{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T23:43:34Z","timestamp":1773272614199,"version":"3.50.1"},"reference-count":20,"publisher":"Wiley","license":[{"start":{"date-parts":[[2024,5,20]],"date-time":"2024-05-20T00:00:00Z","timestamp":1716163200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Federal University of Technology Babura"},{"name":"Women University Swabi"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2024,5,20]]},"abstract":"<jats:p>In this article, we explore the utilization of the Caputo derivative and the Riemann\u2013Liouville (R\u2013L) fractional integral to analyze the optimal auxiliary function method for approximating fractional nonlinear long waves. Approximate long wave equation with a distinct dispersion relation offers the most accurate description of shallow water wave properties. Various methods, including the Adomian decomposition technique, the variational iteration method, the optimum homotopy asymptotic method, and the new iterative technique, have been employed and compared to those obtained using the fractional-order approximate long wave equation. The results of our study indicate that the optimal auxiliary function method is highly successful and practically simple, achieving better and more rapid convergence after just one repetition. This method is recognized as an efficient approach, demonstrating high precision in solving intriguing and intricate problems. Furthermore, it proves to be more time and resource efficient than other relevant analytical techniques, leading to significant savings in both volume and time. Compared to the Adomian decomposition technique, the new iterative technique, the variational iteration method, and the optimum homotopy asymptotic method, the suggested technique is extremely accurate computationally. It is also easy to analyze and solve fractionally linked nonlinear complex phenomena that arise in science and technology. We present the numerical and graphical findings that support these conclusions.<\/jats:p>","DOI":"10.1155\/2024\/8357221","type":"journal-article","created":{"date-parts":[[2024,5,21]],"date-time":"2024-05-21T01:35:08Z","timestamp":1716255308000},"page":"1-10","source":"Crossref","is-referenced-by-count":1,"title":["Utilizing the Optimal Auxiliary Function Method for the Approximation of a Nonlinear Long Wave System considering Caputo Fractional Order"],"prefix":"10.1155","volume":"2024","author":[{"given":"Aaqib","family":"Iqbal","sequence":"first","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan"}]},{"given":"Rashid","family":"Nawaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan"},{"name":"Department of Mathematics, University of South Australia, Adelaide, SA 5000, Australia"}]},{"given":"Hina","family":"Hina","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Women University Swabi, Swabi 23430, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2999-7751","authenticated-orcid":true,"given":"Abdulaziz Garba","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Federal University of Technology, Babura, Jigawa State, Nigeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8034-1475","authenticated-orcid":true,"given":"Homan","family":"Emadifar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai 602 105, Tamil Nadu, India"},{"name":"MEU Research Unit, Middle East University, Amman 11831, Jordan"},{"name":"Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1155\/2022\/6231921"},{"key":"2","first-page":"1","volume-title":"A brief history and exposition of the fundamental theory of fractional calculus","year":"1974"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1155\/2021\/7979365"},{"key":"4","year":"1974"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.3390\/axioms12090881"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1007\/s11082-022-03640-9"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/j.aej.2023.10.030"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1007\/s10773-023-05400-1"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2005.05.017"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.3923\/jas.2011.1416.1420"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1016\/j.icheatmasstransfer.2011.03.025"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2006.02.056"},{"key":"13","volume-title":"Fractional integrals and derivatives","year":"1993"},{"key":"14","volume":"204","year":"2006","journal-title":"Elsevier"},{"key":"15","year":"1998","journal-title":"Elsevier"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1111\/j.1365-246x.1967.tb02303.x"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1002\/mma.3151"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2004.08.012"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.3390\/e17096519"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2006.12.054"}],"container-title":["Complexity"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2024\/8357221.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2024\/8357221.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/complexity\/2024\/8357221.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,5,29]],"date-time":"2024-05-29T15:25:41Z","timestamp":1716996341000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/complexity\/2024\/8357221\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,20]]},"references-count":20,"alternative-id":["8357221","8357221"],"URL":"https:\/\/doi.org\/10.1155\/2024\/8357221","relation":{},"ISSN":["1076-2787","1099-0526"],"issn-type":[{"value":"1076-2787","type":"print"},{"value":"1099-0526","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,20]]}}}