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

Zhi Mao; Huafang Li; Xiaobing Bao; Leilei Wei; Libin Liu; Libo Feng; Zhi Mao; Huafang Li; Xiaobing Bao; Leilei Wei; Libin Liu; Libo Feng

doi:10.3934/nhm.2026019

Networks and Heterogeneous Media

2026, Volume 21, Issue 2: 402-425. doi: 10.3934/nhm.2026019

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Research article

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

1.
School of Data Science, Tongren University, Tongren 554300, China
2.
School of Mathematics and Statistics, Jishou University, Xiangxi 416100, China
3.
School of Big Data and Artificial Intelligence, Chizhou University, Chizhou 247000, China
4.
School of Mathematics and Statistics, Henan University of Technology, Zhengzhou 450001, China
5.
School of Mathematics and Statistics, Nanning Normal University, Nanning 530100, China
6.
School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001, Australia

Received: 28 January 2026 Revised: 09 March 2026 Accepted: 16 March 2026 Published: 24 March 2026

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.
Citation: Zhi Mao, Huafang Li, Xiaobing Bao, Leilei Wei, Libin Liu, Libo Feng. An efficient adaptive moving grid algorithm for time-fractional integrodifferential equations governing viscoelastic nanofluid dynamics[J]. Networks and Heterogeneous Media, 2026, 21(2): 402-425. doi: 10.3934/nhm.2026019

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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.

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