Fractal Dimensions of Fractional Integral of Continuous Functions

Yong Shun LIANG^{1}, Wei Yi SU^{2}

1. Institute of Science, Nanjing University of Science and Technology, Nanjing 210014, P. R. China;
2. Department of Mathematics, Nanjing University, Nanjing 210093, P. R. China

In this paper, we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals. Riemann-Liouville integral of a continuous function f(x) of order v(v>0) which is written as D^{-v}f(x) has been proved to still be continuous and bounded. Furthermore, upper box dimension of D^{-v}f(x) is no more than 2 and lower box dimension of D^{-v}f(x) is no less than 1. If f(x) is a Lipshciz function, D^{-v}f(x) also is a Lipshciz function. While f(x) is differentiable on[0, 1], D^{-v}f(x) is differentiable on[0, 1] too. With definition of upper box dimension and further calculation, we get upper bound of upper box dimension of Riemann-Liouville fractional integral of any continuous functions including fractal functions. If a continuous function f(x) satisfying Hölder condition, upper box dimension of Riemann-Liouville fractional integral of f(x) seems no more than upper box dimension of f(x). Appeal to auxiliary functions, we have proved an important conclusion that upper box dimension of Riemann-Liouville integral of a continuous function satisfying Hölder condition of order v(v>0) is strictly less than 2-v. Riemann-Liouville fractional derivative of certain continuous functions have been discussed elementary. Fractional dimensions of Weyl-Marchaud fractional derivative of certain continuous functions have been estimated.

Yong Shun LIANG,Wei Yi SU. Fractal Dimensions of Fractional Integral of Continuous Functions[J]. Acta Mathematica Sinica, English Series, 2016, 32(12): 1494-1508.

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

[2]

Jumarie, G.:An approach to defferential geometry of fractional order via modified Riemann-Liouville derivative. Acta Math. Sin., Engl. Ser., 28, 1741-1768(2012)

[3]

Liang, Y. S.:The relationship between the Box dimension of the Besicovitch functions and the orders of their fractional calculus. Appl. Math. Comput., 200, 197-207(2008)

[4]

Liang, Y. S.:Box dimensions of Riemann-Liouville fractional integrals of continuous functions of bounded variation. Nonlinear Anal., 72, 4304-4306(2010)

[5]

Liang, Y. S.:Upper Bound estimation of fractal dimensions of fractional integral of continuous functions. Adv. Pure Math., 5, 27-30(2015)

[6]

Liang, Y. S.:Some remarks on continuous functions of unbounded variation. Acta Math. Sin., Chin. Ser., 59, 215-232(2016)

[7]

Liang, Y. S., Su, W. Y.:Connection between the order of fractional calculus and fractional dimensins of a type of fractal functions. Anal. Theory Appl., 23, 354-363(2007)

[8]

Liang, Y. S., Su, W. Y.:Von Koch curve and its fractional calculus. Acta Math. Sin., Chin. Ser., 54, 227-240(2011)

[9]

Liang, Y. S., Su, W. Y.:Riemann-Liouville fractional calculus of 1-dimensional continuous functions. Sci. China Ser. A, 46, 423-438(2016)

[10]

Liang, Y. S., Yao, K.:Fractal dimensions of Riemann-Liouville fractional calculus of linear fractal interpolation functions. Chin. Ann. Math. Ser. A, 38, 1-8(2017)

[11]

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

[12]

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

[13]

Rajkovi?, P. M., Marinkovi?, S. D., Stankovi?, M. C.:A generalization of the concept of q-fractional integrals. Acta Math. Sin., Engl. Ser., 25, 1635-1646(2009)

[14]

Ross, B.:The Fractional Calculus and Its Applications, Springer-Verlag, Berlin, Heidelberg, 1975

[15]

Ruan, H. J., Su, W. Y., Yao, K.:Box dimension and fractional integral of linear fractal interpolation functions. J. Approx. Theory, 161, 187-197(2009)

[16]

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

[17]

Wen, Z. Y.:Mathematical Foundations of Fractal Geometry, Science Technology Education Publication House, Shanghai, 2000

[18]

Yao, K., Liang, Y. S.:Dimension of graphs of fractional derivatives of self-affine curves. Acta Math. Sin., Chin. Ser., 56, 693-698(2013)

[19]

Yao, K., Liang, Y. S., Zhang, F.:On the connection between the order of the fractional derivative and the Hausdorff dimension of a fractal function. Chaos Solitons Fractals, 41, 2538-2545(2009)

[20]

Yao, K., Su, W. Y., Liang, Y. S.:The upper bound of Box dimension of the Weyl-Marchaud derivative of self-affine curves. Anal. Theory Appl., 26, 222-227(2010)

[21]

Yao, K., Su, W. Y., Zhou, S. P.:On the fractional calculus of a type of Weierstrass function. Chin. Ann. Math. Ser. B, 25, 711-716(2004)

[22]

Yao, K., Su, W. Y., Zhou, S. P.:On the fractional derivatives of a fractal function. Acta Math. Sin., Engl. Ser., 22, 719-722(2006)

[23]

Zähle, M.:Fractional differentiation in the self-affine case V-the local degree of differentiability. Math. Nachr., 185, 297-306(1997)

[24]

Zähle, M., Ziezold, H.:Fractional derivatives of Weierstrass-type functions. J. Comput. Appl. Math., 76, 265-275(1996)

[25]

Zhang, Q.:Some remarks on one-dimensional functions and their Riemann-Liouville fractional calculus. Acta Math. Sin., Engl. Ser., 30, 517-524(2014)

[26]

Zhang, Q., Liang, Y. S.:The Weyl-Marchaud fractional derivative of a type of self-affine functions. Appl. Math. Comput., 218, 8695-8701(2012)

[27]

Xie T. F., Zhou, S. P.:Aproximation Theory of Real Functions, Hangzhou University Publication, Hangzhou, 1997