论文标题
与计算机代数嵌入严格，准确地均匀均匀均匀的多尺度多尺度异质PDE
Embed to rigorously and accurately homogenise quasi-periodic multi-scale heterogeneous PDEs, with computer algebra
论文作者
论文摘要
对于微观的异质PDE，本文进一步开发了其宏观数学/渐近均质化的新颖理论和方法。本文特别涵盖了准周期异质性具有有限尺度分离的情况：不需要比例分离限制。动力学系统理论将同质性的集合的慢速分歧归因于异质性的所有相移的慢速分歧。根据准周期异质性内的任何感知尺度分离，可以在一个步骤或两个顺序的步骤中进行均质化：结果是等效的。该理论不仅确保了我们的均匀化的存在和出现，还提供了一种实用的系统方法，可以将均化为任何指定的顺序。对于一类异质性，我们表明宏观均质化可能有效到长度，而长度仅是微观异质性的两倍！该方法提供了一种新的严格而灵活的均质化方法，该方法可能还提供了正确的初始和边界条件，强迫和控制的处理以及不确定性的分析。
For microscale heterogeneous PDEs, this article further develops novel theory and methodology for their macroscale mathematical/asymptotic homogenization. This article specifically encompasses the case of quasi-periodic heterogeneity with finite scale separation: no scale separation limit is required. Dynamical systems theory frames the homogenization as a slow manifold of the ensemble of all phase-shifts of the heterogeneity. Depending upon any perceived scale separation within the quasi-periodic heterogeneity, the homogenization may be done in either one step, or two sequential steps: the results are equivalent. The theory not only assures us of the existence and emergence of the homogenization, it also provides a practical systematic method to construct the homogenization to any specified order. For a class of heterogeneities, we show that the macroscale homogenization is potentially valid down to lengths which are just twice that of the microscale heterogeneity! This methodology provides a new rigorous and flexible approach to homogenization that potentially also provides correct initial and boundary conditions, treatment of forcing and control, and analysis of uncertainty.
