由粗糙噪声驱动的随机时间-空间分数电缆方程的近似
Approximation for stochastic time-space fractional cable equations driven by rough noise
AI总结：
本文提出了一种由粗糙噪声驱动的随机时间-空间分数电缆方程的数值近似方法，通过算子理论和Wong-Zakai近似分析了解的正则性和收敛性，并采用谱伽辽金和后向欧拉卷积四阶方法进行数值计算。
摘要：
时间-空间分数电缆方程源于将广义分数欧姆定律扩展以建模异常扩散过程。本文开发并分析了由粗糙噪声驱动的随机非线性时间-空间分数电缆方程的数值近似方法。该模型包含时间领域的两个非局部项和空间领域的一个非局部项。通过算子理论方法，我们建立了解的存在性、唯一性和正则性。我们还通过Wong-Zakai近似对粗糙噪声进行正则化，从而获得正则化方程的收敛性结果。数值方案在空间上采用标准谱伽辽金方法进行近似，在时间上采用后向欧拉卷积四阶方法进行近似。之后建立了误差估计。
英文摘要：
The time-space fractional cable equation arises from extending the generalized fractional Ohm's law to model anomalous diffusion processes. In this paper, we develop and analyze a numerical approximation for stochastic nonlinear time-space fractional cable equation driven by rough noise. The model features both two nonlocal terms in time and one in space. By an operator theoretic approach, we establish the existence, uniqueness and regularity of solutions. To regularize the rough noise, we introduce a spectral Wong-Zakai approximation and derive its convergence rate. For the fully discrete scheme, we employ the spectral Galerkin method for spatial discretization and the backward Euler convolution quadrature for temporal discretization, and we derive error estimates under explicit parameter conditions. Finally, numerical experiments are presented to validate the theoretical convergence rates.
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