一维连续函数的Riemann-Liouville分数阶微积分

Riemann-Liouville fractional calculus of 1-dimensional continuous functions

本文主要讨论闭区间上一维连续函数的Riemann-Liouville分数阶微积分.首先,证明一维连续有界变差函数的任意阶Riemann-Liouville分数阶积分仍然是连续有界变差函数.其次,给出无界变差点的定义并构造一个含有无界变差点的一维连续无界变差函数.同时证明该无界变差函数的任意阶Riemann-Liouville分数阶积分的分形维数为1.最后,证明对于任意具有有限个无界变差点的一维连续函数,其任意阶Riemann-Liouville分数阶积分的分形维数仍然是1.文中还给出了一些例子的图像和数值结果.

Riemann-Liouville fractional calculus of different orders of 1-dimensional continuous functions is discusses in this paper. Riemann-Liouville fractional integral of 1-dimensional continuous functions of bounded variation of any order still is 1-dimensional continuous functions of bounded variation. Definition of unbounded variation points is given. A 1-dimensional continuous function of unbounded variation based on an unbounded variation point is constructed. We prove that fractal dimension of its Riemann-Liouville fractional integral of any order still is 1. In the end, fractal dimensions of certain 1-dimensional continuous functions of unbounded variation are proved to be 1. Graphs and numerical results of certain example are given.