Anomalous diffusion and new relationships between fractals and fractional integrals

R. Nigmatullin教授

俄罗斯喀山联邦大学 Kazan (Volga region) Federal University

王春雷

2014-04-16 15:40

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In this lecture we want to show how to derive the anomalous diffusion equation containing the fractional derivative. But we want to show also how to find new relationships existing between fractional integral and fractal object studied. Many specialists working in the field of the fractional calculus and its applications simply replace the integer differentiation and integration operators by their non-integer generalizations and do not give any serious justifications for this replacement. What kind of "physics" lies in this mathematical replacement? Is it possible to justify this replacement or not for the given type of fractal and find the proper physical meaning? These or other similar questions are not discussed properly in the current papers related to this subject. In this paper new approach that relates to the procedure of the averaging of smooth functions on a fractal set with fractional integrals is suggested. This approach contains the previous one as a partial case and gives new solutions when the microscopic function entering into the structural-factor does not have finite value at N >> 1 (N is number of self-similar objects). The approach was tested on the spatial Cantor set having M bars with different symmetry. There are cases when the averaging procedure leads to the power-law exponent that does not coincide with the fractal dimension of the self-similar object averaged. These new results will help researches to understand more clearly the meaning of the fractional integral. The limits of applicability of this approach and class of fractal are specified.

尼格马图林教授，科学博士, 俄罗斯喀山联邦大学物理研究所 理论物理系。尼格马图林教授出生于1947年俄罗斯联邦鞑靼斯坦共和国的首府喀山。于1973年在喀山州立大学获得博士学位，然后于1993年在同一所大学获得物理和数学博士学位。 现在是喀山联邦大学（伏尔加区）的物理数学全职教授。1982-1983年期间，在英国Jonscher教授的实验室从事电介质物理研究。从1990年起，成为国际电介质协会的成员。1998年他和法国同事A. Le Mehaute 博士和L. Nivanen博士一起出版了关于分形几何和分数阶微积分的专著。目前的研究兴趣是电介质物理，发展S/N处理分析的新方法以及分数阶微积分和分形几何在不同物理领域的应用。他已经发表了220余篇论文，SCI引用超过1800次，H-因子17。是2012年南京召开第五届分数阶微积分及其应用国际会议的plenary speaker之一，http://em.hhu.edu.cn/fda12/Committees.html