{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T21:46:37Z","timestamp":1767908797199,"version":"3.49.0"},"reference-count":23,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,11,30]],"date-time":"2023-11-30T00:00:00Z","timestamp":1701302400000},"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>Here, we consider the phase field transition system (a nonlinear system of parabolic type) introduced by Caginalp to distinguish between the phases of the material that are involved in the solidification process. We start by investigating the solvability of such boundary value problems in the class Wp1,2(Q)\u00d7W\u03bd1,2(Q). One proves the existence, the regularity, and the uniqueness of solutions, in the presence of the cubic nonlinearity type. On the basis of the convergence of an iterative scheme of the fractional steps type, a conceptual numerical algorithm, alg-frac_sec-ord-varphi_PHT, is elaborated in order to approximate the solution of the nonlinear parabolic problem. The advantage of such an approach is that the new method simplifies the numerical computations due to its decoupling feature. An example of the numerical implementation of the principal step in the conceptual algorithm is also reported. Some conclusions are given are also given as new directions to extend the results and methods presented in the present paper.<\/jats:p>","DOI":"10.3390\/axioms12121098","type":"journal-article","created":{"date-parts":[[2023,11,30]],"date-time":"2023-11-30T09:39:12Z","timestamp":1701337152000},"page":"1098","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Fractional Steps Scheme to Approximate the Phase Field Transition System Endowed with Inhomogeneous\/Homogeneous Cauchy-Neumann\/Neumann Boundary Conditions"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9056-0911","authenticated-orcid":false,"given":"Constantin","family":"Fetecau\u00a0","sequence":"first","affiliation":[{"name":"Academy of Romanian Scientists, 54 Splaiul Independentei, 050094 Bucharest, Romania"}]},{"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"}]},{"given":"Dorin-C\u0103t\u0103lin","family":"Stoicescu","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"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,30]]},"reference":[{"key":"ref_1","unstructured":"Miranville, A., and Moro\u0219anu, C. (2020). Differential Equations & Dynamical Systems, AIMS\u2014American Institute of Mathematical Sciences. Available online: https:\/\/www.aimsciences.org\/fileAIMS\/cms\/news\/info\/28df2b3d-ffac-4598-a89b-9494392d1394.pdf."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Fetec\u0103u, C., and Moro\u015fanu, C. (2023). 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_3","doi-asserted-by":"crossref","unstructured":"Moro\u015fanu, C., and Pav\u0103l, S. (2021). Rigorous Mathematical Investigation of a Nonlocal and Nonlinear Second-Order Anisotropic Reaction-Diffusion Model: Applications on Image Segmentation. Mathematics, 9.","DOI":"10.3390\/math9010091"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.jmaa.2009.03.063","article-title":"Analysis of a two-phase field model for the solidification of an alloy","volume":"357","author":"Boldrini","year":"2009","journal-title":"J. Math. Anal. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1007\/BF00254827","article-title":"An analysis of a phase-field model of a free boundary","volume":"92","author":"Caginalp","year":"1986","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/s10492-009-0008-6","article-title":"On the Caginalp system with dynamic boundary conditions and singular potentials","volume":"54","author":"Cherfils","year":"2009","journal-title":"Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1350024","DOI":"10.1142\/S0219530513500243","article-title":"Attractors for a Caginalp model with a logarithmic potential and coupled dynamic boundary conditions","volume":"11","author":"Conti","year":"2013","journal-title":"Anal. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1475","DOI":"10.1002\/mana.200510560","article-title":"Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials","volume":"280","author":"Grasselli","year":"2007","journal-title":"Math. Nachr."},{"key":"ref_9","first-page":"271","article-title":"Some mathematical models in phase transition","volume":"7","author":"Miranville","year":"2014","journal-title":"Discret. Contin. Dyn. Syst. Ser. S"},{"key":"ref_10","unstructured":"Moro\u015fanu, C. (2012). Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods, Bentham Science Publishers."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"648","DOI":"10.3934\/math.2019.3.648","article-title":"Modeling of the continuous casting process of steel via phase-field transition system. Fractional steps method","volume":"4","year":"2019","journal-title":"AIMS Math."},{"key":"ref_12","first-page":"377","article-title":"Numerical approximation of the phase-field transition system with non-homogeneous Cauchy-Neumann boundary conditions in both unknown functions via fractional steps methods","volume":"3","author":"Ovono","year":"2013","journal-title":"(JAAC) J. Appl. Anal. Comput."},{"key":"ref_13","unstructured":"Pavel, N.H. (1984). Differential Equations, Flow Invariance and Applications, Research Notes in Mathematics; Pitman Advanced Publishing Program."},{"key":"ref_14","first-page":"15","article-title":"Approximating some non\u2013linear equations by a Fractional step scheme","volume":"1","author":"Barbu","year":"1993","journal-title":"Differ. Integral Equ."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/0022-1236(72)90014-6","article-title":"Convergence and approximation of semigroups of nonlinear operators in Banach spaces","volume":"9","author":"Pazy","year":"1972","journal-title":"J. Funct. Anal."},{"key":"ref_16","first-page":"425","article-title":"Product formula for nonlinear semigroups in Hilbert spaces","volume":"58","author":"Kobayashi","year":"1982","journal-title":"Proc. Jpn. Acad."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1016\/j.jmaa.2009.01.065","article-title":"A three-dimensional phase transition model in ferromagnetism: Existence and uniqueness","volume":"355","author":"Berti","year":"2009","journal-title":"J. Math. Anal. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/0167-2789(90)90015-H","article-title":"Thermodynamically consistent models of phase-field type for kinetics of phase transitions","volume":"43","author":"Penrose","year":"1990","journal-title":"Phys. D"},{"key":"ref_19","unstructured":"Rubinstein, L.I. (1971). The Stefan Problem (Translations of Mathematical Monographs), American Mathematical Society."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/S0022-247X(02)00559-0","article-title":"Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions","volume":"279","author":"Sprekels","year":"2003","journal-title":"J. Math. Anal. Appl."},{"key":"ref_21","unstructured":"Temam, R. (1997). Applied Mathematical Sciences, Springer. [2nd ed.]."},{"key":"ref_22","unstructured":"Dantzig, J., Greenwell, A., and Mickalczyk, J. (2001). Continuous Casting: Modeling, The Encyclopedia of Advanced Materials, Pergamon Elsevier Science Ltd."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Axelson, O., and Barker, V. (1984). Finite Element Solution of Boundary Value Problems, Academic Press.","DOI":"10.1016\/B978-0-12-068780-0.50011-X"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/12\/1098\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:35:13Z","timestamp":1760132113000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/12\/1098"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,30]]},"references-count":23,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["axioms12121098"],"URL":"https:\/\/doi.org\/10.3390\/axioms12121098","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,30]]}}}