Anomalous diffusion in pattern-forming systems, and applications

这个项目致力于发展具有反常扩散的系统中的图案形成、非线性动力学和输运理论。一些重要问题的应用，如模式形成反应扩散问题和药物输送问题，将被研究。与常规扩散不同，在反常扩散中，均方位移表现为时间的幂函数。如果指数小于1，则扩散过程比正常扩散(子扩散)慢，如果指数大于1，则比正常(超扩散)快。反常扩散的数学描述涉及积分-微分算子，这些算子必须从适当的连续时间随机游走模型中得到，并导致新的数学问题。具体地说，研究人员研究了(I)增长区域中的图案形成，包括正常扩散和反常扩散，重点研究了当某些扩散系数渐近小时的奇异摄动情况；(Ii)反应反常扩散系统中的图灵模式选择，特别是条纹-斑点选择；(Iii)药物输送问题，其中包括发展具有移动自由边界的次扩散问题的近似解析理论；化学物质的亚扩散传输、电场作用下的透皮给药、离子导入等控制下的生物可降解药物输送装置模型的研究。多年来，可控药物输送作为一种治疗多种疾病的有效方法在医学界引起了极大的关注。各种药物输送装置是基于给定药物向特定器官的质量传递，其中质量传递速率或位置或两者都是根据特定的医学方案规定的。在设计和开发各种受控给药系统方面取得了很大进展，许多人经常服用为控制释放而设计的药物。药物输送系统的数学模型非常重要，因为它可以更好地理解和定量描述控制系统性能的物理、化学和生物过程。在此描述的基础上，可以设计更好的受控药物输送系统。有实验证据表明，药物向生物靶标的扩散不是正常的，而是相当缓慢的，即所谓的次扩散，因为药物分子必须在非常拥挤的环境中扩散。为了更好地了解药物的传递过程，研究人员将研究由次扩散控制的药物转运。此外，研究人员将研究反应扩散系统中的图案形成问题，在反应扩散系统中，次扩散和超扩散都很重要。后者是等离子体、半导体、表面反应和其他许多过程中的典型过程。通过这项研究培训一名博士生是该项目不可或缺的一部分。

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.