Phenomenological modelling of anomalous dynamical systems and flows

传输过程在自然界中是基本的，从宏观的，作为流体表面上的波，通过地质相中的中间尺度的颗粒，到微观尺度的DNA转录。在历史上，均方位移随时间线性增长（菲克扩散定律，1855年）的色散过程被指定为规则的。任何其他依赖都被视为异常。由于建模方程的极端困难，异常输运的分析被回避，因为建模方程是如此强大，以至于任何偏离时间线性色散的实验证据都被认为是测量限制。在过去的50年中，测量技术的合理性和越来越多的应用程序中异常过程的文献积累，从而创建了一个跨学科的领域，需要使用高等数学。异常过程的普遍性要求分析现象学模型，即足够普遍的系统，以重现常见的事件，而无需模拟任何特定的过程。这项研究方案是对这种一般性质的系统进行研究。** 要研究的方程组是基于现有的规则对应物构建的，其结构和性质在很大程度上已经了解。然后，根据最近的实验结果（显示异常行为）对所述方程进行雅阁。这些研究的突出成果是识别和解释规则和异常现象之间的主要差异，以支持实验结果。分析这些范例对于分类和理解生物学、地质学、化学和其他自然科学中发生的复杂输运过程以及促进数学工具的完备性至关重要。该研究计划将寻求揭示支撑异常分散过程的统一原则，从而推动多学科的知识。学员将掌握广泛的分析和可视化技术，并获得跨学科合作的经验。总的来说，这将增加加拿大数学界对这些主题的国际研究的参与。作为一名跨学科研究人员，我也致力于解决环境科学中的经典解决方案。项目包括为填埋场威尔斯井开发新的井口结构，模拟其中的流动;填埋场管网中的气体流动理论，控制和优化压力分布;分析大坝运行引起的河流生命迁移。学员将接触到与行业专业人士的直接合作，并在该环境中获得宝贵的经验。与学术界合作，工业公司将有利于其人员的素质和改善工程实践。

Transport processes are fundamental in nature, from macroscopic, as a wave on the surface of a fluid, through an intermediate scale of particles percolating in geologic facies, and to the miscroscopic scale of DNA transcription. Historically dispersion processes, wherein the mean square displacement grew linearly in time (Fick's diffusion law, 1855) were designated as regular. Any other dependence was deemed an anomaly. Analysis of anomalous transport was eschewed due to extreme difficulty of the modelling equations, so formidable that any experimental evidence of deviation from time-linear dispersion was excused as a measurement limitation. In the last 50 years soundness of measuring techniques and accrual of documentation on anomalous processes in ever growing number of applications thus created an inter-disciplinary field, requiring use of advanced mathematics. The ubiquity of anomalous processes calls for analysis of phenomenological models, i.e. systems general enough to reproduce common occurrences without modelling any particular process. This research programme is to undertake studies of systems of such generic nature. ******The systems of equations to be investigated are constructed based on existing regular counterparts, whose structure and properties are largely understood. Said equations are then to be modified in accord with recent experimental results evincing anomalous behaviour. The salient outcomes of such studies are identification and explication of principal differences between regular and anomalous phenomena, in support of experimental findings. Analysis of such paradigms is crucial to the classification and understanding of complex transport processes occurring in biology, geology, chemistry and other natural sciences, as well as to the advancement of completeness of mathematical tools. This research programme will seek to reveal unifying principles underpinning anomalous dispersion processes, thus forwarding the knowledge in multiple disciplines. Trainees will master a wide spectrum of analysis and visualisation techniques, and obtain experience in inter-disciplinary collaborations. Overall this will increase the Canadian mathematical community' participation in the internationally based research on these topics.******As an inter-disciplinary researcher I also work on problems defying classic solutions in environmental sciences. Projects include development of novel wellhead configuration for landfill wells, modelling of flow therein; theory of gas flow in a landfill pipe network, control and optimisation of pressure distribution; analysis of river life migration due to dam operation. Trainees will be exposed to direct cooperation with industry professionals and gain valuable experience in that environment. Working with academics, industrial companies will benefit the quality of their personnel and improve engineering practices.