{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T16:13:29Z","timestamp":1783095209111,"version":"3.54.6"},"reference-count":20,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,17]],"date-time":"2022-12-17T00:00:00Z","timestamp":1671235200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deputyship for Research &amp; Innovation, Ministry of Education in Saudi Arabia","award":["IF2\/PSAU\/2022\/01\/21495"],"award-info":[{"award-number":["IF2\/PSAU\/2022\/01\/21495"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, the pantograph delay differential equation y\u2032(t)=ay(t)+byct subject to the condition y(0)=\u03bb is reanalyzed for the real constants a, b, and c. In the literature, it has been shown that the pantograph delay differential equation, for \u03bb=1, is well-posed if c&lt;1, but not if c&gt;1. In addition, the solution is available in the form of a standard power series when \u03bb=1. In the present research, we are able to determine the solution of the pantograph delay differential equation in a closed series form in terms of exponential functions. The convergence of such a series is analysed. It is found that the solution converges for c\u2208(\u22121,1) such that ba&lt;1 and it also converges for c&gt;1 when a&lt;0. For c=\u22121, the exact solution is obtained in terms of trigonometric functions, i.e., a periodic solution with periodicity 2\u03c0b2\u2212a2 when b&gt;a. The current results are introduced for the first time and have not been reported in the relevant literature.<\/jats:p>","DOI":"10.3390\/axioms11120741","type":"journal-article","created":{"date-parts":[[2022,12,19]],"date-time":"2022-12-19T08:41:41Z","timestamp":1671439301000},"page":"741","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Analytical and Numerical Simulations of a Delay Model: The Pantograph Delay Equation"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7266-1893","authenticated-orcid":false,"given":"Essam Roshdy","family":"El-Zahar","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences and Humanities, Prince Sattam bin Abdulaziz University, Alkharj 11942, Saudi Arabia"},{"name":"Department of Basic Engineering Science, Faculty of Engineering, Menofia University, Shebin El-Kom 32511, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1122-6297","authenticated-orcid":false,"given":"Abdelhalim","family":"Ebaid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1093\/imamat\/8.3.271","article-title":"On a Functional Differential Equation","volume":"8","author":"Fox","year":"1971","journal-title":"IMA J. Appl. Math."},{"key":"ref_2","first-page":"891","article-title":"The functional-differential equation y\u2032(x) = ay(\u03bbx) + by(x)","volume":"77","author":"Kato","year":"1971","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1006\/jmaa.1997.5483","article-title":"The pantograph equation in the complex plane","volume":"213","author":"Derfel","year":"1997","journal-title":"J. Math. Anal. 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