Upper Bound Estimation of Fractal Dimensions of Fractional Integral of Continuous Functions

Yongshun Liang^*

Faculty of Science, Nanjing University of Science and Technology, Nanjing, China.

DOI: 10.4236/apm.2015.51003                

Abstract

Fractional integral of continuous functions has been discussed in the present paper. If the order of Riemann-Liouville fractional integral is v, fractal dimension of Riemann-Liouville fractional integral of any continuous functions on a closed interval is no more than 2 - v.

Keywords

Box Dimension, Riemann-Liouville Fractional Calculus, Fractal Function

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Liang, Y.S. (2010) Box Dimension of Riemann-Liouville Fractional Integral of Continuous Function of Bounded Variation. Nonlinear Analysis Series A: Theory, Method and Applications, 72, 2758-2761.
[2]	Yao, K. and Liang, Y.S. (2010) The Upper Bound of Box Dimension of the Weyl-Marchaud Derivative of Self-Affine Curves. Analysis Theory and Application, 26, 222-227. http://dx.doi.org/10.1007/s10496-010-0222-9
[3]	Liang, Y.S. and Su, W.Y. (2011) The Von Koch Curve and Its Fractional Calculus. Acta Mathematic Sinica, Chinese Series [in Chinese], 54, 1-14.
[4]	Zhang, Q. and Liang, Y.S. (2012) The Weyl-Marchaud Fractional Derivative of a Type of Self-Affine Functions. Applied Mathematics and Computation, 218, 8695-8701. http://dx.doi.org/10.1016/j.amc.2012.01.077
[5]	Falconer, J. (1990) Fractal Geometry: Mathematical Foundations and Applications. John Wiley Sons Inc., New York.
[6]	Oldham, K.B. and Spanier, J. (1974) The Fractional Calculus. Academic Press, New York.