时空耦合分数阶非线性扩散模型的高精度数值求解及微观生物分子输运仿真
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聂佳磊
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江苏海洋大学，江苏连云港，222005
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摘要：生物微尺度环境中，受大分子缠结、空间受限及介质黏弹性特性影响，生物分子输运过程呈现显著的反常扩散特征，传统整数阶扩散方程难以精准刻画其含记忆效应、非局部特性的动力学演化规律。针对这一问题，本文以生物分子输运为数理应用载体，构建一类含非线性源项的时间 -空间分数阶非线性扩散方程，依托Caputo分数阶微分算子与 Riemann-Liouville积分算子完成方程规范化建模。结合有限差分法与预估 -校正迭代思想，改进传统分数阶扩散方程数值离散格式，有效解决经典算法求解非线性项时精度不足、收敛速度较慢的问题。通过严格的数理推导，完成改进数值格式的稳定性、收敛性证明，并推导误差阶数的定量关系。基于构建的数值算法开展多组参数仿真实验，分析分数阶阶数、非线性系数对生物分子输运时空演化规律的影响。仿真结果表明，改进算法具备更高的计算精度与稳定性，可精准表征生物分子反常输运的动力学特性，为微尺度生物分子输运的数理建模与数值预测提供可靠的理论支撑。
关健词：分数阶扩散方程；非线性源项；有限差分法；数值解法；稳定性分析；生物分子反常输运 |
High-precision Numerical Solution of Spatiotemporally Coupled Fractional Nonlinear Diffusion Model and Simulation of Microscopic iomolecular Transport
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Jialei Nie
Jiangsu Ocean University, Lianyungang Jiangsu 222005, China
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Abstract：In microscale biological environments, biomolecular transport exhibits prominent anomalous diffusion behaviors due to macromolecular entanglement, spatial confinement and medium viscoelasticity. Traditional integer-order diffusion equations fail to accurately describe the dynamic evolution laws with memory effects and non-local characteristics. Taking biomolecular transport as a mathematical and physical application carrier, this paper constructs a time-space fractional nonlinear diffusion equation with nonlinear source terms and establishes a standardized model based on Caputo fractional differential operators and Riemann-Liouville integral operators. Combined with the finite difference method and predictor-corrector iteration theory, an improved numerical discretization scheme is proposed to solve the problems of low accuracy and slow convergence in traditional nonlinear term calculation. Strict mathematical derivations are performed to verify the stability and convergence of the improved scheme, and the quantitative relationship of error orders is deduced. A series of parametric numerical simulations are carried out to analyze the influences of fractional orders and nonlinear coefficients on the spatiotemporal evolution of biomolecular transport. The numerical results demonstrate that the proposed algorithm achieves higher computational accuracy and stability, which can precisely characterize the dynamic properties of biomolecular anomalous diffusion and provide a reliable theoretical framework for mathematical modeling and numerical prediction of microscale biomolecular transport.
Keywords : fractional diffusion equation; nonlinear source term; finite difference method; numerical solution; stability analysis; biomolecular anomalous diffusion
参考文献
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