{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T11:28:43Z","timestamp":1774438123527,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,6]],"date-time":"2022-12-06T00:00:00Z","timestamp":1670284800000},"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>The present manuscript examines different forms of Initial-Value Problems (IVPs) featuring various types of Ordinary Differential Equations (ODEs) by proposing a proficient modification to the famous standard Adomian decomposition method (ADM). The present paper collected different forms of inverse integral operators and further successfully demonstrated their applicability on dissimilar nonlinear singular and nonsingular ODEs. Furthermore, we surveyed most cases in this very new method, and it was found to have a fast convergence rate and, on the other hand, have high precision whenever exact analytical solutions are reachable.<\/jats:p>","DOI":"10.3390\/axioms11120698","type":"journal-article","created":{"date-parts":[[2022,12,6]],"date-time":"2022-12-06T01:43:48Z","timestamp":1670291028000},"page":"698","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Modified Adomian Method through Efficient Inverse Integral Operators to Solve Nonlinear Initial-Value Problems for Ordinary Differential Equations"],"prefix":"10.3390","volume":"11","author":[{"given":"Mariam","family":"AL-Mazmumy","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah P. O. Box 80327, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aishah A.","family":"Alsulami","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah P. O. Box 80327, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huda O.","family":"Bakodah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah P. O. Box 80327, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nawal","family":"Alzaid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah P. O. Box 80327, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/0895-7177(90)90125-7","article-title":"A review of the decomposition method and some recent results for nonlinear equations","volume":"13","author":"Adomian","year":"1990","journal-title":"Math. Comput. Model."},{"key":"ref_2","first-page":"77","article-title":"A reliable modification of Adomian decomposition method","volume":"102","author":"Wazwaz","year":"1999","journal-title":"Appl. Math. Comput."},{"key":"ref_3","first-page":"1685","article-title":"Adomian decomposition method with Chebyshev polynomials","volume":"175","author":"Hosseini","year":"2006","journal-title":"Appl. Math. 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A Method for the Solution of Coupled System of Emden\u2013Fowler\u2013Type Equations. Symmetry, 14.","DOI":"10.3390\/sym14050843"},{"key":"ref_8","first-page":"45","article-title":"A new method for solving singular initial value problems in the second-order ordinary differential equations","volume":"128","author":"Wazwaz","year":"2002","journal-title":"Appl. Math. Comput."},{"key":"ref_9","first-page":"543","article-title":"Adomian decomposition method for a reliable treatment of the Emden\u2013Fowler equation","volume":"161","author":"Wazwaz","year":"2005","journal-title":"Appl. Math. Comput."},{"key":"ref_10","first-page":"117","article-title":"Modified Adomian decomposition method for specific second order ordinary differential equations","volume":"186","author":"Hosseini","year":"2007","journal-title":"Appl. Math. 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