Phenomenological modelling of anomalous dynamical systems and flows

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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.