# Conformable Derivative and Fractal Derivative of Functions on the Interval [0,1]

## Keywords:

Conformable derivative, Fractal calculus, Fractal dimension, Fα-derivative## Abstract

Recently, a calculus-based fractal, called F^{α}-calculus, has been developed which involve F

^{α}-integral, conjugate to the Riemann integral, and Fα-derivative, conjugate to ordinary derivative, of orders α, 0< α <1, where α is the dimension of F. In F

^{α}-calculus the staircase function has a special role. In this paper we obtain fractal Taylor series for fractal elementary functions, sine, cosine, exponential function, etc. and then we compare the graph of these fractal functions with their counterparts in standard calculus on the interval [0,1]. Then, the main part of the paper is discussed about the transition from continuous state to a discrete state when we do fractal differentiation in which characteristic function χC

^{(x)}appears. Since Fα-derivative is local, we compare it with a conformable derivative which is also local. Moreover, fractal differential equations for fractal sine, cosine, sine hyperbolic, cosine hyperbolic and exponential functions are deduced. We also represent Pythagorean trigonometric identity for sin and cosine, and hyperbolic sine and hyperbolic cosine in F

^{α}-calculus, respectively

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## References

Mandelbrot, B.B.The Fractal Geometry of Nature. W. H. Freeman and Company, 1977.

Falconer, K.Fractal Geometry: Mathematical Foundations and Applications. Wiley; 2nded., 2007.

Nicoletti, D. Properties of the Weierstrass function in the time and frequency domains.Chaos, Solitons&Fractals, 5(1):1-8, 1995.

Chalice, D. R. A Characterization of the Cantor function.The American Mathematical Monthly, 98(3):255-258,1991.

Private, A. and Gangal, A.D. Calculus on fractal subsets of the real line - I: formulation.Fractals, 17(01):53-81,2009.

Private, A. and Gangal, A.D. Calculus on fractal subsets of Real Line II: conjugacy with ordinary calculus.Fractals, 19(03):271-290, 2011.

Private, A., Satin, S., and Gangal, A.D. Caclulus on fractal curves inRnFractals, 19(01):15-27, 2011.

Golmankhaneh, A. K., C. Tun ̧c, Sumudu transform in fractal calculus, Applied Mathematics, and Computation.,350:386-401, 2019.

Golmankhaneh, A. K., Statistical Mechanics Involving Fractal Temperature, Fractal Fract, 3 (2019), 20;https://doi.org/10.3390/fractalfract3020020.

Golmankhaneh, A. K., Balankin, A. S., Sub- and super-diffusion on Cantor sets: Beyond the paradox.PhysicsLetters A, 382(14):960-967, 2018.

Golmankhaneh, A. K., Baleanu, D., Calculus on fractals.Fractional Dynamics, Sciendo Migration, 2015.Chapter:307-332.

Jafari, F. K., Asgari, M. S. and Pishkoo, A., Fractal calculus for fractal materials, Fractal Fract, 3 (2019), 8;https://doi.org/10.3390/fractalfract3010008.

Golmankhaneh, A. K., Baleanu, D., Diffraction from fractal grating Cantor sets.Journal of Modern Optics,63(14):1364-1369, 2016.

Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264 (2014), 65-70.

Zhao, D., Luo, M., General conformable fractional derivative and its physical interpretation, Calcolo, 54 (2017)903-917, DOI 10.1007/s10092-017-0213-8.

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## How to Cite

*To Physics Journal*,

*4*, 82-90. Retrieved from https://purkh.com/index.php/tophy/article/view/573