{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T15:47:38Z","timestamp":1768837658968,"version":"3.49.0"},"reference-count":40,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,3,14]],"date-time":"2022-03-14T00:00:00Z","timestamp":1647216000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this research study, a novel computational algorithm for solving a second-order singular functional differential equation as a generalization of the well-known Lane\u2013Emden and differential-difference equations is presented by using the Bessel bases. This technique depends on transforming the problem into a system of algebraic equations and by solving this system the unknown Bessel coefficients are determined and the solution will be known. The method is tested on several test examples and proves to provide accurate results as compared to other existing methods from the literature. The simplicity and robustness of the proposed technique drive us to investigate more of their applications to several similar problems in the future.<\/jats:p>","DOI":"10.3390\/axioms11030133","type":"journal-article","created":{"date-parts":[[2022,3,15]],"date-time":"2022-03-15T03:06:10Z","timestamp":1647313570000},"page":"133","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["An Effective Approximation Algorithm for Second-Order Singular Functional Differential Equations"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6116-4928","authenticated-orcid":false,"given":"Mohammad","family":"Izadi","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman 76169-14111, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9277-8092","authenticated-orcid":false,"given":"Hari M.","family":"Srivastava","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada"},{"name":"Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan"},{"name":"Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan"},{"name":"Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0557-8536","authenticated-orcid":false,"given":"Waleed","family":"Adel","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt"},{"name":"Faculty of Engineering, Universit\u00e9 Fran\u00e7aise d\u2019Egypte, Ismailia Desert Road, El Shorouk, Cairo, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"065004","DOI":"10.1088\/0031-8949\/78\/06\/065004","article-title":"The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics","volume":"78","author":"Dehghan","year":"2008","journal-title":"Phys. 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