{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:35:48Z","timestamp":1760146548428,"version":"build-2065373602"},"reference-count":56,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T00:00:00Z","timestamp":1731456000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Shaanxi Province of China","award":["2024JC-YBMS-014","SLGQD2017-08"],"award-info":[{"award-number":["2024JC-YBMS-014","SLGQD2017-08"]}]},{"name":"Research Foundation for the Doctoral Program of Shaanxi University of Technology","award":["2024JC-YBMS-014","SLGQD2017-08"],"award-info":[{"award-number":["2024JC-YBMS-014","SLGQD2017-08"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper presents an efficient numerical manifold method for solving the Burgers\u2019 equation. To improve accuracy and streamline the solution process, we apply a nonlinear function transformation technique that reformulates the original problem into a linear diffusion equation. We utilize a dual cover mesh along with an explicit multi-step time integration method for spatial and temporal discretization, respectively. Constant cover functions are employed across mathematical covers, interconnected by a linear weight function for each manifold element. The full discretization formulation is derived using the Galerkin weak form. To efficiently compute the inverse of the symmetric positive definite mass matrix, we employ the Crout algorithm. The performance and convergence of our method are rigorously evaluated through several benchmark numerical tests. Extensive comparisons with exact solutions and alternative methods demonstrate that our approach delivers an accurate, stable, and efficient computational scheme for the Burgers\u2019 equation.<\/jats:p>","DOI":"10.3390\/sym16111521","type":"journal-article","created":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T06:23:16Z","timestamp":1731478996000},"page":"1521","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Explicit Runge\u2013Kutta Numerical Manifold Method for Solving the Burgers\u2019 Equation via the Hopf\u2013Cole Transformation"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8829-8168","authenticated-orcid":false,"given":"Yue","family":"Sun","sequence":"first","affiliation":[{"name":"School of Mathematics and Computer Science, Shaanxi University of Technology, Hanzhong 723001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-2414-6100","authenticated-orcid":false,"given":"Qian","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Shaanxi University of Technology, Hanzhong 723001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tao","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Shaanxi University of Technology, Hanzhong 723001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9679-1446","authenticated-orcid":false,"given":"Longquan","family":"Yong","sequence":"additional","affiliation":[{"name":"School of Mathematics and Computer Science, Shaanxi University of Technology, Hanzhong 723001, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/S0065-2156(08)70100-5","article-title":"A mathematical model illustrating the theory of turbulence","volume":"1","author":"Burgers","year":"1948","journal-title":"Adv. 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