关于某些连续函数的分数阶微积分的分形维数估计

doi:10.12677/PM.2021.114061

期刊菜单

关于某些连续函数的分数阶微积分的分形维数估计
A Remark on Fractal Dimension Estimation of Fractional Calculus of Certain Continuous Functions

DOI: 10.12677/PM.2021.114061, PDF, HTML, 下载: 333 浏览: 460 国家自然科学基金支持
作者: 王含西：纽约大学文理学院，美国纽约
关键词: 分形函数；Riemann-Liouville 分数阶微积分；分形维数；The Fractal Function； The Riemann-Liouville Fractional Calculus； The Fractal Dimension

摘要: 在本文中，我们对分形函数的定义进行了初步的研究，接着讨论了分形函数分数阶微积分的分形维数估计。我们使用新方法进行的估计表明分形函数的分形维数和分数阶微积分的阶之间存在一定关系。如果分形函数满足 Hölder 条件，则这种分形函数的 Riemann-Liouville 分数阶积分的上 Box 维数小于这些分形函数的上 Box 维数。这就意味着一个重要的结论：分形函数的 Riemann-Liouville 分数阶微积分的上 Box 维数不会增加。

Abstract: In the present paper, we make research on the deﬁnition of fractal functions elementary. Then we discuss the fractal dimensions of fractional calculus of fractal functions. The estimation using a new method shows certain relationship between the fractal dimensions of fractal functions and orders of fractional calculus. If the fractal function satisﬁes the Hölder condition, the upper Box dimension of the Riemann-Liouville fractional integral of such fractal functions has been proved to be less than the upper Box dimension of those fractal functions. This means an important conclusion that the upper Box dimension of the Riemann-Liouville fractional integral of such fractal functions will not increase.

文章引用：王含西. 关于某些连续函数的分数阶微积分的分形维数估计[J]. 理论数学, 2021, 11(4): 477-484. https://doi.org/10.12677/PM.2021.114061

参考文献

[1]	Tatom, F.B. (1995) The Relationship between Fractional Calculus and Fractals. Fractals, 3, 217-229. https://doi.org/10.1142/S0218348X95000175
[2]	Zähle, M. and Ziezold, H. (1996) Fractional Derivatives of Weierstrass-Type Functions. Journal of Computational and Applied Mathematics, 76, 265-275. https://doi.org/10.1016/S0377-0427(96)00110-0
[3]	Zähle, M. (1997) Fractional Diﬀerentiation in the Self-Aﬃne Case. V—The Local Degree of Diﬀerentiability. Mathematische Nachrichten, 185, 297-306. https://doi.org/10.1002/mana.3211850117
[4]	Kolwankar, K.M. and Gangal, A.D. (1996) Fractional Diﬀerentiability of Nowhere Diﬀeren- tiable Functions and Dimensions. Chaos, 6, 505-513. https://doi.org/10.1063/1.166197
[5]	Kolwankar, K.M. and Gangal, A.D. (1998) Local Fractional Derivatives and Fractal Functions of Several Variables. Physics.
[6]	Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Diﬀerential Equations. John Wiley Sons Inc., New York.
[7]	姚奎, 苏维宜, 周颂平. 关于一类 Weierstrass 函数的分数阶微积分函数 [J]. 数学年刊, 2004, 25(6): 711-716.
[8]	Yao, K., Su, W.Y. and Zhou, S.P. (2005) On the Connection between the Order of the Frac- tional Calculus and the Dimension of a Fractal Function. Chaos, Solitons and Fractals, 23, 621-629. https://doi.org/10.1016/j.chaos.2004.05.037
[9]	Yao, K., Su, W.Y. and Zhou, S.P. (2006) On the Fractional Derivatives of a Fractal Function. Acta Mathematica Sinica, 22, 719-722. https://doi.org/10.1007/s10114-005-0644-z
[10]	Ruan, H.J., Su, W.Y. and Yao, K. (2009) Box Dimension and Fractional Integral of linear Fractal Interpolation Functions. Journal of Approximation Theory, 161, 187-197. https://doi.org/10.1016/j.jat.2008.08.012
[11]	Zhou, S.P., He, G.L. and Xie, T.F. (2004) On a Class of Fractals: The Constructive Structure. Chaos, Solitons and Fractals, 19, 1099-1104. https://doi.org/10.1016/S0960-0779(03)00282-0
[12]	Liang, Y.S. and Su, W.Y. (2007) The Relationship between the Fractal Dimensions of a Type of Fractal Functions and the Order of Their Fractional Calculus. Chaos, Solitons and Fractals, 34, 682-692. https://doi.org/10.1016/j.chaos.2006.01.124
[13]	Liang, Y.S. (2007) Connection between the Order of Fractional Calculus and Fractal Dimen- sions of a Type of Fractal Functions. Analysis Theory and Its Application, 23, 354-363. https://doi.org/10.1007/s10496-007-0354-8
[14]	Liang, Y.S. (2008) The Relationship between the Box Dimension of the Besicovitch Functions and the Orders of Their Fractional Calculus. Applied Mathematics and Computation, 200, 297-307. https://doi.org/10.1016/j.amc.2007.11.014
[15]	Liang, Y.S. (2010) Box Dimensions of Riemann-Liouville Fractional Integrals of Continuous Functions of Bounded Variation. Nonlinear Analysis: Theory, Methods and Applications, 72, 4304-4306. https://doi.org/10.1016/j.na.2010.02.007
[16]	Liang, Y.S., Yao, K. and Xiao, W. (2008) The Fractal Dimensions of Graphs of the Weierstrass Function with the Weyl-Marchaud Fractional Derivative. Journal of Physics: Conference Se- ries, 96, Article ID: 012111. https://doi.org/10.1088/1742-6596/96/1/012111
[17]	Yao, K., Liang, Y.S. and Fang, J.X. (2008) The Fractal Dimensions of Graphs of the Weyl- Marchaud Fractional Derivative of the Weierstrass-Type Function. Chaos, Solitons and Frac- tals, 35, 106-115. https://doi.org/10.1016/j.chaos.2007.04.017
[18]	Yao, K., Liang, Y.S. and Zhang, F. (2009) On the Connection between the Order of the Fractional Derivative and the Hausdorﬀ Dimension of a Fractal Function. Chaos, Solitons and Fractals, 41, 2538-2545. https://doi.org/10.1016/j.chaos.2008.09.053
[19]	梁永顺, 苏维宜. Koch 曲线及其分数阶微积分 [J]. 数学学报, 2011, 54(2): 227-240.
[20]	Falconer, K.J. (1990) Fractal Geometry: Mathematical Foundations and Applications. John Wiley Sons Inc., New York.
[21]	Oldham, K.B. and Spanier, J. (1974) The Fractional Calculus. Academic Press, New York.

为你推荐

友情链接