{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T12:40:20Z","timestamp":1780663220875,"version":"3.54.1"},"reference-count":32,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T00:00:00Z","timestamp":1733270400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The time-fractional coupled Korteweg\u2013De Vries equations (TFCKdVEs) serve as a vital framework for modeling diverse real-world phenomena, encompassing wave propagation and the dynamics of shallow water waves on a viscous fluid. This paper introduces a precise and resilient numerical approach, termed the Conformable Hyperbolic Non-Polynomial Spline Method (CHNPSM), for solving TFCKdVEs. The method leverages the inherent symmetry in the structure of TFCKdVEs, exploiting conformable derivatives and hyperbolic non-polynomial spline functions to preserve the equations\u2019 symmetry properties during computation. Additionally, first-derivative finite differences are incorporated to enhance the method\u2019s computational accuracy. The convergence order, determined by studying truncation errors, illustrates the method\u2019s conditional stability. To validate its performance, the CHNPSM is applied to two illustrative examples and compared with existing methods such as the meshless spectral method and Petrov\u2013Galerkin method using error norms. The results underscore the CHNPSM\u2019s superior accuracy, showcasing its potential for advancing numerical computations in the domain of TFCKdVEs and preserving essential symmetries in these physical systems.<\/jats:p>","DOI":"10.3390\/sym16121610","type":"journal-article","created":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T10:07:10Z","timestamp":1733306830000},"page":"1610","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Hyperbolic Non-Polynomial Spline Approach for Time-Fractional Coupled KdV Equations: A Computational Investigation"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1567-0264","authenticated-orcid":false,"given":"Miguel","family":"Vivas-Cortez","sequence":"first","affiliation":[{"name":"Faculty of Exact and Natural Sciences, School of Physical Sciences and Mathematics, Pontifical Catholic University of Ecuador, Av. 12 de Octubre 1076 y Roca, Sede Quito 17-01-2184, Ecuador"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0206-3828","authenticated-orcid":false,"given":"Majeed A.","family":"Yousif","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Zakho, Duhok 42001, Iraq"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6837-8075","authenticated-orcid":false,"given":"Pshtiwan Othman","family":"Mohammed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah 46001, Iraq"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2855-7535","authenticated-orcid":false,"given":"Alina Alb","family":"Lupas","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9927-2388","authenticated-orcid":false,"given":"Ibrahim S.","family":"Ibrahim","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education, University of Zakho, Duhok 42001, Iraq"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8833-6585","authenticated-orcid":false,"given":"Nejmeddine","family":"Chorfi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1080\/00207390410001686571","article-title":"A brief historical introduction to fractional calculus","volume":"35","author":"Debnath","year":"2004","journal-title":"Int. J. Math. Educ. Sci. Technol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"828","DOI":"10.1108\/HFF-07-2016-0278","article-title":"Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm","volume":"28","year":"2018","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1122\/1.549724","article-title":"A theoretical basis for the application of fractional calculus to viscoelasticity","volume":"27","author":"Bagley","year":"1983","journal-title":"J. Rheol."},{"key":"ref_4","first-page":"214","article-title":"Fractional derivatives for physicists and engineers","volume":"27","author":"Uchaikin","year":"2013","journal-title":"Chaos"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Agarwal, P., Baleanu, D., Chen, Y., Momani, S., and Machado, T. (2018). Fractional Calculus, Springer.","DOI":"10.1007\/978-981-15-0430-3"},{"key":"ref_6","unstructured":"Podlubny, I. (1999). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press."},{"key":"ref_7","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations, Elsevier\/North-Holland."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/j.cam.2014.01.002","article-title":"A new definition of fractional derivative","volume":"264","author":"Khalil","year":"2014","journal-title":"J. Comput. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"25343","DOI":"10.3934\/math.20231293","article-title":"Application of trigonometric B-spline functions for solving Caputo time fractional gas dynamics equation","volume":"8","author":"Noureen","year":"2023","journal-title":"AIMS Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1080\/16583655.2022.2089396","article-title":"Consistent travelling waves solutions to the non-linear time fractional Klein\u2013Gordon and Sine-Gordon equations through extended tanh-function approach","volume":"16","author":"Sadiya","year":"2022","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-018-1743-3","article-title":"Non-polynomial spline method for the time-fractional nonlinear Schr\u00f6dinger equation","volume":"2018","author":"Li","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1016\/j.matcom.2023.11.033","article-title":"The fractional non-polynomial spline method: Precision and modeling improvements","volume":"218","author":"Yousif","year":"2024","journal-title":"Math. Comput. Simul."