The Effect of the Caputo Fractional Derivative on Polynomiography

This paper presents the visualization process of finding the roots of a complex polynomial - which is called polynomiography - by the Caputo frac tional 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 polynomi als p(z) = z5 − 1 and p(z) = −2z4 + z3 + z2 − 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 pat terns. The obtained patterns show that the proposed method has potential artistic application.