Koch曲线及其分数阶微积分

梁永顺, 苏维宜

数学学报 ›› 2011, Vol. 54 ›› Issue (2) : 227-240.

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PDF(630 KB)
数学学报 ›› 2011, Vol. 54 ›› Issue (2) : 227-240. DOI: 10.12386/A2011sxxb0024
论文

Koch曲线及其分数阶微积分

    梁永顺1, 苏维宜2
作者信息 +

Von Koch Curve and Its Fractional Calculus

    Yong Shun LIANG1, Wei Yi SU2
Author information +
文章历史 +

摘要

给出了Koch曲线的一个复值表达式, 并且估计了该表达式 的分数阶微积分的分形维数, 同时给出了此表达式的Weyl--Marchaud分数阶导数的图像. 进一步讨论了Koch曲线的图像与某类自仿分形函数图像的联系. 最后证明了这类自仿分形函数的分形维数与其分数阶微积分的分形维数成立着 线性关系, 一个特殊例子的图像和数值结果在文中给出.

 

Abstract

An analytic expression of von Koch curve has been given. Based on this complex-valued function, we give estimation of fractal dimension of its fractional calculus. Graphs of Weyl-Marchaud fractional derivative of this function have been given. Such function can also be transferred into certain self-affine fractal function. Finally, we set up the linear connection between fractal dimension of this function and order of fractional calculus. Graphs and numerical results of certain examples have been shown.

 

关键词

Koch曲线 / 复值函数 / 分形函数

Key words

von Koch curve / complex-valued function / fractal function

引用本文

导出引用
梁永顺, 苏维宜. Koch曲线及其分数阶微积分. 数学学报, 2011, 54(2): 227-240 https://doi.org/10.12386/A2011sxxb0024
Yong Shun LIANG, Wei Yi SU. Von Koch Curve and Its Fractional Calculus. Acta Mathematica Sinica, Chinese Series, 2011, 54(2): 227-240 https://doi.org/10.12386/A2011sxxb0024

参考文献


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[7] Liang Y. S., Connection between the order of fractional calculus and fractional dimensins of a type of fractal functions, Analysis Theory and its Application, 2007, 23: 354-363.

[8] Miller K. S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equation, New York: John Wiley. Sons. Inc., 1993.

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基金

国家自然科学基金项目(10171045,10571084);南京理工大学校科研发展基金项目(XKF09033)及自主科研专项计划一般项目(2010GJPY081

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