{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:11:31Z","timestamp":1760058691695,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T00:00:00Z","timestamp":1745193600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Graduate Studies and Scientific Research at Qassim University","award":["QU-APC-2025"],"award-info":[{"award-number":["QU-APC-2025"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This article investigates and analyzes the diverse patterns of Julia sets generated by new classes of generalized exponential and sine rational functions. Using a generalized viscosity approximation-type iterative method, we derive escape criteria to visualize the Julia sets of these functions. This approach enhances existing algorithms, enabling the visualization of intricate fractal patterns as Julia sets. We graphically illustrate the variations in size and shape of the images as the iteration parameters change. The new fractals obtained are visually appealing and attractive. Moreover, we observe fascinating behavior in Julia sets when certain input parameters are fixed, while the values of n and m vary. We believe the conclusions of this study will inspire and motivate researchers and enthusiasts with a strong interest in fractal geometry.<\/jats:p>","DOI":"10.3390\/axioms14040322","type":"journal-article","created":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T20:38:26Z","timestamp":1745267906000},"page":"322","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Generation of Julia Sets for a Novel Class of Generalized Rational Functions via Generalized Viscosity Iterative Method"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8447-0709","authenticated-orcid":false,"given":"Iqbal","family":"Ahmad","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, College of Engineering, Qassim University, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7712-3947","authenticated-orcid":false,"given":"Ahmad","family":"Almutlg","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, Qassim University, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1169","DOI":"10.1007\/s11071-013-1032-3","article-title":"Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography","volume":"74","author":"Muthukumar","year":"2013","journal-title":"Nonlinear Dyn."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1080\/17513472.2021.1943998","article-title":"Iterated inversion system: An algorithm for efficiently visualizing Kleinian groups and extending the possibilities of fractal art","volume":"15","author":"Nakamura","year":"2021","journal-title":"J. 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