{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:29:32Z","timestamp":1760059772386,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T00:00:00Z","timestamp":1751846400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This article introduces two enhanced techniques: the Natural Transform Iterative Method (NTIM) and the Optimal Auxiliary Function Method (OAFM). These approaches provide a close approximation for solving fractional-order Navier\u2013Stokes equations, which are widely employed in domains such as biology, ecology, and applied sciences. By comparing the solutions derived from these methods to exact solutions, it is clear that they provide accurate and efficient outcomes. These findings highlight the straightforward yet effective use of these methodologies in modeling engineering systems. Navier\u2013Stokes equations have numerous practical uses, including analyzing fluid flow in pipelines and channels, predicting weather patterns, and constructing aircraft and vehicles.<\/jats:p>","DOI":"10.3390\/axioms14070521","type":"journal-article","created":{"date-parts":[[2025,7,7]],"date-time":"2025-07-07T10:03:27Z","timestamp":1751882607000},"page":"521","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Comprehensive Study on the Applications of NTIM and OAFM in Analyzing Fractional Navier\u2013Stokes Equations"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-8596-4895","authenticated-orcid":false,"given":"Siddiq Ur","family":"Rehman","sequence":"first","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan"},{"name":"Department of Petroleum Engineering, The University of Adelaide, North Terrace Campus, 230 North Terrace, Adelaide 5005, Australia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4773-8446","authenticated-orcid":false,"given":"Rashid","family":"Nawaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan"},{"name":"School of Mathematical and Statistical Sciences, University of South Australia, St Bernards Road Magill, Adelaide 5072, Australia"}]},{"given":"Faisal","family":"Zia","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0911-9943","authenticated-orcid":false,"given":"Nick","family":"Fewster-Young","sequence":"additional","affiliation":[{"name":"School of Mathematical and Statistical Sciences, University of South Australia, St Bernards Road Magill, Adelaide 5072, Australia"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18576\/pfda\/020101","article-title":"Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels","volume":"2","author":"Caputo","year":"2016","journal-title":"Prog. 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