{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T06:42:49Z","timestamp":1778222569846,"version":"3.51.4"},"reference-count":18,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T00:00:00Z","timestamp":1684368000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Arab Open University","award":["AOURG-2023-008"],"award-info":[{"award-number":["AOURG-2023-008"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This work purposes to establish two small numerical modifications for the Fractional Euler method (FEM) and the Modified Fractional Euler Method (MFEM) to deal with fractional initial value problems. Two such modifications, which are named Improved Modified Fractional Euler Method 1 (IMFEM 1) and Improved Modified Fractional Euler Method 2 (IMFEM 2), endeavor to further enhance FEM and MFEM in terms of attaining more accuracy. By utilizing certain theoretical results, the resultant error bounds of the proposed methods are analyzed and estimated. Several numerical comparisons are carried out to validate the efficiency of our proposed methods.<\/jats:p>","DOI":"10.3390\/axioms12050488","type":"journal-article","created":{"date-parts":[[2023,5,18]],"date-time":"2023-05-18T07:35:50Z","timestamp":1684395350000},"page":"488","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["New Algorithms for Dealing with Fractional Initial Value Problems"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8443-8848","authenticated-orcid":false,"given":"Iqbal M.","family":"Batiha","sequence":"first","affiliation":[{"name":"Department of Mathematics, Al Zaytoonah University, Amman 11733, Jordan"},{"name":"Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, United Arab Emirates"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0694-1680","authenticated-orcid":false,"given":"Ahmad A.","family":"Abubaker","sequence":"additional","affiliation":[{"name":"Faculty of Computer Studies, Arab Open University, Riyadh 11681, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4348-6197","authenticated-orcid":false,"given":"Iqbal H.","family":"Jebril","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al Zaytoonah University, Amman 11733, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9960-2591","authenticated-orcid":false,"given":"Suha B.","family":"Al-Shaikh","sequence":"additional","affiliation":[{"name":"Faculty of Computer Studies, Arab Open University, Riyadh 11681, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Khaled","family":"Matarneh","sequence":"additional","affiliation":[{"name":"Faculty of Computer Studies, Arab Open University, Riyadh 11681, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,18]]},"reference":[{"key":"ref_1","unstructured":"Aho, A., Hopcroft, J.E., and Ullman, J.D. 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Anal."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/488\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:37:18Z","timestamp":1760125038000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/5\/488"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,18]]},"references-count":18,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2023,5]]}},"alternative-id":["axioms12050488"],"URL":"https:\/\/doi.org\/10.3390\/axioms12050488","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,18]]}}}