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学者姓名：张梦晨
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Abstract ：
Based on the mechanism of physical memory effects, an improved fractional Zener constitutive model is proposed to describe the boundary layer characteristics of viscoelastic fluids in magnetohydrodynamic (MHD) environments. The model employs a mixed-order structure where instantaneous elastic response maintains integer-order characteristics, while delayed elastic response and viscous flow introduce fractional operators. This design simultaneously captures the stress relaxation and strain lag characteristics of the fluid. The established model is numerically solved using the finite difference method together with a fast algorithm, and the regulatory mechanisms on the fluid boundary layer characteristics are investigated. The research reveals that the proposed model achieves more comprehensive characterization of boundary layer behavior compared to conventional models, while exhibiting unique saturation effects in magnetic field regulation. The fractional parameter α governs stress relaxation, with larger values enhancing viscous behavior and producing thicker boundary layers. Parameter β controls strain memory intensity, promoting greater flow mobility. Rheologically, a larger stress relaxation time λ1 enhances elastic character and compacts boundary layers, while an increased strain lag time λ2 extends microstructural adjustment time, manifesting as thicker boundary layers. This study establishes correlations between model parameters and physical phenomena, providing a theoretical framework for understanding MHD viscoelastic fluid boundary layer behavior. © 2025 Author(s).
Keyword ：
Boundary layer flow Boundary layer flow Boundary layers Boundary layers Finite difference method Finite difference method Magnetohydrodynamics Magnetohydrodynamics Relaxation time Relaxation time Viscoelasticity Viscoelasticity
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| GB/T 7714 | Shen, Ming , Zhou, Zi , Chen, Hui et al. A novel physically motivated Zener model for magnetohydrodynamics viscoelastic fluids with differential-integral fractional operator [J]. | Physics of Fluids , 2025 , 37 (7) . |
| MLA | Shen, Ming et al. "A novel physically motivated Zener model for magnetohydrodynamics viscoelastic fluids with differential-integral fractional operator" . | Physics of Fluids 37 . 7 (2025) . |
| APA | Shen, Ming , Zhou, Zi , Chen, Hui , Fang, Hui , Zhang, Mengchen . A novel physically motivated Zener model for magnetohydrodynamics viscoelastic fluids with differential-integral fractional operator . | Physics of Fluids , 2025 , 37 (7) . |
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The purpose of this research is to establish the generalised fractional Bloch-Torrey equation for better simulating anomalous diffusion in heterogeneous biological tissues. The introduction of the distributed-order time fractional derivative allows for an improved interpretation of the complex diffusion behaviours with multi-scale effects. The use of variable coefficients in the model increases its applicability for describing the spatial heterogeneity evident in the cellular structures. The proposed distributed-order time and space fractional Bloch-Torrey equation is discretised in time and space by the ������2-1 ������ formula and the finite element method, respectively. The stability and convergence analyses of these numerical methods are provided. To further improve the computational efficiency, a reduced-order extrapolation scheme is developed. We verify the effectiveness of the proposed methods by numerical examples. Moreover, the coupled fractional dynamic system solution behaviour is explored on a human brain-like domain divided into the white matter and grey matter regions. Compared with the model having constant coefficients, solution behaviours suggest that variable diffusion coefficients offer an effective way to differentiate the distinct diffusion phenomena evolving in different tissue micro-environments. Furthermore, to evaluate the impacts of the weight function in the distributed-order operator, we choose three types of beta distributions with the same mean but different values of the variance. The results indicate that the larger value of the variance leads to a more remarkable fluctuation and a slower decay of the transverse magnetisation. This generalised distributed-order fractional model may provide further insights into capturing anomalous diffusion in heterogeneous media.
