数学学报 2011, 54(2) 227-240 DOI:      ISSN: 0583-1431 CN: 11-2038/O1

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本文关键词相关文章
Koch曲线
复值函数
分形函数
本文作者相关文章
梁永顺
苏维宜
Koch曲线及其分数阶微积分
梁永顺1, 苏维宜2
1. 南京理工大学理学院 南京 210094;
2. 南京大学数学系 南京 210094
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.

 

关键词 Koch曲线   复值函数   分形函数  
MSC2000 O172
Von Koch Curve and Its Fractional Calculus
Yong Shun LIANG1, Wei Yi SU2
1. Institute of Science, Nanjing University of Science and Technology, Nanjing 210094, P. R. China;
2. Department of Mathematics, Nanjing University, Nanjing 210094, P. R. China
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.

 

Keywords: von Koch curve   complex-valued function   fractal function  
收稿日期 2010-05-20 修回日期 2010-10-22 网络版发布日期  
DOI:
基金项目:

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

通讯作者:
作者简介:
作者Email: liangyongshun@gmail.com; suqiu@nju.edu.cn

参考文献:


[1] Liang Y. S., The relationship between the Box dimension of the Besicovitch functions and the orders of their fractional calculus, Applied Mathematics and Computation, 2008, 200: 197-207.

[2] Tatom F. B., The relationship between fractional calculus and fractals, Fractals, 1995, 3(1): 217-229.

[3] Yao K., Su W. Y., Zhou S. P., On the fractional derivatives of a fractal function, Acta Mathematica Sinica, English Series, 2006, 22(3): 719-722.

[4] Z¨ahle M., Ziezold H., Fractional derivatives of Weierstrass-type functions, J. Computational and Appl. Math., 1996, 76: 265-275.

[5] Oldham K. B., Spanier J., The Fractional Calculus, New York: Academic Press, 1974.

[6] Liang Y. S., Su W. Y., The relationship between the fractal dimensions of a type of fractal functions and the order of their fractional calculus, Chaos, Solitons and Fractals, 2007, 34: 682-692.

[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.

[9] Falconer J., Fractal Geometry: Mathematical Foundations and Applications, New York: John Wiley Sons Inc., 1990.

[10] Hu T. Y., Lau K. S., Fractal dimensions and singularities of the Weierstrass type functions, Trans. Amer. Math. Soc., 1993, 335(2): 649-665.

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