{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T13:00:40Z","timestamp":1773925240519,"version":"3.50.1"},"reference-count":35,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,19]],"date-time":"2022-05-19T00:00:00Z","timestamp":1652918400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we consider the one-dimensional transport equation with delay and advanced arguments. A maximum principle is proven for the problem considered. As an application of the maximum principle, the stability of the solution is established. It is also proven that the solution\u2019s discontinuity propagates. Finite difference methods with linear interpolation that are conditionally stable and unconditionally stable are presented. This paper presents applications of unconditionally stable numerical methods to symmetric delay arguments and differential equations with variable delays. As a consequence, the matrices of the difference schemes are asymmetric. An illustration of the unconditional stable method is provided with numerical examples. Solution graphs are drawn for all the problems.<\/jats:p>","DOI":"10.3390\/sym14051046","type":"journal-article","created":{"date-parts":[[2022,5,20]],"date-time":"2022-05-20T00:18:11Z","timestamp":1653005891000},"page":"1046","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Stable Difference Schemes with Interpolation for Delayed One-Dimensional Transport Equation"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5758-4703","authenticated-orcid":false,"given":"Karthick","family":"Sampath","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, Tamilnadu 603203, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2102-5322","authenticated-orcid":false,"given":"Subburayan","family":"Veerasamy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, Tamilnadu 603203, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0634-2370","authenticated-orcid":false,"given":"Ravi","family":"Agarwal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Texas A & M University-Kingsville, Kingsville, TX 78363, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1608","DOI":"10.1137\/S1064827502409717","article-title":"An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays","volume":"25","author":"Huang","year":"2004","journal-title":"SIAM J. 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