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（High-precision Numerical Solution of Spatiotemporally Coupled Fractional Nonlinear Diffusion Model and Simulation of Microscopic iomolecular Transport）
TL;DR
An improved numerical discretization scheme is proposed to solve the problems of low accuracy and slow convergence in traditional nonlinear term calculation and provide a reliable theoretical framework for mathematical modeling and numerical prediction of microscale biomolecular transport.
Abstract
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.