论文标题
时间多尺度方法及其对分数微分方程系统的分析
A temporal multiscale method and its analysis for a system of fractional differential equations
论文作者
论文摘要
在本文中，研究了具有多个时间尺度的分数普通微分方程的非线性系统。我们对解决方案的有效长期计算感兴趣。主要的挑战是如何以较低的计算成本获得耦合问题的解决方案。我们对非线性系统进行多尺度方法分析，该方法是快速系统具有周期性施加力，而慢速方程式包含分数衍生物，以简化动脉粥样硬化并具有斑块生长。得出局部周期方程以近似原始系统，并给出了误差估计。然后，设计了有限的差异方法，以近似原始问题和近似问题。我们构建了四个示例，包括三个具有精确解决方案的示例，以及一个遵循原始问题设置的一个示例，以测试所提出方法的准确性和计算效率。据观察，与完全分离的仿真相比，计算时间大大减少了，多尺度方法的性能非常好。时间尺度分离越大，多尺度方法的有效性就越大。
In this paper, a nonlinear system of fractional ordinary differential equations with multiple scales in time is investigated. We are interested in the effective long-term computation of the solution. The main challenge is how to obtain the solution of the coupled problem at a lower computational cost. We analysize a multiscale method for the nonlinear system where the fast system has a periodic applied force and the slow equation contains fractional derivatives as a simplication of the atherosclerosis with a plaque growth. A local periodic equation is derived to approximate the original system and the error estimates are given. Then a finite difference method is designed to approximate the original and the approximate problems. We construct four examples, including three with exact solutions and one following the original problem setting, to test the accuracy and computational efficiency of the proposed method. It is observed that, the computational time is very much reduced and the multiscale method performs very well in comparison to fully resolved simulation for the case of small time scale separation. The larger the time scale separation is, the more effective the multiscale method is.
