嵌入到严格和准确地均匀准周期多尺度异质偏微分方程，与计算机代数
Embed to rigorously and accurately homogenise quasi-periodic multi-scale heterogeneous PDEs, with computer algebra
摘要：
对于微尺度非均质偏微分方程，本文进一步发展了新的理论和方法，其宏观尺度的数学/渐近均匀化。本文特别包括有限尺度分离的准周期非均匀性的情况：不需要尺度分离限制。动力系统理论将均匀化定义为非均匀性所有相移的集合的慢流形。根据准周期不均匀性内的任何感知尺度分离，均匀化可以在一个步骤或两个连续步骤中完成：结果是等效的。这一理论不仅为我们提供了均匀化的存在和产生的保证，而且为我们提供了一种构造任意阶均匀化的实用的系统方法。对于一类的异质性，我们表明，宏观均匀化是潜在的有效的长度，这只是微观异质性的两倍！这种方法提供了一种新的严格和灵活的方法来均匀化，也可能提供正确的初始和边界条件，强迫和控制的治疗，以及不确定性分析。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
