{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T22:08:01Z","timestamp":1773180481686,"version":"3.50.1"},"reference-count":20,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,4,23]],"date-time":"2024-04-23T00:00:00Z","timestamp":1713830400000},"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>The paper concerns a nonlinear second-order system of coupled PDEs, having the principal part in divergence form and subject to in-homogeneous dynamic boundary conditions, for both \u03b8(t,x) and \u03c6(t,x). Two main topics are addressed here, as follows. First, under a certain hypothesis on the input data, f1, f2, w1, w2, \u03b1, \u03be, \u03b80, \u03b10, \u03c60, and \u03be0, we prove the well-posedness of a solution \u03b8,\u03b1,\u03c6,\u03be, which is \u03b8(t,x),\u03b1(t,x)\u2208Wp1,2(Q)\u00d7Wp1,2(\u03a3), \u03c6(t,x),\u03be(t,x)\u2208W\u03bd1,2(Q)\u00d7Wp1,2(\u03a3), \u03bd=min{q,\u03bc}. According to the new formulation of the problem, we extend the previous results, allowing the new mathematical model to be even more complete to describe the diversity of physical phenomena to which it can be applied: interface problems, image analysis, epidemics, etc. The main goal of the present paper is to develop an iterative scheme of fractional-step type in order to approximate the unique solution to the nonlinear second-order system. The convergence result is established for the new numerical method, and on the basis of this approach, a conceptual algorithm, alg-frac_sec-ord_u+varphi_dbc, is elaborated. The benefit brought by such a method consists of simplifying the computations so that the time required to approximate the solutions decreases significantly. Some conclusions are given as well as new research topics for the future.<\/jats:p>","DOI":"10.3390\/axioms13050286","type":"journal-article","created":{"date-parts":[[2024,4,23]],"date-time":"2024-04-23T08:08:27Z","timestamp":1713859707000},"page":"286","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["On the Convergence of an Approximation Scheme of Fractional-Step Type, Associated to a Nonlinear Second-Order System with Coupled In-Homogeneous Dynamic Boundary Conditions"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9056-0911","authenticated-orcid":false,"given":"Constantin","family":"Fetec\u0103u","sequence":"first","affiliation":[{"name":"Academy of Romanian Scientists, 54 Splaiul Independentei, 050094 Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7250-3131","authenticated-orcid":false,"given":"Costic\u0103","family":"Moro\u015fanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, \u201cAlexandru Ioan Cuza\u201d University, Bd. Carol I, 11, 700506 Ia\u015fi, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0660-269X","authenticated-orcid":false,"given":"Silviu-Dumitru","family":"Pav\u0103l","sequence":"additional","affiliation":[{"name":"Faculty of Automatic Control and Computer Engineering, Technical University \u201cGheorghe Asachi\u201d of Ia\u015fi, Dimitrie Mangeron, nr. 27, 700050 Ia\u015fi, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"148","DOI":"10.3934\/dcdss.2022203","article-title":"A qualitative analysis of a second-order anisotropic phase-field transition system endowed with a general class of nonlinear dynamic boundary conditions","volume":"16","author":"Berinde","year":"2023","journal-title":"Discret. Contin. Dyn. Syst. Ser. 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Fractional Step Scheme to Approximate a Non-Linear Second-Order Reaction\u2013Diffusion Problem with Inhomogeneous Dynamic Boundary Conditions. Axioms, 12.","DOI":"10.3390\/axioms12040406"},{"key":"ref_19","first-page":"149","article-title":"Asymptotic behavior of a phase-field system with dynamic boundary conditions","volume":"Volume 521","author":"Favini","year":"2006","journal-title":"Differential Equations: Inverse and Direct Problems"},{"key":"ref_20","first-page":"425","article-title":"Product formula for nonlinear semigroups in Hilbert spaces","volume":"58","author":"Kobayashi","year":"1982","journal-title":"Proc. Jpn. Acad."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/5\/286\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:32:41Z","timestamp":1760106761000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/5\/286"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,23]]},"references-count":20,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["axioms13050286"],"URL":"https:\/\/doi.org\/10.3390\/axioms13050286","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,23]]}}}