## Abstract and Applied Analysis

### Fractional Calculus of Fractal Interpolation Function on $\mathrm{[0,b]}\mathrm{(b>0)}$

XueZai Pan

#### Abstract

The paper researches the continuity of fractal interpolation function’s fractional order integral on $[0,+\infty )\mathrm{}$ and judges whether fractional order integral of fractal interpolation function is still a fractal interpolation function on $\mathrm{[0,b]}\mathrm{(b>0)}$ or not. Relevant theorems of iterated function system and Riemann-Liouville fractional order calculus are used to prove the above researched content. The conclusion indicates that fractional order integral of fractal interpolation function is a continuous function on $[0,+\infty )\mathrm{}$ and fractional order integral of fractal interpolation is still a fractal interpolation function on the interval $\mathrm{[0,b]}$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 640628, 5 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412278611

Digital Object Identifier
doi:10.1155/2014/640628

Mathematical Reviews number (MathSciNet)
MR3198224

#### Citation

Pan, XueZai. Fractional Calculus of Fractal Interpolation Function on $\mathrm{[0,b]}\mathrm{(b&gt;0)}$. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 640628, 5 pages. doi:10.1155/2014/640628. https://projecteuclid.org/euclid.aaa/1412278611

#### References

• B. B. Mandelbrot, “How long is the coast of Britain? Statistical self-similarity and fractional dimension,” Science, vol. 156, no. 3775, pp. 636–638, 1967.
• B. B. Mandelbrot and J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Review, vol. 10, pp. 422–437, 1968.
• B. B. Mandelbrot, The Fractal Geometry of Nature, Macmillan Press, New York, NY, USA.
• B. B. Mandelbrot, D. E. Passoja, and A. J. Paullay, “Fractal character of fracture surfaces of metals,” Nature, vol. 308, no. 5961, pp. 721–722, 1984.
• B. B. Mandelbrot, Fractal Objects: form, Opportunity, and Dimension, World Publishing Corporation, Beijing, China, 1999.
• M. F. Barnsley, “Lecture notes on iterated function systems,” in Chaos and Fractals, vol. 39 of Proceedings Symposia in Applied Mathematics, pp. 127–144, American Mathematical Society, Providence, RI, USA, 1989.
• M. F. Barnsley, “Fractal functions and interpolation,” Constructive Approximation, vol. 2, no. 4, pp. 303–329, 1986.
• P. R. Massopust, “Fractal surfaces,” Journal of Mathematical Analysis and Applications, vol. 151, no. 1, pp. 275–290, 1990.
• P. R. Massopust, Fractal Functions, Fractal Surfaces, and Wavelets, Academic Press, Orlando, Fla, USA, 1995.
• Z. G. Feng and L. Wang, “$\delta$-variation properties of fractal interpolation functions,” Journal of Jiangsu University, vol. 26, no. 1, pp. 49–52, 2005 (Chinese).
• Z. Feng, “Variation and Minkowski dimension of fractal interpolation surface,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 322–334, 2008.
• Z. G. Feng and Y. L. Huang, “Variation and box-counting dimension of fractal interpolation surfaces based on the fractal interpolation function,” Chinese Journal of Engineering Mathematics, vol. 29, no. 3, pp. 393–398, 2012 (Chinese).
• S. G. Li and J. T. Wu, Fractals and Wavelets, Science Press, Beijing, China, 2002.
• Q. W. Ran and X. Y. Tan, Wavelet Analysis, Fractional Fourier Transformation and Application, National Defence Industry Press, Beijing, China, 2002.
• A. P. Mark, Introduction to Fourier Analysis and Wavelets, Machine Press, Beijing, China, 2003.
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
• M. F. Barnsley, Fractals Everywhere, Elsevier, Singapore, 2nd edition, 2009.
• Z. Sha and H. J. Ruan, Fractals and Fitting, Zhenjiang University Press, Hangzhou, China, 2005. \endinput