Fractalization of Fractional Integral and Composition of Fractal Splines

Year 2023, Volume: 5 Issue: 4, 318 - 325, 31.12.2023

Gowrisankar Arulprakash

https://doi.org/10.51537/chaos.1334407

Abstract

The present study perturbs the fractional integral of a continuous function $f$ defined on a real compact interval, say $(\mathcal{I}^vf)$ using a family of fractal functions $(\mathcal{I}^vf)^\alpha$ based on the scaling parameter $\alpha$. To elicit this phenomenon, a fractal operator is proposed in the space of continuous functions, an analogue to the existing fractal interpolation operator which perturbs $f$ giving rise to $\alpha$-fractal function $f^\alpha$. In addition, the composition of $\alpha$-fractal function with the linear fractal function is discussed and the composition operation on the fractal interpolation functions is extended to the case of differentiable fractal functions.

Keywords

Fractional integral, α-fractal function, Error estimation, Composite fractal functions

References

• Agathiyan, A., A. Gowrisankar, and T. Priyanka, 2022 Construction of new fractal interpolation functions through integration method. Results in Mathematics 77: 122.
• Akhtar, M. N., M. Prasad, and M. Navascués, 2016 Box dimensions of α-fractal functions. Fractals 24: 1650037.
• Akhtar, M. N., M. Prasad, and M. Navascués, 2017 Box dimension of α-fractal function with variable scaling factors in subintervals. Chaos, Solitons & Fractals 103: 440–449.
• Balasubramani, N., M. Prasad, and S. Natesan, 2020 Shape preserving α-fractal rational cubic splines. Calcolo 57: 21.
• Banerjee, A., M. N. Akhtar, and M. Navascués, 2023 Local α-fractal interpolation function. The European Physical Journal Special Topics pp. 1–8.
• Banerjee, S., D. Easwaramoorthy, and A. Gowrisankar, 2021 Fractal Functions, Dimensions and Signal Analysis. Springer, Cham.
• Barnsley, M., 1986 Fractal functions and interpolation. Constructive Approximation 2: 303–329.
• Barnsley, M. and A. Harrington, 1989 The calculus of fractal interpolation functions. Journal of Approximation Theory 57: 14–34.
• Çimen, M., Z. Garip, O. Boyraz, I. Pehlivan, M. Yildiz, et al., 2020 An interface design for calculation of fractal dimension. Chaos Theory and Applications 2: 3–9.
• Dai, Z. and S. Liu, 2023 Construction and box dimension of the composite fractal interpolation function. Chaos, Solitons & Fractals 169: 113255.
• Falconer, K., 2004 Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons.
• Fortin, C., R. Kumaresan, W. Ohley, and S. Hoefer, 1992 Fractal dimension in the analysis of medical images. IEEE Engineering in Medicine and Biology Magazine 11: 65–71.
• Gowrisankar, A. and M. Prasad, 2019 Riemann-Liouville calculus on quadratic fractal interpolation function with variable scaling factors. The Journal of Analysis 27: 347–363.
• Gowrisankar, A. and R. Uthayakumar, 2016 Fractional calculus on fractal interpolation for a sequence of data with countable iterated function system. Mediterranean Journal of Mathematics 13: 3887–3906.
• Massopust, P., 2022a Fractal interpolation: From global to local, to nonstationary and quaternionic. Frontiers of Fractal Analysis. Recent Advances and Challenges; CRC Press: Boca Raton, FL, USA pp. 25–49.
• Massopust, P., 2022b Fractal interpolation over nonlinear partitions. Chaos, Solitons & Fractals 162: 112503.
• Navascués, M., 2005 Fractal polynomial interpolation. Zeitschrift fur Analysis und ihre Anwendung 24: 401–418.
• Navascués, M., 2010 Fractal approximation. Complex Analysis and Operator Theory 4: 953–974.
• Navascués, M., C. Pacurar, and V. Drakopoulos, 2022 Scale-free fractal interpolation. Fractal and Fractional 6: 602.
• Navascués, M. and M. Sebastián, 2006 Smooth fractal interpolation. Journal of Inequalities and Applications 2006: 1–20.
• Pan, X., 2014 Fractional calculus of fractal interpolation function on [0, b](b > 0). In Abstract and Applied Analysis 2014.
• Priyanka, T. and A. Gowrisankar, 2021a Analysis on weylmarchaud fractional derivative for types of fractal interpolation function with fractal dimension. Fractals 29: 2150215.
• Priyanka, T. and A. Gowrisankar, 2021b Riemann-Liouville fractional integral of non-affine fractal interpolation function and its fractional operator. The European Physical Journal Special Topics 230: 37889–3805.
• Ruan, H.-J., W.-Y. Su, and K. Yao, 2009 Box dimension and fractional integral of linear fractal interpolation functions. Journal of Approximation Theory 161: 187–197.
• Samko, S., A. Kilbas, and O. Marichev, 1993 Fractional integrals and derivatives.
• Sanjuán, M. A., 2021 Unpredictability, uncertainty and fractal structures in physics.