Modeling, Analysis, and Computation for Problems from Biology and Industry

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My long-term research objective is to develop evolution differential equation based mathematical models and numerical methods for problems motivated by industrial applications and other scientific disciplines. As the source of these problems evolves, my research program maintains a balanced approach between continuing existing research topics and developing new research directions, which is reflected in the current proposal. This proposal aims at analyzing and developing new mathematical models, methods and algorithms for three classes of problems with complex structures: (i) multi-scale systems of reaction- and electro-diffusion equations applied to physiology and pathology, (ii) transport equations applied to coupled mass transfer and fluid flow problems, and (iii) Hamilton-Jacobi-Bellman equation applied to optimal control in finance. 1. Spreading depression (SD) of brain cell electric activities is a relatively new area for me, but also an under-developed area from modeling point of view. - Cell membrane dynamics (1A) and local field potential (1B) during SD are of theoretical interest. - Asymptotic analysis (1C) of Poisson-Nernst-Planck (PNP) system and homogenization (1D) of PNP system are challenging problems with multiple time and spatial scales. - The objective of ion channel model and computation (1E) is to derive approximate flux formula that are more accurate than the current Hodgkin-Huxley and Goldman-Hodgkin-Katz models. 2. Extensional flow of polymer fibers is a new direction. - A combined initial- and boundary-value problem will be theoretically analyzed, which has not yet been investigated (2A). - Stability analysis will be carried out for the effects of temperature and chemical reaction (2B). 3. Extension of the immersed boundary framework to mass transfer problems is one of my ongoing research topics. - A method will be developed for interface permeable to both solvent and solute (3A: Navier-Stokes equations coupled with mass transfer equation). - This method will be extended to ion transport coupled with fluid flows at immersed interfaces (3B: Navier-Stokes with PNP equations). 4. Risk modeling and control in personal finance and retirement planning is a long-term research topic of mine. - Product allocation (4A) & taxation and borrowing constraints (4B) are technically challenging compared to the typical optimal control lifecycle framework but can still be formulated using the Hamilton-Jacobi-Bellman equation. - Incorporating income/health shocks into a lifecycle model (4C) involves modelling and is more open ended. Nevertheless, we hope to make some progress by starting with simple jump processes such as Poisson one for shocks and a recursive utility function that separates risk aversion and intertemporal substitution.