{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:44:19Z","timestamp":1759970659462,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,30]],"date-time":"2025-01-30T00:00:00Z","timestamp":1738195200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Universidad Cat\u00f3lica de Temuco; Universidad del B\u00edo-B\u00edo (Chile); the National Agency for Research and Development, ANID-Chile","award":["1230560","LPR23-03"],"award-info":[{"award-number":["1230560","LPR23-03"]}]},{"name":"Universidad Tecnol\u00f3gica Metropolitana","award":["1230560","LPR23-03"],"award-info":[{"award-number":["1230560","LPR23-03"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this article, we are interested in studying and analyzing the heat conduction phenomenon in a multi-layered solid. We consider the physical assumptions that the dual-phase-lag model governs the heat flow on each solid layer. We introduce a one-dimensional mathematical model given by an initial interface-boundary value problem, where the unknown is the solid temperature. More precisely, the mathematical model is described by the following four features: the model equation is given by a dual-phase-lag equation at the inside each layer, an initial condition for temperature and the temporal derivative of the temperature, heat flux boundary conditions, and the interfacial condition for the temperature and heat flux conditions between the layers. We discretize the mathematical model by a finite difference scheme. The numerical approach has similar features to the continuous model: it is considered to be the accuracy of the dual-phase-lag model on the inside each layer, the initial conditions are discretized by the average of the temperature on each discrete interval, the inside of each layer approximation is extended to the interfaces by using the behavior of the continuous interface conditions, and the inside each layer approximation on the boundary layers is extended to state the numerical boundary conditions. We prove that the finite difference scheme is unconditionally stable and unconditionally convergent. In addition, we provide some numerical examples.<\/jats:p>","DOI":"10.3390\/axioms14020105","type":"journal-article","created":{"date-parts":[[2025,1,30]],"date-time":"2025-01-30T04:01:39Z","timestamp":1738209699000},"page":"105","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Numerical Solution of Mathematical Model of Heat Conduction in Multi-Layered Nanoscale Solids"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6602-7378","authenticated-orcid":false,"given":"An\u00edbal","family":"Coronel","sequence":"first","affiliation":[{"name":"Departamento de Ciencias B\u00e1sicas\u2014Centro de Ciencias Exactas CCE-UBB, Facultad de Ciencias, Universidad del B\u00edo-B\u00edo, Campus Fernando May, Chill\u00e1n 3780000, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-2550-4400","authenticated-orcid":false,"given":"Ian","family":"Hess","sequence":"additional","affiliation":[{"name":"Departamento de Ciencias Matem\u00e1ticas y F\u00edsicas, Facultad de Ingenier\u00eda, Universidad Cat\u00f3lica de Temuco, Temuco 4813302, Chile"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2794-029X","authenticated-orcid":false,"given":"Fernando","family":"Huancas","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Facultad de Ciencias Naturales, Matem\u00e1ticas y del Medio Ambiente, Universidad Tecnol\u00f3gica Metropolitana, Las Palmeras 3360, \u00d1u\u00f1oa-Santiago 7750000, Chile"}]},{"given":"Jos\u00e9","family":"Chiroque","sequence":"additional","affiliation":[{"name":"Departamento Acad\u00e9mico de Matem\u00e1ticas, Facultad de Ciencias F\u00edsicas y Matem\u00e1ticas, Universidad Nacional Pedro Ruiz Gallo, Avenida Juan XXIII 391, Lambayeque 14013, Peru"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"927350","DOI":"10.1155\/2009\/927350","article-title":"Conjugate Problems in Convective Heat Transfer: Review","volume":"2009","author":"Dorfman","year":"2009","journal-title":"Math. 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