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There are many non-smooth objects in nature, such as coastline, rock fracture, cross section, whose differentiabilities cannot be described by ordinary calculus and methods in Euclidean geometry. The local fractional derivative is one of the potential tools to investigate the non-smooth problems. This study revisits the non-smooth curves generated from the fractional integrals and Cantor-like set. From the view of the fractional differentiable functions, the differentiabilities of the non-smooth curves are derived by using a binomial expansion.
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Keywords:
- fractals /
- fractional differentiability /
- non-smooth curves /
- global behaviors
[1] Mandelbrot B B 1983 The Fractal Geometry of Nature (WH Freeman)
[2] Sun X, Wu Z Q 2001 Acta Phys. Sin. 50 2126 (in Chinese) [孙霞, 吴自勤2001 物理学报 50 2126]
[3] Yu B M, Li J H 2001 Fractals 9 365
[4] Xie H P, Gao F 2000 Int. J. Rock Mech. Min. Sci. 37 477
[5] Yu Z G, Anh V, Lau K S 2001 Phys. Rev. E 64 031903
[6] Zhu J M, Ma Z Y, Zheng C L 2004 Acta Phys. Sin. 53 3248 (in Chinese) [朱加民, 马正义, 郑春龙 2004 物理学报 53 3248]
[7] Barnsley M F, Demko S 1985 Proc. R. Soc. London, Ser. A 399 243
[8] Feng Z G, Xie H P 1998 Fractals 6 269
[9] Xie H P, Sun H Q 1997 Fractals 5 625
[10] Kolwankar K M, Gangal A D 1996 Chaos 6 505
[11] Kolwankar K M, Gangal A D 1997 Pramana-J. Phys. 48 49
[12] Kolwankar K M 2004 Fractals 12 375
[13] Wu G C 2011 Math. Comput. Model. 54 2104
[14] Wu G C, Lee E W M 2010 Phys. Lett. A 374 2506
[15] Wu G C, Zhang S 2011 Phys. Lett. A 375 5
[16] Chen Y, Yan Y, Zhang K W 2010 J. Math. Anal. Appl. 362 17
[17] Carpinteri A, Sapora A 2010 ZAMM-Z. ANGEW. MATH. ME. 90 203
[18] Podlubny I 2001 Arxiv preprint math/0110241
[19] Qiu W Y, Lu J 2000 Phys. Lett. A 272 353
[20] Ren F Y, Liang J R, Wang, X T, Qiu, W Y 2003 Chaos, Soliton. Fractal. 16 107
[21] Ruan H J, Su W Y, Yao K 2009 J. Approx. Theory. 161 187
[22] Yao K, Su W Y, Zhou S P 2004 Chin. Annal. Math. 25 711
[23] Yao K, Su W Y, Zhou S P 2006 Acta. Math. Sin. 22 719
[24] Liang Y S, Su W Y 2007 Chaos, Soliton. Fract. 34 682
[25] Liang Y S 2007 Anal.Theory Appl. 23 354
[26] Rutman R S 1995 Theor. Math. Phys. 105 1509
[27] Tarasov V E 2005 Ann. Phys-New York 318 286
[28] Ni Z X 2001 J. Fuyang Teach. Coll. 18 40
[29] Wu G C 2011 Commun.Frac.Calc. 2 27
[30] Yang X J 2009 WorlD Sci．Tech. R & D 31 920
[31] Jumarie G 2006 Comput. Math. Appl. 51 1367
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[1] Mandelbrot B B 1983 The Fractal Geometry of Nature (WH Freeman)
[2] Sun X, Wu Z Q 2001 Acta Phys. Sin. 50 2126 (in Chinese) [孙霞, 吴自勤2001 物理学报 50 2126]
[3] Yu B M, Li J H 2001 Fractals 9 365
[4] Xie H P, Gao F 2000 Int. J. Rock Mech. Min. Sci. 37 477
[5] Yu Z G, Anh V, Lau K S 2001 Phys. Rev. E 64 031903
[6] Zhu J M, Ma Z Y, Zheng C L 2004 Acta Phys. Sin. 53 3248 (in Chinese) [朱加民, 马正义, 郑春龙 2004 物理学报 53 3248]
[7] Barnsley M F, Demko S 1985 Proc. R. Soc. London, Ser. A 399 243
[8] Feng Z G, Xie H P 1998 Fractals 6 269
[9] Xie H P, Sun H Q 1997 Fractals 5 625
[10] Kolwankar K M, Gangal A D 1996 Chaos 6 505
[11] Kolwankar K M, Gangal A D 1997 Pramana-J. Phys. 48 49
[12] Kolwankar K M 2004 Fractals 12 375
[13] Wu G C 2011 Math. Comput. Model. 54 2104
[14] Wu G C, Lee E W M 2010 Phys. Lett. A 374 2506
[15] Wu G C, Zhang S 2011 Phys. Lett. A 375 5
[16] Chen Y, Yan Y, Zhang K W 2010 J. Math. Anal. Appl. 362 17
[17] Carpinteri A, Sapora A 2010 ZAMM-Z. ANGEW. MATH. ME. 90 203
[18] Podlubny I 2001 Arxiv preprint math/0110241
[19] Qiu W Y, Lu J 2000 Phys. Lett. A 272 353
[20] Ren F Y, Liang J R, Wang, X T, Qiu, W Y 2003 Chaos, Soliton. Fractal. 16 107
[21] Ruan H J, Su W Y, Yao K 2009 J. Approx. Theory. 161 187
[22] Yao K, Su W Y, Zhou S P 2004 Chin. Annal. Math. 25 711
[23] Yao K, Su W Y, Zhou S P 2006 Acta. Math. Sin. 22 719
[24] Liang Y S, Su W Y 2007 Chaos, Soliton. Fract. 34 682
[25] Liang Y S 2007 Anal.Theory Appl. 23 354
[26] Rutman R S 1995 Theor. Math. Phys. 105 1509
[27] Tarasov V E 2005 Ann. Phys-New York 318 286
[28] Ni Z X 2001 J. Fuyang Teach. Coll. 18 40
[29] Wu G C 2011 Commun.Frac.Calc. 2 27
[30] Yang X J 2009 WorlD Sci．Tech. R & D 31 920
[31] Jumarie G 2006 Comput. Math. Appl. 51 1367
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