The Effect of the Caputo Fractional Derivative on Polynomiography

Document Type : Original Scientific Paper


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



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.


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