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Journal of Applied Mathematics and Computation

ISSN Online: 2576-0653 ISSN Print: 2576-0645 CODEN: JAMCEZ
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ArticleOpen Access http://dx.doi.org/10.26855/jamc.2020.12.003

Riemann-Liouville Fractional Calculus of Blancmange Curve and Cantor Functions

Srijanani Anurag Prasad

Department of Mathematics and Statistics, Indian Institute of Technology Tirupati, India.

*Corresponding author: Srijanani Anurag Prasad

Published: October 22,2020

Abstract

Riemann-Liouville fractional calculus of Blancmange Curve and Cantor Func-tions are studied in this paper. In this paper, Blancmange Curve and Cantor func-tion defined on the interval is shown to be Fractal Interpolation Functions with appropriate interpolation points and parameters. Then, using the properties of Fractal Interpolation Function, the Riemann-Liouville fractional integral of Blancmange Curve and Cantor function are described to be Fractal Interpolation Function passing through a different set of points. Finally, using the conditions for the fractional derivative of order ν of a FIF, it is shown that the fractional deriva-tive of Blancmange Curve and Cantor function is not a FIF for any value of ν.

Keywords

Fractal, Interpolation, Iterated Function System, fractional integral, fractional de-rivative, Blancmange Curve, Cantor function

References

[1] Barnsley, M. F. (1986). Fractal functions and interpolation. Constructive Approximation, 2, 303-329. 

[2] Barnsley, M. F., Elton, J., Hardin, D., and Massopust, P. (1989). Hidden variable fractal interpolation functions. SIAM Journal of Mathematical Analysis, 20(5), 1218-1242.

[3] Chand, A. K. B. and Kapoor, G. P. (2007). Smoothness analysis of coalescence hidden variable fractal interpolation functions. International Journal of Non-Linear Science, 3, 15-26.

[4] Kapoor, G. P. and Prasad, S. A. (2014). Multiresolution analysis based on coalescence hidden-variable fractal in-terpolation functions. International Journal of Computational Mathematics, Article ID 531562.

[5] Navascués, M. A. and Sebastián, M. V. (2004). Generalization of Hermite functions by fractal interpolation. Jour-nal of Approximation Theory, 131, 19-29.

[6] Kapoor, G. P. and Prasad, S. A. (2014). Convergence of cubic spline super fractal interpolation functions. Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society, 22(1), 1-7.

[7] Kapoor, G. P. and Prasad, S. A. (2015). Super fractal interpolation functions. International Journal of Non-Linear Science, 19(1), 20-29.

[8] Kempfle, S. and Schaefer, I. (1999). Functional Calculus Method Versus Riemann-Liouville Approach. Fractional Calculus and Applied Analysis, 2(4), 415-428.

[9] Changpin Li, Deliang Qian, and Yang Quan Chen. (2011). On Riemann-Liouville and Caputo Derivatives. Discrete Dynamics in Nature and Society, Article ID 562494.

[10] Edmundo Capelas de Oliveira and José António Tenreiro Machado. (2014). A Review of Definitions for Fractional Derivatives and Integral. Mathematical Problems in Engineering, Article ID: 238459.

[11] Hilfer, R. (2008). Threefold Introduction to Fractional Derivatives, In. Anomalous Transport: Foundations and Applications, Wiley-VCH, Weinheim.

[12] Huo-Jun Ruan, Wei-Yi Su, and Kui Yao. (2009). Box dimension and fractional integral of linear fractal interpola-tion functions. Journal of Approximation Theory, 161, 187-197.

[13] XueZai Pan. (2014). Fractional Calculus of Fractal Interpolation Function on [0; b]. Abstract and Applied Analysis, Article ID: 640628. 

[14] Prasad, S. A. (2017). Fractional calculus of coalescence hidden-variable fractal interpolation functions. Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society, 25(2), Article ID 1750019.

How to cite this paper

Riemann-Liouville Fractional Calculus of Blancmange Curve and Cantor Functions

How to cite this paper: Srijanani Anurag Prasad. (2020) Riemann-Liouville Fractional Calculus of Blancmange Curve and Cantor Functions. Journal of Applied Mathematics and Computation, 4(4), 123-129.

DOI: http://dx.doi.org/10.26855/jamc.2020.12.003