{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:16:13Z","timestamp":1760238973306,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,21]],"date-time":"2020-09-21T00:00:00Z","timestamp":1600646400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00eda y Competitividad","doi-asserted-by":"publisher","award":["CGL2017-89804-R"],"award-info":[{"award-number":["CGL2017-89804-R"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we obtain exact solutions and continuous numerical approximations for mixed problems of coupled systems of diffusion equations with delay. Using the method of separation of variables, and based on an explicit expression for the solution of the separated vector initial-value delay problem, we obtain exact infinite series solutions that can be truncated to provide analytical\u2013numerical solutions with prescribed accuracy in bounded domains. Although usually implicit in particular applications, the method of separation of variables is deeply correlated with symmetry ideas.<\/jats:p>","DOI":"10.3390\/sym12091560","type":"journal-article","created":{"date-parts":[[2020,9,21]],"date-time":"2020-09-21T08:18:01Z","timestamp":1600676281000},"page":"1560","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Exact Solutions and Continuous Numerical Approximations of Coupled Systems of Diffusion Equations with Delay"],"prefix":"10.3390","volume":"12","author":[{"given":"Elia","family":"Reyes","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, University of Alicante, Apdo. 99, 03080 Alicante, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4331-4619","authenticated-orcid":false,"given":"M. \u00c1ngeles","family":"Castro","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, University of Alicante, Apdo. 99, 03080 Alicante, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Antonio","family":"Sirvent","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, University of Alicante, Apdo. 99, 03080 Alicante, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0753-7826","authenticated-orcid":false,"given":"Francisco","family":"Rodr\u00edguez","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, University of Alicante, Apdo. 99, 03080 Alicante, Spain"},{"name":"Multidisciplinary Institute for Environmental Studies (IMEM), University of Alicante, Apdo. 99, 03080 Alicante, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kolmanovskii, V., and Myshkis, A. 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Stability Analysis and Robust Control of Time-Delay Systems, Springer.","DOI":"10.1007\/978-3-642-03037-6"},{"key":"ref_6","first-page":"431","article-title":"Sur une forme de l\u2019\u00e9quation de la chaleur \u00e9liminant le paradoxe d\u2019une propagation instantan\u00e9e","volume":"247","author":"Cattaneo","year":"1958","journal-title":"C. R. Acad. Sci."},{"key":"ref_7","first-page":"3154","article-title":"Les paradoxes de la th\u00e9orie continue de l\u2019\u00e9quation de la chaleur","volume":"246","author":"Vernotte","year":"1958","journal-title":"C. R. Acad. Sci."},{"key":"ref_8","first-page":"2190","article-title":"Some possible complications in the phenomena of thermal conduction","volume":"252","author":"Vernotte","year":"1961","journal-title":"C. R. Acad. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1103\/RevModPhys.61.41","article-title":"Heat waves","volume":"61","author":"Joseph","year":"1989","journal-title":"Rev. Mod. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Tzou, D.Y. (2015). Macro-to Microscale Heat Transfer: The Lagging Behavior, John Wiley & Sons. [2nd ed.].","DOI":"10.1002\/9781118818275"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1416","DOI":"10.1007\/s10765-015-1913-4","article-title":"Macro-to Nanoscale Heat and Mass Transfer: The Lagging Behavior","volume":"36","author":"Ghazanfarian","year":"2015","journal-title":"Int. J. Thermophys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/j.mcm.2003.10.046","article-title":"Analytic solution of mixed problems for the generalized diffusion equation with delay","volume":"40","author":"Company","year":"2004","journal-title":"Math. Comput. Model."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1016\/j.camwa.2008.02.011","article-title":"Analytic-numerical solutions of diffusion mathematical models with delays","volume":"56","author":"Reyes","year":"2008","journal-title":"Comput. Math. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Miller, W. (1984). Symmetry and Separation of Variables, Cambridge University Press.","DOI":"10.1017\/CBO9781107325623"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/0898-1221(96)00099-5","article-title":"Accurate continuous numerical solutions of time dependent mixed partial differential problems","volume":"32","author":"Almenar","year":"1996","journal-title":"Comput. Math. Appl."},{"key":"ref_16","first-page":"1","article-title":"Exact and Analytic-Numerical Solutions of Strongly Coupled Mixed Diffusion Problems","volume":"43","author":"Navarro","year":"2000","journal-title":"Proc. Edinb. Math. Soc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/S0895-7177(97)00082-4","article-title":"Mixed Problems for the Time-Dependent Telegraph Equation: Continuous Numerical Solutions with A Priori Error Bounds","volume":"25","author":"Almenar","year":"1997","journal-title":"Math. Comput. Model."},{"key":"ref_18","first-page":"99","article-title":"On a Class of Linear Partial Differential Equations with Retarded Argument in Time","volume":"15","author":"Scott","year":"1969","journal-title":"Bul. Inst. Politeh. Din Iasi"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1155\/S0161171297000239","article-title":"Boundary value problems for the diffusion equation with piecewise continuous time delay","volume":"20","author":"Wiener","year":"1997","journal-title":"Int. J. Math. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1007\/s11072-009-0075-3","article-title":"Solution of one heat equation with delay","volume":"12","author":"Khusainov","year":"2009","journal-title":"Nonlinear Oscil."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1841","DOI":"10.1016\/j.mcm.2010.11.074","article-title":"Exact and analytic-numerical solutions of bidimensional lagging models of heat conduction","volume":"54","author":"Escolano","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3178","DOI":"10.1016\/j.amc.2012.09.050","article-title":"Exact solutions and numerical approximations of mixed problems for the wave equation with delay","volume":"219","author":"Roales","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1023\/A:1024871503542","article-title":"On the Absolute Exponential Stability of Solutions of Systems of Linear Parabolic Differential Equations with One Delay","volume":"6","author":"Kushnir","year":"2012","journal-title":"Nonlinear Oscill."},{"key":"ref_24","unstructured":"Golub, G., and Van Loan, C.F. (1989). Matrix Computations, Johns-Hopkins University Press."},{"key":"ref_25","unstructured":"Olver, F.W., Lozier, D.W., Boisvert, R.F., and Clark, C.W. (2010). NIST Handbook of Mathematical Functions, Cambridge University Press."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1016\/j.amc.2019.03.048","article-title":"Applying the Random Variable Transformation method to solve a class of random linear differential equation with discrete delay","volume":"356","author":"Caraballo","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/s00009-019-1370-6","article-title":"Lp-calculus approach to the random autonomous linear differential equation with discrete delay","volume":"16","author":"Calatayud","year":"2019","journal-title":"Mediterr. J. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/j.amc.2018.06.029","article-title":"Exact and nonstandard numerical schemes for linear delay differential models","volume":"338","author":"Castro","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1038","DOI":"10.3390\/math7111038","article-title":"Exact and nonstandard finite difference schemes for coupled linear delay differential systems","volume":"7","author":"Castro","year":"2019","journal-title":"Mathematics"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Calatayud, J., Cort\u00e9s, J.C., Jornet, M., and Rodr\u00edguez, F. (2020). Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay. Mathematics, 8.","DOI":"10.3390\/math8091417"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1560\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:12:07Z","timestamp":1760177527000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/9\/1560"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,21]]},"references-count":30,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["sym12091560"],"URL":"https:\/\/doi.org\/10.3390\/sym12091560","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,9,21]]}}}