{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T14:48:17Z","timestamp":1779202097295,"version":"3.51.4"},"reference-count":28,"publisher":"Wiley","license":[{"start":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T00:00:00Z","timestamp":1663891200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2022,9,23]]},"abstract":"<jats:p>We introduce the delayed Mittag-Leffler type matrix functions, delayed fractional cosine, and delayed fractional sine and use the Laplace transform to obtain an analytical solution to the IVP for a Hilfer type fractional linear time-delay system<jats:inline-formula><a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><a:msubsup><a:mrow><a:mi>D<\/a:mi><\/a:mrow><a:mrow><a:mn>0<\/a:mn><a:mo>,<\/a:mo><a:mi>t<\/a:mi><\/a:mrow><a:mrow><a:mi>\u03bc<\/a:mi><a:mo>,<\/a:mo><a:mi>\u03bd<\/a:mi><\/a:mrow><\/a:msubsup><a:mi>z<\/a:mi><a:mfenced open=\"(\" close=\")\"><a:mrow><a:mi>t<\/a:mi><\/a:mrow><\/a:mfenced><a:mo>+<\/a:mo><a:mi>A<\/a:mi><a:mi>z<\/a:mi><a:mfenced open=\"(\" close=\")\"><a:mrow><a:mi>t<\/a:mi><\/a:mrow><\/a:mfenced><a:mo>+<\/a:mo><a:mi>\u03a9<\/a:mi><a:mi>z<\/a:mi><a:mfenced open=\"(\" close=\")\"><a:mrow><a:mi>t<\/a:mi><a:mo>\u2212<\/a:mo><a:mi>h<\/a:mi><\/a:mrow><\/a:mfenced><a:mo>=<\/a:mo><a:mi>f<\/a:mi><a:mfenced open=\"(\" close=\")\"><a:mrow><a:mi>t<\/a:mi><\/a:mrow><\/a:mfenced><\/a:math><\/jats:inline-formula>of order<jats:inline-formula><k:math xmlns:k=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><k:mn>1<\/k:mn><k:mo>&lt;<\/k:mo><k:mi>\u03bc<\/k:mi><k:mo>&lt;<\/k:mo><k:mn>2<\/k:mn><\/k:math><\/jats:inline-formula>and type<jats:inline-formula><m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><m:mn>0<\/m:mn><m:mo>\u2264<\/m:mo><m:mi>\u03bd<\/m:mi><m:mo>\u2264<\/m:mo><m:mn>1<\/m:mn><\/m:math><\/jats:inline-formula>, with nonpermutable matrices<jats:inline-formula><o:math xmlns:o=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><o:mi>A<\/o:mi><\/o:math><\/jats:inline-formula>and<jats:inline-formula><q:math xmlns:q=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><q:mi>\u03a9<\/q:mi><\/q:math><\/jats:inline-formula>. Moreover, we study Ulam-Hyers stability of the Hilfer type fractional linear time-delay system. Obtained results extend those for Caputo and Riemann-Liouville type fractional linear time-delay systems with permutable matrices and new even for these fractional delay systems.<\/jats:p>","DOI":"10.1155\/2022\/2661343","type":"journal-article","created":{"date-parts":[[2022,9,23]],"date-time":"2022-09-23T19:20:35Z","timestamp":1663960835000},"page":"1-7","source":"Crossref","is-referenced-by-count":5,"title":["Analytical Solution of the Fractional Linear Time-Delay Systems and their Ulam-Hyers Stability"],"prefix":"10.1155","volume":"2022","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3943-1732","authenticated-orcid":true,"given":"Nazim I.","family":"Mahmudov","sequence":"first","affiliation":[{"name":"Department of Mathematics, Eastern Mediterranean University, Famagusta, 99628 Northern Cyprus, Mersin 10, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1007\/s11072-008-0030-8"},{"issue":"1","key":"2","first-page":"101","article-title":"Linear autonomous time-delay system with permutation matrices solving","volume":"17","author":"D. Y. Khusainov","year":"2003","journal-title":"Studies of the University of Zilina Mathematical Series"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2009.10.025"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2006.08.104"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1007\/s11253-013-0765-y"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1155\/2010\/108218"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1137\/070689085"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2005.05.021"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1155\/ade\/2006\/80825"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2015.07.008"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1186\/1687-1847-2013-139"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1155\/2014\/320476"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2016.09.004"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.14232\/ejqtde.2020.1.53"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2018.05.015"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2019.01.001"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.14232\/ejqtde.2020.1.53"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2016.09.019"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2022.112267"},{"issue":"2","key":"20","first-page":"238","article-title":"Analytical approach to a class of Bagley-Torvik equations","volume":"11","author":"N. I. Mahmudov","year":"2020","journal-title":"TWMS Journal of Applied and Engineering Mathematics"},{"issue":"3","key":"21","first-page":"339","article-title":"Some mathematical problems and their solutions for the oscillating systems with liquid dampers: a review","volume":"30","author":"A. F. Aliev","year":"2021","journal-title":"Applied and Computational Mathematics"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.3390\/math10071089"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2017.09.015"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2020.106666"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1007\/s13540-022-00062-y"},{"key":"26","doi-asserted-by":"publisher","DOI":"10.1002\/mma.5446"},{"key":"27","doi-asserted-by":"publisher","DOI":"10.3390\/math8091493"},{"key":"28","doi-asserted-by":"crossref","DOI":"10.1155\/2022\/2661343","article-title":"Analytical solution of the fractional linear time-delay systems and their Ulam-Hyers stability","author":"N. I. Mahmudov","year":"2022"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2022\/2661343.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2022\/2661343.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2022\/2661343.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,20]],"date-time":"2023-02-20T08:36:04Z","timestamp":1676882164000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/jam\/2022\/2661343\/"}},"subtitle":[],"editor":[{"given":"Yansheng","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2022,9,23]]},"references-count":28,"alternative-id":["2661343","2661343"],"URL":"https:\/\/doi.org\/10.1155\/2022\/2661343","relation":{},"ISSN":["1687-0042","1110-757X"],"issn-type":[{"value":"1687-0042","type":"electronic"},{"value":"1110-757X","type":"print"}],"subject":[],"published":{"date-parts":[[2022,9,23]]}}}