Keyword ：
Distributed-order fractional operator Distributed-order fractional operator Finite element method Finite element method Fractional Bloch-Torrey equation Fractional Bloch-Torrey equation Variable coefficients Variable coefficients
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| GB/T 7714 | Zhang, Mengchen , Liu, Fawang , Turner, Ian W. et al. Numerical simulation of the distributed-order time-space fractional Bloch-Torrey equation with variable coefficients [J]. | APPLIED MATHEMATICAL MODELLING , 2024 , 129 : 169-190 . |
| MLA | Zhang, Mengchen et al. "Numerical simulation of the distributed-order time-space fractional Bloch-Torrey equation with variable coefficients" . | APPLIED MATHEMATICAL MODELLING 129 (2024) : 169-190 . |
| APA | Zhang, Mengchen , Liu, Fawang , Turner, Ian W. , Anh, Vo V. . Numerical simulation of the distributed-order time-space fractional Bloch-Torrey equation with variable coefficients . | APPLIED MATHEMATICAL MODELLING , 2024 , 129 , 169-190 . |
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This study explores the 2D stretching flow of a hybrid nanofluid over a curved surface influenced by a magnetic field and reactions. A steady laminar flow model is created with curvilinear coordinates, considering thermal radiation, suction, and magnetic boundary conditions. The nanofluid is made of water with copper and MWCNTs as nanoparticles. The equations are transformed into nonlinear ODEs and solved numerically. The model’s accuracy is confirmed by comparing it with published data. Results show that fluid velocity increases, temperature decreases, and concentration increases with the curvature radius parameter. The hybrid nanofluid is more sensitive to magnetic field changes in velocity, while the nanofluid is more sensitive to magnetic boundary coefficient changes. These insights can optimize heat and mass transfer in industrial processes like chemical reactors and wastewater treatment.
Keyword ：
Hybrid Nanofluids Hybrid Nanofluids Improved Shooting Method Improved Shooting Method Induced Magnetic Field Induced Magnetic Field Stretching Curved Surface Stretching Curved Surface
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| GB/T 7714 | Ming Shen , Yunhua Zheng , Yihong Liu et al. Hybrid Nanofluid Flow over a Stretching Curved Surface with Induced Magnetic Field and Homogeneous-Heterogeneous Reactions [J]. | Journal of Applied Mathematics and Physics , 2024 , 12 (10) : 3638-3654 . |
| MLA | Ming Shen et al. "Hybrid Nanofluid Flow over a Stretching Curved Surface with Induced Magnetic Field and Homogeneous-Heterogeneous Reactions" . | Journal of Applied Mathematics and Physics 12 . 10 (2024) : 3638-3654 . |
| APA | Ming Shen , Yunhua Zheng , Yihong Liu , Hui Chen , Mengchen Zhang . Hybrid Nanofluid Flow over a Stretching Curved Surface with Induced Magnetic Field and Homogeneous-Heterogeneous Reactions . | Journal of Applied Mathematics and Physics , 2024 , 12 (10) , 3638-3654 . |
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The pioneering work in finance by Black, Scholes and Merton during the 1970s led to the emergence of the Black-Scholes (B-S) equation, which offers a concise and transparent formula for determining the theoretical price of an option. The establishment of the B-S equation, however, relies on a set of rigorous assumptions that give rise to several limitations. The non-local property of the fractional derivative (FD) and the identification of fractal characteristics in financial markets have paved the way for the introduction and rapid development of fractional calculus in finance. In comparison to the classical B-S equation, the fractional B-S equations (FBSEs) offer a more flexible representation of market behavior by incorporating long-range dependence, heavy-tailed and leptokurtic distributions, as well as multifractality. This enables better modeling of extreme events and complex market phenomena, The fractional B-S equations can more accurately depict the price fluctuations in actual financial markets, thereby providing a more reliable basis for derivative pricing and risk management. This paper aims to offer a comprehensive review of various FBSEs for pricing European options, including associated solution techniques. It contributes to a deeper understanding of financial model development and its practical implications, thereby assisting researchers in making informed decisions about the most suitable approach for their needs.
Keyword ：
analytic solution analytic solution European option European option fractional Black-Scholes equation fractional Black-Scholes equation fractional derivative fractional derivative numerical simulation numerical simulation
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| GB/T 7714 | Zhang, Hongmei , Zhang, Mengchen , Liu, Fawang et al. Review of the Fractional Black-Scholes Equations and Their Solution Techniques [J]. | FRACTAL AND FRACTIONAL , 2024 , 8 (2) . |
| MLA | Zhang, Hongmei et al. "Review of the Fractional Black-Scholes Equations and Their Solution Techniques" . | FRACTAL AND FRACTIONAL 8 . 2 (2024) . |
| APA | Zhang, Hongmei , Zhang, Mengchen , Liu, Fawang , Shen, Ming . Review of the Fractional Black-Scholes Equations and Their Solution Techniques . | FRACTAL AND FRACTIONAL , 2024 , 8 (2) . |
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