Anomalous diffusion in pattern-forming systems, and applications

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This project is devoted to development of a theory of pattern formation, nonlinear dynamics and transport in systems with anomalous diffusion. Applications to a number of significant problems, such as pattern-forming reaction-diffusion problems and drug delivery problems, will be investigated. Unlike regular diffusion, in anomalous diffusion the mean square displacement behaves as a power function of time. If the exponent is less than one the diffusion process is slower than normal diffusion (subdiffusion), and if the exponent is greater than one it is faster than normal (superdiffusion). Mathematical description of anomalous diffusion involves integro-differential operators which have to be derived from appropriate continuous time random walk models and leads to novel mathematical problems. Specifically, the investigators study(i) Pattern formation in growing domains, both for normal diffusion and anomalous diffusion, focusing on the singular perturbation case when some of the diffusion coefficients are asymptotically small;(ii) Turing pattern selection in reaction - anomalous diffusion systems, in particular, the stripes-spots selection;(iii) Drug delivery problems which include development of an approximate analytic theory of subdiffusive problems with moving free boundaries; study of models of bioerodible controlled drug delivery devices governed by subdiffusive transport of the chemicals, transdermal drug release in the presence of an electric field i.e. accompanied by iontophoresis and others.Controlled drug delivery has been attracting a great deal of attention in the medical community for years as an efficient way of providing treatment for a wide class of diseases. Various drug delivery devices are based on mass transfer of the given drug towards particular organs, in which either the mass transfer rate, or place, or both are prescribed according to certain medical protocols. Much progress has been achieved in the design and development of various controlled drug delivery systems, and many people routinely take medicine designed for controlled release. Mathematical modeling of drug delivery systems is very important since it can provide a better understanding and a quantitative description of the physical, chemical and biological processes governing the performance of the systems. On the basis of this description, better controlled drug delivery systems can be designed. There exists experimental evidence that drug diffusion toward the biological target is not normal but rather slow, so-called sub-diffusion, as the drug molecule has to diffuse through a very crowded environment. The investigators will study drug transport governed by sub-diffusion in order to obtain better understanding of drug delivery processes. In addition, the investigators will study problems of pattern formation in reaction-diffusion systems where subdiffusion, as well as superdiffusion are important. The latter is typical of some processes in plasmas, semiconductors, surface reactions and many others. Training of a PhD student through this research is an integral part of the project.