{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:34:51Z","timestamp":1760150091035,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,19]],"date-time":"2023-10-19T00:00:00Z","timestamp":1697673600000},"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 paper calculates numerical solutions of an extended three-coupled Korteweg\u2013de Vries system by the q-homotopy analysis transformation method (q-HATM), which is a hybrid of the Laplace transform and the q-homotopy analysis method. Multiple investigations inspecting planetary oceans, optical cables, and cosmic plasma have employed the KdV model, significantly contributing to its development. The uniqueness, convergence, and maximum absolute truncation error of this algorithm are demonstrated. A numerical simulation has been performed to validate the accuracy and validity of the proposed approach. With high accuracy and few algorithmic processes, this algorithm supplies a series solution in the form of a recursive relation.<\/jats:p>","DOI":"10.3390\/axioms12100990","type":"journal-article","created":{"date-parts":[[2023,10,19]],"date-time":"2023-10-19T07:15:36Z","timestamp":1697699736000},"page":"990","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["High-Performance Computational Method for an Extended Three-Coupled Korteweg\u2013de Vries System"],"prefix":"10.3390","volume":"12","author":[{"given":"Panpan","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiufang","family":"Feng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,19]]},"reference":[{"key":"ref_1","unstructured":"Leung, A.W. (2013). Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering, Springer Science Business Media."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1007\/s11082-018-1459-3","article-title":"Soliton solutions and stability analysis for some conformable nonlinear partial differential equations in mathematical physics","volume":"50","author":"Inc","year":"2018","journal-title":"Opt. Quantum Electron."},{"key":"ref_3","first-page":"553","article-title":"Exact Traveling Wave Solutions of Nonlinear PDEs in Mathematical Physics Using the Modified Simple Equation Method","volume":"8","author":"Zayed","year":"2013","journal-title":"Appl. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1002\/cnm.857","article-title":"Numerical solution of the oxygen diffusion in absorbing tissue with a moving boundary","volume":"22","author":"Boureghda","year":"2006","journal-title":"Commun. Numer. Methods Eng."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Botmart, T., Alotaibi, B.M., Shah, R., El-Sherif, L.S., and El-Tantawy, S.A. (2022). A Reliable Way to Deal with the Coupled Fractional Korteweg-De Vries Equations within the Caputo Operator. Symmetry, 14.","DOI":"10.3390\/sym14112452"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Aljahdaly, N.H., Akg\u00fcl, A., Shah, R., Mahariq, I., and Kafle, J. (2022). A comparative analysis of the fractional-order coupled Korteweg\u2013De Vries equations with the Mittag\u2013Leffler law. J. Math., 8876149\u20138876178.","DOI":"10.1155\/2022\/8876149"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Shah, N.A., Alyousef, H.A., El-Tantawy, S.A., Shah, R., and Chung, J.D. (2022). Analytical investigation of fractional-order Korteweg-De-Vries-type equations under Atangana-Baleanu-Caputo operator: Modeling nonlinear waves in a plasma and fluid. Symmetry, 14.","DOI":"10.3390\/sym14040739"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"108775","DOI":"10.1016\/j.aml.2023.108775","article-title":"AKNS type reduced integrable bi-Hamiltonian hierarchies with four potentials","volume":"145","author":"Ma","year":"2023","journal-title":"Appl. Math. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1180","DOI":"10.1134\/S0040577923080093","article-title":"Four-component integrable hierarchies of Hamiltonian equations with (m+n+2)th-order Lax pairs","volume":"216","author":"Ma","year":"2023","journal-title":"Theor. Math. Phys."},{"key":"ref_10","first-page":"499","article-title":"On the homotopy analysis method for nonlinear problems","volume":"147","author":"Liao","year":"2004","journal-title":"Appl. Math. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2455","DOI":"10.1016\/j.nonrwa.2008.05.003","article-title":"Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method","volume":"10","author":"Liao","year":"2009","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1111\/j.1467-9590.2007.00387.x","article-title":"A general approach to obtain series solutions of nonlinear differential equations","volume":"119","author":"Liao","year":"2007","journal-title":"Stud. Appl. