The Effect of the Caputo Fractional Derivative on Polynomiography

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 $p(z)=z^5-1$ and $ p(z)=-2z^4+z^3+z^2-2z-1 $ on the complex plane and compute the MNI and CAI parameters‎.‎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

• caputo‎ ‎fractional‎ ‎derivative
• ‎ ‎root-finding‎ ‎method
• ‎ ‎newton's method‎
• ‎polynomiography