非线性乘性分数噪声驱动的分数阶扩散波系统的数值算法与动力学行为

Fractional diffusion wave systems are derived from many fields of science, such as computational biology, acoustics, finance and so on. However, the systems discussed in scientific research are often affected by a number of uncertain (random) factors from the external environment. As an important class of colored noises, the fractional Brownian motion-type noise appears naturally in the modeling of many physical and social phenomena. At present, few researches have been done on numerical algorithms and dynamic behaviors of stochastic time-space fractional diffusion wave systems driven by nonlinear multiplicative fractional noise. Especially, the results about the Hurst parameter of fractional Brownian motion H<1/2 are even less. This project aims to study the numerical algorithms and dynamic behaviors of stochastic time-space fractional diffusion wave systems driven by nonlinear multiplicative fractional noise. The main contents are as follows: (1) the long-time efficient and robust numerical algorithms for the time-space fractional wave system driven by nonlinear multiplicative fractional noise (H<1/2) are designed and its dynamic behavior is discussed; (2) the numerical algorithms and dynamic behavior of the time-space fractional diffusion system driven by nonlinear multiplicative fractional noise (0<H<1) are studied; (3) the numerical algorithms of the time-space fractional diffusion wave system with delay and nonlinear multiplicative fractional noise (0<H<1) are explored and a description of its dynamic behavior is given.

分数阶扩散波系统源于计算生物学、声学、金融等科学领域。然而，在科学研究中所讨论的系统往往会受到来自外部环境的不确定（随机）因素的影响。分数布朗运动型噪声作为一类重要的有色噪声，在许多物理和社会现象的建模中自然出现。目前，有关非线性乘性分数噪声驱动的随机时空分数阶扩散波系统的数值算法和动力学行为的研究工作较为稀少，特别是分数布朗运动的Hurst参数H<1/2时的相关结果更是甚少。本项目拟对非线性乘性分数噪声驱动的随机时空分数阶扩散波系统的数值算法和动力学行为展开研究。具体内容如下：（1）为非线性乘性分数噪声（H<1/2）驱动的时空分数阶波系统设计长时间上高效稳健的数值算法并探讨其动力学行为；（2）研究非线性乘性分数噪声（0<H<1）驱动的时空分数阶扩散系统的数值算法与其动力学性态；（3）探究含时滞和非线性乘性分数噪声（0<H<1）的时空分数阶扩散波系统的数值算法并给出其动力学行为描述。

分数阶扩散波系统源于计算生物学、声学、金融等科学领域。然而，在科学研究中所讨论的系统往往会受到来自外部环境的不确定（随机）因素的影响。分数布朗运动型噪声作为一类重要的有色噪声，在许多物理和社会现象的建模中自然出现。本项目针对非线性乘性分数高斯噪声驱动的随机时空分数阶扩散波系统的数值算法和动力学行为展开了研究。我们深入研究了乘性白噪声驱动的具有无界时滞的随机分数阶偏泛函微分方程解的存在性，得到了在均方拓扑下非局部系统紧的极限集以及紧的、极小的向前吸引集；进一步利用随机积分项的非紧性测度的新结果和多值映射的不动点定理，研究了具有非线性乘性噪声和分数高斯噪声的时间分数阶随机时滞发展包含问题mild解的全局存在性，其中解决了计算随机积分项非紧性测度的难题；同时，我们研究了由无穷维乘性白噪声和分数阶高斯噪声驱动的半线性随机时空分数波方程的Galerkin有限元逼近法，提供了一个完整的解理论，包括存在性、唯一性和正则性。进一步得到了正则化随机分数阶波方程的完整正则性理论，对空间算子采用标准有限元近似，并建立了建模误差和正则化问题解的近似误差的均方先验估计，其中我们克服了非线性乘性分数高斯噪声所带来的困难。此外，我们还研究了不可压缩Navier-Stokes方程的快速数值算法，为后续研究随机Navier-Stokes方程打好基础。