{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T16:05:50Z","timestamp":1768838750648,"version":"3.49.0"},"reference-count":32,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,14]],"date-time":"2023-09-14T00:00:00Z","timestamp":1694649600000},"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>In this article, we investigate the utilization of Riemann\u2013Liouville\u2019s fractional integral and the Caputo derivative in the application of the Optimal Auxiliary Function Method (OAFM). The extended OAFM is employed to analyze fractional non-linear coupled ITO systems and non-linear KDV systems, which feature equations of a fractional order in time. We compare the results obtained for the ITO system with those derived from the Homotopy Perturbation Method (HPM) and the New Iterative Method (NIM), and for the KDV system with the Laplace Adomian Decomposition Method (LADM). OAFM demonstrates remarkable convergence with a single iteration, rendering it highly effective. In contrast to other existing analytical approaches, OAFM emerges as a dependable and efficient methodology, delivering high-precision solutions for intricate problems while saving both computational resources and time. Our results indicate superior accuracy with OAFM in comparison to HPM, NIM, and LADM. Additionally, we enhance the accuracy of OAFM through the introduction of supplementary auxiliary functions.<\/jats:p>","DOI":"10.3390\/axioms12090881","type":"journal-article","created":{"date-parts":[[2023,9,15]],"date-time":"2023-09-15T04:06:13Z","timestamp":1694750773000},"page":"881","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A New Extension of Optimal Auxiliary Function Method to Fractional Non-Linear Coupled ITO System and Time Fractional Non-Linear KDV System"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4773-8446","authenticated-orcid":false,"given":"Rashid","family":"Nawaz","sequence":"first","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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aaqib","family":"Iqbal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hina","family":"Bakhtiar","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Women University Swabi, Swabi 23430, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wissal Audah","family":"Alhilfi","sequence":"additional","affiliation":[{"name":"Marine Science Centre, University of Basrah, Basrah 61001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0911-9943","authenticated-orcid":false,"given":"Nicholas","family":"Fewster-Young","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of South Australia, Adelaide, SA 5000, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2959-4212","authenticated-orcid":false,"given":"Ali Hasan","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah 61001, Iraq"},{"name":"Institute of Mathematics, University of Debrecen, Pf. 400, H-4002 Debrecen, Hungary"},{"name":"College of Engineering Technology, National University of Science and Technology, Nasiriyah 64001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ana Danca","family":"Po\u021bclean","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Technical University of Cluj-Napoca, Str. Memorandumului nr. 28, 400114 Cluj-Napoca, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,14]]},"reference":[{"key":"ref_1","first-page":"69","article-title":"On Riemann-Liouville integral of ultra-hyperbolic type","volume":"Volume 16","author":"Nozaki","year":"1964","journal-title":"Kodai Mathematical Seminar Reports"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Baleanu, D., G\u00fcven\u00e7, Z.B., and Machado, J.T. (2010). New Trends in Nanotechnology and Fractional Calculus Applications, Springer.","DOI":"10.1007\/978-90-481-3293-5"},{"key":"ref_3","unstructured":"Caputo, M. (1969). Elasticita e Dissipazione, Zanichelli."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s10659-011-9346-1","article-title":"A fractional model of continuum mechanics","volume":"107","author":"Drapaca","year":"2012","journal-title":"J. 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