{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T14:01:38Z","timestamp":1770732098036,"version":"3.49.0"},"reference-count":42,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T00:00:00Z","timestamp":1745193600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European Union\u2014NextGenerationEU","award":["BG-RRP-2.013-0001"],"award-info":[{"award-number":["BG-RRP-2.013-0001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Numerical solutions of turbulent mass-transfer diffusion present challenges due to the nonlinearity of the elliptic\u2013parabolic degeneracy of the mathematical models. Our main result in this paper concerns the development and implementation of an efficient high-order numerical method that is fourth-order accurate in space and second-order accurate in time for computing both the solution and its gradient for a Barenblatt-type equation. First, we reduce the original Neumann boundary value problem to a Dirichlet problem for the equation of the solution gradient. This problem is then solved by a compact fourth-order spatial approximation. To implement the numerical discretization, we employ Newton\u2019s iterative method. Then, we compute the original solution while preserving the order of convergence. Numerical test results confirm the efficiency and accuracy of the proposed numerical scheme.<\/jats:p>","DOI":"10.3390\/axioms14040319","type":"journal-article","created":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T20:38:26Z","timestamp":1745267906000},"page":"319","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["High-Order Approximations for a Pseudoparabolic Equation of Turbulent Mass-Transfer Diffusion"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0360-3651","authenticated-orcid":false,"given":"Miglena N.","family":"Koleva","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Natural Sciences and Education, \u201cAngel Kanchev\u201d University of Ruse, 8 Studentska Str., 7017 Ruse, Bulgaria"}]},{"given":"Lubin G.","family":"Vulkov","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Statistics, Faculty of Natural Sciences and Education, \u201cAngel Kanchev\u201d University of Ruse, 8 Studentska Str., 7017 Ruse, Bulgaria"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1286","DOI":"10.1016\/0021-8928(60)90107-6","article-title":"Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]","volume":"24","author":"Barenblatt","year":"1960","journal-title":"J. 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