Phenomenological modelling of anomalous dynamical systems and flows

运输过程在本质上是基本的，从宏观的，作为流体表面的波，通过地质相中尺度的颗粒渗透，到微观尺度的DNA转录。在历史上，均方位移随时间线性增长的弥散过程(菲克扩散定律，1855)被指定为规则的。任何其他依赖被认为是一种反常现象。由于模型方程的极端困难，对异常输运的分析被回避了，因为建模方程是如此令人敬畏，以至于任何偏离时间线性色散的实验证据都被辩解为测量限制。在过去的50年里，在越来越多的应用中，测量技术的稳健性和关于异常过程的文件的积累，因此形成了一个跨学科的领域，需要使用高级数学。异常过程的无处不在要求对现象学模型进行分析，即系统足够普遍，可以在不对任何特定过程进行建模的情况下重现常见情况。这一研究方案是对这种类属性质的系统进行研究。*要研究的方程组是在现有的正则方程组的基础上构造的，其结构和性质已被很好地理解。然后，根据最近表现出异常行为的实验结果对这些方程进行修改。这些研究的显著结果是识别和解释正常现象和异常现象之间的主要差异，以支持实验结果。对这些范例的分析对于分类和理解发生在生物学、地质学、化学和其他自然科学中的复杂运输过程以及提高数学工具的完备性至关重要。这项研究计划将寻求揭示支持异常弥散过程的统一原则，从而在多个学科中推进知识。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。总体而言，这将增加加拿大数学界对这些主题的国际研究的参与。*作为一名跨学科的研究人员，我还致力于环境科学中违反经典解决方案的问题。项目包括为垃圾填埋井开发新的井口结构、其中的流动建模、垃圾填埋场管网中的气体流动理论、压力分布的控制和优化、大坝运行导致的河流生命迁移分析。学员将接触到与行业专业人士的直接合作，并在那种环境中获得宝贵的经验。与学术界合作，工业公司将有利于提高其员工的素质，并改进工程实践。

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.