基于GPU的几类分数阶微分方程的并行算法研究及其实现

As fractional order differential equations have been widely used in more and more fields, the efficient and fast numerical methods for fractional differential equations have become particularly important. Usually, the numerical methods for fractional differential equations have huge computational loads because of memory effect of the fractional order operators. In this project, we will focus on solving this problem, and the detailed contents include the following two parts: (1) Through constructing the CPU-GPU heterogeneous system, the design and implementation of efficient parallel algorithms for the system of fractional order ordinary differential system will be studied by using three kinds of techniques; (2) Combined with domain decomposition method, spatial discretization of the time fractional diffusion equation will lead to a system of fractional ordinarily differential equations, which can be solved by the efficient parallel algorithms. Hence, it is possible to obtain the efficient parallel algorithms for the time fractional diffusion equation based on the CPU-GPU heterogeneous system. The outcomes of this project will effectively promote the rapid development of numerical methods for fractional differential equations. And these results will have important scientific significance and practical significance.

随着分数阶微分方程在众多的领域内得到广泛应用，高效快速的数值求解变得尤为重要。由于分数阶导数具有记忆性，数值求解分数阶微分方程会面临很大的计算量。本课题将致力于此方面的研究，期待取得重要进展。主要内容包括：（1）搭建GPU和CPU异构计算系统，利用基于方法分割、系统分割、时间分割三种研究策略设计分数阶常微分系统的高效并行算法并实现；（2）结合区域分解等技术对时间分数阶扩散方程进行空间离散，得到分数阶常微分系统，进一步结合分数阶常微分系统的并行算法，获得基于GPU的时间分数阶扩散方程的高效并行算法。本项目获得的成果将能有助于推动分数阶微分方程并行数值算法的发展，具有重要的科学意义和实践意义。

随着分数阶微分方程模型更深入地应用在很多不同的领域，分数阶延迟微分模型也不断地获得关注。 本项目研究了几类分数阶中立型延迟微分方程解的依赖性以及渐近稳定性，并通过数值实验验证了理论结果，同时也针对一类含两个分数阶导数的非线性分数阶微分方程设计了高阶数值算法，并给出了相应的理论分析以及数值实验。进一步，为了提高精度，研究了几类分数阶延迟微分方程的高阶Jacobi配置方法，并给出了相应的理论分析。针对含延迟项的分数阶扩散微分方程设计了高效的数值算法，理论结果和数值结果都表明设计的算法是非常有效的。另外，针对几类非线性延迟系统和脉冲切换系统，给出两个新的系统有限时间稳定的充分条件。进一步，完成了对非线性分数阶脉冲切换系统的渐近稳定性研究。本项目获得的结果能为分数阶延迟微分模型更好地应用在各个领域打下了一定的基础，也能为未来分数阶泛函微分方程（包括分数阶常微分延迟方程和分数阶偏微分延迟方程等）的数值解的研究提供一些参考。