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/0020-7462(94)00054-E","article-title":"An approximate solution technique which does not depend upon small parameters: A special example","volume":"30","author":"Liao","year":"1995","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_14","first-page":"51","article-title":"The Q-homotopy analysis method (QHAM)","volume":"8","author":"Huseen","year":"2012","journal-title":"Int. J. Appl. Math. Mech."},{"key":"ref_15","first-page":"8363","article-title":"The improved homotopy analysis method for the Thomas\u2013Fermi equation","volume":"218","author":"Zhao","year":"2012","journal-title":"Appl. Math Comput."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Mesloub, S., and Alsaud, H. (2023). Exploring the Efficiency of the q-Homotopy Analysis Transform Method for Solving a Fractional Initial Boundary Value Problem with a Nonlocal Condition. Axioms, 12.","DOI":"10.3390\/axioms12080790"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Shah, R., Khan, H., and Baleanu, D. (2019). Fractional Whitham\u2013Broer\u2013Kaup equations within modified analytical approaches. Axioms, 8.","DOI":"10.3390\/axioms8040125"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1016\/j.matcom.2021.05.030","article-title":"A new computational technique for the analytic treatment of time-fractional Emden Fowler equations","volume":"190","author":"Malagi","year":"2021","journal-title":"Math. Comput. Simul."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1016\/j.matcom.2019.06.005","article-title":"A novel technique for (2+1)-dimensional time-fractional coupled Burgers equations","volume":"166","author":"Veeresha","year":"2019","journal-title":"Math. Comput. Simul."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"481","DOI":"10.12988\/ijcms.2013.13048","article-title":"On Convergence of q-Homotopy Analysis Method","volume":"8","author":"Huseen","year":"2013","journal-title":"Int. J. Contemp. Math. Sci."},{"key":"ref_21","first-page":"124637","article-title":"An efficient technique for nonlinear time-fractional Klein-Fock-Gordon equation","volume":"364","author":"Veeresha","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2250229","DOI":"10.1142\/S0219887822502292","article-title":"Singular manifold, auto-B\u00e4cklund transformations and symbolic-computation steps with solitons for an extended three-coupled Korteweg-de Vries system","volume":"19","author":"Gao","year":"2022","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Gao, X.Y., Guo, Y.J., Shan, W.R., and Zhou, T.Y. (2023). Report on an extended three-coupled Korteweg-de Vries system. Ric. Mat.","DOI":"10.1007\/s11587-023-00769-x"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s12346-021-00512-7","article-title":"Hetero-B\u00e4cklund transformation, bilinear forms and N solitons for a generalized three-coupled Korteweg-de Vries system","volume":"20","author":"Gao","year":"2021","journal-title":"Qual. Theory Dyn. Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1088\/0253-6102\/63\/1\/05","article-title":"New Exact Solutions of Fractional Zakharov-Kuznetsov and Modified Zakharov-Kuznetsov Equations Using Fractional Sub-Equation Method","volume":"63","author":"Ray","year":"2015","journal-title":"Commun. Theor. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"055003","DOI":"10.1088\/1572-9494\/ab7707","article-title":"Analytical and approximate solutions of (2+1)-dimensional time-fractional Burgers-Kadomtsev-Petviashvili equation","volume":"72","author":"Senol","year":"2020","journal-title":"Commun. Theor. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2240019","DOI":"10.1142\/S0218348X22400199","article-title":"Numerical simulations for the variable order two-dimensional reaction sub-diffusion equation: Linear and Nonlinear","volume":"30","author":"Adel","year":"2022","journal-title":"Fractals"},{"key":"ref_28","first-page":"286","article-title":"Generalized Taylor\u2019s formula","volume":"186","author":"Odibat","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_29","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1007\/BF00973476","article-title":"On the integral mean value theorem","volume":"34","author":"Nikonorov","year":"1993","journal-title":"Sib. Math. J."},{"key":"ref_31","first-page":"215","article-title":"A new tool to study real dynamics: The convergence plane","volume":"248","author":"Magrenan","year":"2014","journal-title":"Appl. Math. Comput."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/990\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:09:47Z","timestamp":1760130587000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/10\/990"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,19]]},"references-count":31,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["axioms12100990"],"URL":"https:\/\/doi.org\/10.3390\/axioms12100990","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,10,19]]}}}