{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T00:33:39Z","timestamp":1772757219071,"version":"3.50.1"},"reference-count":30,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2019,9,9]],"date-time":"2019-09-09T00:00:00Z","timestamp":1567987200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The aim of this paper is to investigate experimentally and to present visually the dynamics of the processes in which in the standard Newton\u2019s root-finding method the classic derivative is replaced by the fractional Riemann\u2013Liouville or Caputo derivatives. These processes applied to polynomials on the complex plane produce images showing basins of attractions for polynomial zeros or images representing the number of iterations required to obtain polynomial roots. These latter images were called by Kalantari as polynomiographs. We use both: the colouring by roots to present basins of attractions, and the colouring by iterations that reveal the speed of convergence and dynamic properties of processes visualised by polynomiographs.<\/jats:p>","DOI":"10.3390\/sym11091143","type":"journal-article","created":{"date-parts":[[2019,9,9]],"date-time":"2019-09-09T04:12:40Z","timestamp":1568002360000},"page":"1143","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":31,"title":["Visual Analysis of the Newton\u2019s Method with Fractional Order Derivatives"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9434-9307","authenticated-orcid":false,"given":"Krzysztof","family":"Gdawiec","sequence":"first","affiliation":[{"name":"Institute of Computer Science, University of Silesia, B\u0229dzi\u0144ska 39, 41-200 Sosnowiec, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5920-5161","authenticated-orcid":false,"given":"Wies\u0142aw","family":"Kotarski","sequence":"additional","affiliation":[{"name":"Institute of Computer Science, University of Silesia, B\u0229dzi\u0144ska 39, 41-200 Sosnowiec, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0492-772X","authenticated-orcid":false,"given":"Agnieszka","family":"Lisowska","sequence":"additional","affiliation":[{"name":"Institute of Computer Science, University of Silesia, B\u0229dzi\u0144ska 39, 41-200 Sosnowiec, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,9,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Mandelbrot, B. 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