{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:16:21Z","timestamp":1760231781925,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,7]],"date-time":"2022-10-07T00:00:00Z","timestamp":1665100800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"INEGI-LAETA","award":["UIDB\/50022\/2020"],"award-info":[{"award-number":["UIDB\/50022\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper proposes a new numerical method for solving single time-delayed stochastic differential equations via orthogonal functions. The basic principles of the technique are presented. The new method is applied to approximate two kinds of stochastic differential equations with additive and multiplicative noise. Excellence computational burden is achieved along with a O(h2) convergence rate, which is better than former methods. Two examples are examined to illustrate the validity and efficiency of the new technique.<\/jats:p>","DOI":"10.3390\/sym14102085","type":"journal-article","created":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T03:32:56Z","timestamp":1665459176000},"page":"2085","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A New Approach to Approximate Solutions of Single Time-Delayed Stochastic Integral Equations via Orthogonal Functions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2274-795X","authenticated-orcid":false,"given":"Seyyedeh N.","family":"Kiaee","sequence":"first","affiliation":[{"name":"Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj 31499-68111, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6113-2208","authenticated-orcid":false,"given":"Morteza","family":"Khodabin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj 31499-68111, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3162-6212","authenticated-orcid":false,"given":"Reza","family":"Ezzati","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj 31499-68111, Iran"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7359-4370","authenticated-orcid":false,"given":"Ant\u00f3nio M.","family":"Lopes","sequence":"additional","affiliation":[{"name":"LAETA\/INEGI, Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bazighifan, O., Ali, A.H., Mofarreh, F., and Raffoul, Y.N. 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