University of Kashan Mathematics Interdisciplinary Research 2538-3639 8 4 2023 12 01 The Effect of the Caputo Fractional Derivative on Polynomiography 347 358 114036 10.22052/mir.2022.246736.1367 EN Morteza Bisheh-Niasar Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, ‎Kashan‎, ‎I‎. ‎R‎. ‎Iran Journal Article 2022 08 10 ‎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 $p(z)=z^5-1$ and $ p(z)=-2z^4+z^3+z^2-2z-1 $ on the complex plane and compute the MNI and CAI parameters‎.<br />‎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. https://mir.kashanu.ac.ir/article_114036_37ed4aa322e076928a274e768845b94d.pdf