关于分数次导数的几点注记  

Fractional derivatives from the perspective of the Minkowski's space-time

作　　者：吴宗敏[1] WU ZongMin

机构地区：[1]复旦大学数学科学学院,上海200433

出　　处：《中国科学：数学》2017年第10期1335-1344,共10页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:91330201和11461161006)资助项目

摘　　要：本文对经典的分数次导数从定义开始进行了分析,进而提出了一种基于Minkowski时空观的修正定义,使得这个新定义的分数次导数保持了局部性、平移不变性、可以作为Taylor展开的系数和具备Leibniz法则等经典导数的性质,以及可以通过局部分析的方法用于分数次微分方程建模.进一步地,进行了事件进程函数分析,即通过现时的函数信息,回溯由分数次截断多项式表示的事件进程函数的历史发展过程,同时应用于预测将来的事件发展趋向,得到了一些有趣的结果.最后给出了新定义的分数次导数的数值计算公式及微分方程建模的例子.This paper deals with the fractional derivatives from the beginning of the definition.Based on the conception of the Minkowski＇s space-time,a revised version of the definition is given.The new version of the definition possesses the properties of locality,shift invariant,served as the coefficients of the Taylor＇s expansions,Leibnitz＇s rule etc.,which the classical derivative have.The new fractional derivatives can be got by local differential analysis and to model the fractional differential equations,too.Moreover,we take the fractional derivative to analyze the functions of time processes,which are expressed by generalized Taylor＇s expansions,and to look back upon the histories based on the information of nowadays.The theory is applied to the prediction and gets some interesting results,too.Finally,numerical formula for calculating the fractional derivatives and some examples of the mathematical modeling for fractional differential equation are given.

关 键 词：分数次导数 事件进程函数分析 推广的Taylor展开 MINKOWSKI时空 

分 类 号：O172.1[理学—数学]