# The Effect of the Caputo Fractional Derivative on Polynomiography

Document Type : Original Scientific Paper

Author

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, I. R. Iran

10.22052/mir.2022.246736.1367

Abstract

This paper presents the visualization process of finding the roots of a complex polynomial - which is called polynomiography - by the Caputo fractional derivative. In this work, we substitute the variable-order Caputo fractional derivative for classic derivative in Newton’s iterative method. To investigate the proposed root-finding method, we apply it for two polynomials p(z) = z5 −1 and p(z) = −2z4 + z3 + z2 −2z−1 on the complex plan and compute the MNI and CAI parameters. Presented examples show that through the expressed process, we can obtain very interesting fractal patterns. The obtained patterns show that the proposed method has potential artistic application.

Keywords

#### References

[1] R. Agarwal, D. ORegan and D. Sahu, Iterative construction of xed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (1) (2007) 61 - 79.
[2] A. Akgül, A. Cordero and J. R. Torregrosa, A fractional Newton method with
2αth-order of convergence and its stability, Appl. Math. Letters 98 (2019) 344 - 351.
[3] M. Bisheh-Niasar and K. Gdawiec, Bisheh-NiasarSaadatmandi root finding method via the S-iteration with periodic parameters and its polynomiography, Math. Comput. Simul. 160 (2019) 1 - 12.
[4] M. Bisheh-Niasar and A. Saadatmandi, Some novel Newton-type methods for solving nonlinear equations,
Bol. Soc. Parana. Mat. 38 (3) (2020) 111 - 123.
[5] G. Candelario, A. Cordero, J. R. Torregrosa and M. P. Vassileva, An optimal and low computational cost fractional Newton-type method for solving nonlinear equations,
Appl. Math. Letters 124 (2022) 107650.
[6] G. Candelario, A. Cordero and J. R. Torregrosa, Multipoint fractional iterative methods with
(2α + 1)th-order of convergence for solving nonlinear problems, Mathematics 8 (3) (2020) 452.
[7] A. Cordero, C. Jordán and J. R. Torregrosa, One-point Newton-type iterative methods: A unified point of view,
J. Comput. Appl. Math. 275 (2015) 366 - 374.
[8] A. Cordero, I. Girona and J. R. Torregrosa, A variant of chebyshevs method with
3αth-order of convergence by using fractional derivatives, Symmetry 11 (8) 1017.
[9] K. Gdawiec, W. Kotarski and A. Lisowska, Polynomiography based on the nonstandard Newton-like root nding methods,
Abstr. Appl. Anal. 2015 (2015) 797594.
[10] K. Gdawiec and W. Kotarski, Polynomiography for the polynomial innity norm via Kalantaris formula and nonstandard iterations,
Appl. Math. Comput. 307 (2017) 17 - 30.
[11] K. Gdawiec, Polynomiography and various convergence tests, In: V. Skala (ed.), WSCG 2013 Communication Papers Proceedings, pages 15-20, Plzen, Czech Republic, 2013.
[12] K. Gdawiec, W. Kotarski and A. Lisowska, Visual analysis of the Newtons method with fractional order derivatives,
Symmetry 11 (9) (2019) 1143.
[13] K. Gdawiec, W. Kotarski and A. Lisowska, Newtons method with fractional derivatives and various iteration processes via visual analysis,
Numer. Algor. 86 (3) (2021) 953 - 1010.
[14] S. Ishikawa, Fixed points by a new iteration method,
Proc. Amer. Math. Soc. 44 (3) (1974) 147150.
[15] B. Kalantari, Polynomiography and applications in art, education and science,
Comput. Graph. 28 (3) (2004) 417430.
[16] B. Kalantari, Polynomial Root-Finding and Polynomiography, World Scientific Publishing Co. Pt. Ltd., Singapore, 2009.
[17] W. Mann, Mean value methods in iteration, 2nd ed.,
Proc. Amer. Math. Soc. 4 (3) (1953) 506 - 510.
[18] V. Torkashvand, T. Lotfi and M. A. Fariborzi Araghi, A new family of adaptive methods with memory for solving nonlinear equations,
Math. Sci. 13 (2019) 1 - 20.