An efficient adaptive grid method for a system of singularly perturbed convection-diffusion problems with Robin boundary conditions, Adv. Differ. Equ., 2021(2021), 1-13. https://doi.org/10.1186/s13662-020-03166-y

[31] M. Sheikholeslami, A. Ghasemi, Solidification heat transfer of nanofluid in existence of thermal radiation by means of FEM, Int. J. Heat Mass Transfer., 123(2018), 418-431. https://doi.org/10.1016/j.ijheatmasstransfer.2018.02.095

[32] E. Abu-Nada, Application of nanofluids for heat transfer enhancement of separated flows encountered in a backward facing step, Int. J. Heat Fluid Flow, 29(2008), 242-249. https://doi.org/10.1016/j.ijheatfluidflow.2007.07.001

[33] B. Jin, Fractional calculus, in Fractional Differential Equations , Springer, Cham, (2021), 19-58. https://doi.org/10.1007/978-3-030-76043-4

[34] H. Schiessel, C. Friedrich, A. Blumen, Fractional powers of infinitesimal generators of semigroups, in Applications of Fractional Calculus in Physics (Ed. R. Hilfer) , World scientific, Singapore, (2000), 131-170. https://doi.org/10.1142/9789812817747

[35] H. Schiessel, R. Metzler, A. Blumen, T. F. Nonnenmacher, Generalized viscoelastic models: Their fractional equations with solutions, J. Phys. A: Math. Gen., 28(1995), 6567. https://doi.org/10.1088/0305-4470/28/23/012

[36] C. Friedrich, Relaxation and retardation functions of the maxwell model with fractional derivatives, Rheol. Acta, 30(1991), 151-158. https://doi.org/10.1007/BF01134604

[37] W. Tan, W. Pan, M. Xu, A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates, Int. J. Nonlinear Mech., 38(2003), 645-650. https://doi.org/10.1016/S0020-7462(01)00121-4

[38] M. Shen, L. Chen, M. Zhang, F. Liu, A renovated Buongiorno’s model for unsteady Sisko nanofluid with fractional Cattaneo heat flux, Int. J. Heat Mass Transfer., 126(2018), 277-286. https://doi.org/10.1016/j.ijheatmasstransfer.2018.05.131

[39] M. A. Ezzat, Thermoelectric MHD non-Newtonian fluid with fractional derivative heat transfer, Phys. B, 405(2010), 4188-4194. https://doi.org/10.1016/j.physb.2010.07.009

[40] Z. Wang, S. Vong, Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation, J. Comput. Phys., 277(2014), 1-15. https://doi.org/10.1016/j.jcp.2014.08.012

[41] M. Stynes, E. O Riordan, J. L. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55(2017), 1057-1079. https://doi.org/10.1137/16M1082329

[42] P. Roul, A robust adaptive moving mesh technique for a time-fractional reaction-diffusion model, Commun. Nonlinear Sci. Numer. Simul., 109(2022), 106290. https://doi.org/10.1016/j.cnsns.2022.106290

[43] J. Huang, Z. Cen, J. Zhao, An adaptive moving mesh method for a time-fractional Black-Scholes equation, Adv. Differ. Equ., 2019(2019), 516. https://doi.org/10.1186/s13662-019-2453-1

[44] Z. Cen, L. Liu, A. Xu, A second-order adaptive grid method for a nonlinear singularly perturbed problem with an integral boundary condition, J. Comput. Appl. Math., 385(2021), 113205. https://doi.org/10.1016/j.cam.2020.113205

[45] M. Cakir, Y. Ekinci, E. Cimen, A numerical approach for solving nonlinear Fredholm integro-differential equation with boundary layer, Comput. Appl. Math., 41(2022), 259. https://doi.org/10.1007/s40314-022-01933-z

[46] M. Cakir, B. Gunes, A fitted operator finite difference approximation for singularly perturbed Volterra-Fredholm integro-differential equations, Mathematics, 10(2022), 3560. https://doi.org/10.3390/math10193560

[47] S. Priyadarshana, A. Padhan, J. Mohapatra, Adaptive grid based moving mesh algorithms for singularly perturbed second-order Volterra integro differential equations, Quaest. Math., 48(2024), 1007-1028. https://doi.org/10.2989/16073606.2025.2462939

[48] N. Kopteva, M. Stynes, A robust adaptive method for a quasilinear one-dimensional convection-diffusion problem, SIAM J. Numer. Anal., 39(2001), 1446-1467. https://doi.org/10.1137/S003614290138471X

[49] W. S. Manebo, M. M. Woldaregay, T. G. Dinka, G. F. Duressa, A computational approach to solving a second-order singularly perturbed Fredholm integro-differential equation with discontinuous source term, Numer. Algor., 97(2024), 1415-1430. https://doi.org/10.1007/s11075-024-01756-5

[50] S. Jiang, J. Zhang, Q. Zhang, Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21(2015), 650-678. https://doi.org/10.4208/cicp.OA-2016-0136

[51] X. Gu, S. Wu, A parallel-in-time iterative algorithm for Volterra partial integro-differential problems with weakly singular kernel, J. Comput. Phys., 417(2020), 109576. https://doi.org/10.1016/j.jcp.2020.109576

[52] J. Buongiorno, Convective transport in nanofluids, J. Heat Transfer, 128(2006), 240-250. https://doi.org/10.1115/1.2150834

[53] F. Mebarek-Oudina, Preeti, A. S. Sabu, H. Vaidya, R. W. Lewis, S. Areekara, et al., Hydromagnetic flow of magnetite-water nanofluid utilizing adapted Buongiorno model, Int. J. Mod. Phys. B, 38(2024), 2450003. https://doi.org/10.1142/S0217979224500036