{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T22:43:28Z","timestamp":1775083408048,"version":"3.50.1"},"reference-count":37,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2022,5,26]],"date-time":"2022-05-26T00:00:00Z","timestamp":1653523200000},"content-version":"vor","delay-in-days":145,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2022,1]]},"abstract":"<jats:p>\n                    In this paper, DSEK model with fractional derivatives of the Atangana\u2010Baleanu Caputo (ABC) is proposed. This paper gives a brief overview of the ABC fractional derivative and its attributes. Fixed point theory has been used to establish the uniqueness and existence of solutions for the fractional DSEK model. According to this theory, we will define two operators based on Lipschitzian and prove that they are contraction mapping and relatively compact. Ulam\u2010Hyers stability theorem is implemented to prove the fractional DSEK model\u2019s stability in Banach space. Also, fractional Euler\u2019s numerical method is derived for initial value problems with ABC fractional derivative and implemented on fractional DSEK model. The symmetric properties contribute to determining the appropriate method for finding the correct solution to fractional differential equations. The numerical solutions generated using fractional Euler\u2019s method have been plotted for different values of\n                    <jats:italic>\u03b1<\/jats:italic>\n                    where\n                    <jats:italic>\u03b1<\/jats:italic>\n                    \u2208 (0,1] and different step sizes\n                    <jats:italic>h<\/jats:italic>\n                    . Result discussion will be given, describing the changes that occur due to the step size\n                    <jats:italic>h<\/jats:italic>\n                    .\n                  <\/jats:p>","DOI":"10.1155\/2022\/4475491","type":"journal-article","created":{"date-parts":[[2022,5,26]],"date-time":"2022-05-26T14:35:07Z","timestamp":1653575707000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Euler\u2019s Numerical Method on Fractional DSEK Model under ABC Derivative"],"prefix":"10.1155","volume":"2022","author":[{"given":"Fareeha Sami","family":"Khan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Khalid","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7251-9608","authenticated-orcid":false,"given":"Omar","family":"Bazighifan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2009-3905","authenticated-orcid":false,"given":"A.","family":"El-Mesady","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2022,5,26]]},"reference":[{"key":"e_1_2_10_1_2","doi-asserted-by":"publisher","DOI":"10.7498\/aps.54.1113"},{"key":"e_1_2_10_2_2","volume-title":"Fractional Differential Equations","author":"Podlubny I.","year":"1990"},{"key":"e_1_2_10_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.molliq.2007.03.006"},{"key":"e_1_2_10_4_2","first-page":"1033","article-title":"Singular diffusion in fractal porous media","volume":"21","author":"Wang Z. 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