一类分形插值函数的分数阶微积分

A Class of Fractional Order Calculus of Fractal Interpolation Function

介绍了分形插值函数和迭代函数系统以及v阶黎曼-刘维尔分数阶积分、微分的概念和相关定理.由于分形插值函数满足应用分数阶微积分处理问题的条件,所以利用这些概念及分步积分的方法讨论了折线段分形插值函数的分数阶积分的连续性,可微性及哪些点是不可微的,进一步说明了该插值函数分数阶微分的连续性并指出其不连续点,用黎曼-刘维尔分数阶微积分与分形插值函数结合起来研究,目的是想设法跟经典微积分一样,能找出函数上在该点的微积分的具体的实际应用意义.这些理论为研究分形插值函数的分数阶微积分的实际应用意义提供了一些理论基础.

The thesis introduces the concept and relative theories of fractal interpolation function and the ReimannLiouville fractional integral of order v and fractional differential. Fractal interpolation functions satisfy the conditions of fractional calculus and fractional differential, so the thesis discusses continuous property; differential property and non- differentiable points of fold line fractal interpolation function＇s v order integral. Further indicate the continuous property of the interpolation function＇ s p order differential and points out uncontinuous points by applying these theories. The purpose of studing the thesis is that we can find actual means of fractional calculus of points on functions as classic calculus. These theories provide some theoretical foundations for researching actual application means of fractal interpolation function＇s fractional order calculus.