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., Saad, K.M., and Hamanah, W.M. (2022). Certain new models of the multi-space fractal-fractional Kuramoto-Sivashinsky and Korteweg-de Vries equations. Mathematics, 10.","DOI":"10.3390\/math10071089"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1007\/s11075-018-0613-0","article-title":"Numerical solutions to time-fractional stochastic partial differential equations","volume":"82","author":"Zou","year":"2019","journal-title":"Numer. Algorithms"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"101879","DOI":"10.1016\/j.asej.2022.101879","article-title":"Conformable double Sumudu transformations: An efficient approximation solutions to the fractional coupled Burger\u2019s equation","volume":"14","author":"Mohamed","year":"2023","journal-title":"Ain Shams Eng. J."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1243","DOI":"10.1016\/j.asej.2020.03.016","article-title":"Smooth expansion to solve high-order linear conformable fractional PDEs via residual power series method: Applications to physical and engineering equations","volume":"11","author":"Oqielat","year":"2020","journal-title":"Ain Shams Eng. J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.28924\/2291-8639-21-2023-5","article-title":"Computational approach for a singularly perturbed differential equations with mixed shifts using a non-polynomial spline","volume":"21","author":"Ragula","year":"2023","journal-title":"Int. J. Anal. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1120","DOI":"10.1002\/num.22570","article-title":"Numerical solutions for solving model time-fractional Fokker\u2013Planck equation","volume":"37","author":"Mahdy","year":"2021","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1134\/S106192082103002X","article-title":"Lie symmetry and exact solution of the time-fractional Hirota-Satsuma Korteweg-de Vries system","volume":"28","author":"Srivastava","year":"2021","journal-title":"Russ. J. Math. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1580","DOI":"10.3934\/math.2022092","article-title":"Localized modes in time-fractional modified coupled Korteweg-de Vries equation with singular and non-singular kernels","volume":"7","author":"Khan","year":"2022","journal-title":"AIMS Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1186\/s13662-020-02625-w","article-title":"A reliable technique to study nonlinear time-fractional coupled Korteweg\u2013de Vries equations","volume":"2020","author":"Akinyemi","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"226","DOI":"10.1016\/j.matcom.2022.12.028","article-title":"Petrov\u2013Galerkin approximation of time-fractional coupled Korteweg\u2013de Vries equation for propagation of long wave in shallow water","volume":"207","author":"Arifeen","year":"2023","journal-title":"Math. Comput. Simul."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1007\/s40995-021-01065-9","article-title":"Radial basis functions collocation method for numerical solution of coupled Burgers\u2019 and Korteweg-de Vries equations of fractional order","volume":"45","author":"Hussain","year":"2021","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1897","DOI":"10.1016\/j.asej.2016.10.010","article-title":"A numerical solution of time-fractional coupled Korteweg-de Vries equation by using spectral collection method","volume":"9","author":"Albuohimad","year":"2018","journal-title":"Ain Shams Eng. J."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Alzahrani, A.B.M., and Alhawael, G. (2023). Novel computations of the time-fractional coupled Korteweg\u2013de Vries equations via non-singular kernel operators in terms of the natural transform. Symmetry, 15.","DOI":"10.3390\/sym15112010"},{"key":"ref_26","first-page":"321","article-title":"Meshless spectral method for solution of time-fractional coupled KdV equations","volume":"341","author":"Hussain","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2564","DOI":"10.1080\/00207160.2021.1906422","article-title":"Crank\u2013Nicolson finite difference method for time-fractional coupled KdV equation","volume":"98","author":"Kawala","year":"2021","journal-title":"Int. J. Comput. Math."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Saad, K.M., and Srivastava, H.M. (2023). Numerical solutions of the multi-space fractional-order coupled Korteweg-de Vries equation with several different kernels. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7100716"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1016\/j.rinp.2016.06.003","article-title":"Lie symmetry analysis, conservation laws and exact solutions of the seventh-order time fractional Sawada\u2013Kotera\u2013Ito equation","volume":"6","author":"Khalique","year":"2016","journal-title":"Results Phys."},{"key":"ref_30","first-page":"439","article-title":"Lie symmetry analysis of the time fractional KdV-type equation","volume":"233","author":"Hu","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.chaos.2017.04.020","article-title":"Lie symmetries, symmetry reductions and conservation laws of time fractional modified Korteweg\u2013de Vries (mkdv) equation","volume":"100","author":"Akbulut","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"100166","DOI":"10.1016\/j.exco.2024.100166","article-title":"Symmetry analysis, exact solutions and conservation laws of time fractional Caudrey\u2013Dodd\u2013Gibbon equation","volume":"6","author":"Yu","year":"2024","journal-title":"Ex. Counterexamples"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1610\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:46:59Z","timestamp":1760114819000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1610"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,4]]},"references-count":32,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["sym16121610"],"URL":"https:\/\/doi.org\/10.3390\/sym16121610","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,4]]}}}