{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:09:57Z","timestamp":1760144997036,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,4]],"date-time":"2024-06-04T00:00:00Z","timestamp":1717459200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Shaqra University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper investigates the inhomogeneous version of the pantograph equation. The current model includes the exponential function as the inhomogeneous part of the pantograph equation. The Maclaurin series expansion (MSE) is a well-known standard method for solving initial value problems; it may be easier than any other approaches. Moreover, the MSE can be used in a straightforward manner in contrast to the other analytical methods. Thus, the MSE is extended in this paper to treat the inhomogeneous pantograph equation. The solution is obtained in a closed series form with an explicit formula for the series coefficients and the convergence of the series is proved. Also, the analytic solutions of some models in the literature are recovered as special cases of the present work. The accuracy of the results is examined through several comparisons with the available exact solutions of some classes in the relevant literature. Finally, the residuals are calculated and then used to validate the accuracy of the present approximations for some classes which have no exact solutions.<\/jats:p>","DOI":"10.3390\/axioms13060377","type":"journal-article","created":{"date-parts":[[2024,6,5]],"date-time":"2024-06-05T08:43:54Z","timestamp":1717577034000},"page":"377","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Analytical and Numerical Investigation for the Inhomogeneous Pantograph Equation"],"prefix":"10.3390","volume":"13","author":[{"given":"Faten","family":"Aldosari","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, Shaqra University, P.O. Box 15572, Shaqra 11961, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/j.cam.2005.12.015","article-title":"A Taylor method for numerical solution of generalized pantograph equations with linear functional argument","volume":"200","author":"Sezera","year":"2007","journal-title":"J. Comput. Appl. 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