Fractional calculus, fractal geometry, and stochastic processes.

摘要

Fractional calculus studies integrals and derivatives of fractional order. It has a long history, yet still remains a purely theoretical study lacking broad applications. The goal of this thesis is to help change this situation by establishing the links of fractional calculus to other concept in mathematics and physics.; First the classical definitions of fractional derivatives are surveyed and their common characteristics and distinctions are analyzed. Then a new definition of fractional derivative is proposed. Unlike the traditional fractional derivatives, this new definition retains the local property of integral order derivatives and can measure the smoothness of fractal dimensional curves.; Next we focus on fractional differential equations. The fractional ordinary differential equation is reviewed and the Maple package FracCalc is developed to help solve a particular class of fractional differential equations, as well as calculate fractional integrals and derivatives. The new type of fractional partial differential equation is proposed and its fundamental properties are investigated using Fourier analysis and operator semigroup theory.; The rest of the thesis is devoted to the connections between fractional calculus and stochastic processes, fractal geometry and physics. First the solutions to the fractional differential equations are expressed in terms of stable distributions and a class of fractional differential equations is formulated which generates all stable distributions. With the help of fractional derivatives some new expressions about the stable densities are found. A random walk is found whose macroscopic behavior can be represented by the fractional diffusion equation. Based on this random walk a new computer model for fractal growth---fractional diffusion limited aggregation (FDLA) is designed. A comparative study of two types of fractional diffusion equations and the telegrapher's equation shows that fractional diffusion equations have a unique characteristic---the entropy production paradoxically increases as the equation changes from being irreversible to being reversible. This discovery enlightens our understanding of the difficult coexistence of stochastic and deterministic behaviors.