{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,2]],"date-time":"2025-08-02T17:56:44Z","timestamp":1754157404687,"version":"3.41.2"},"reference-count":6,"publisher":"Emerald","issue":"5","license":[{"start":{"date-parts":[[2009,6,12]],"date-time":"2009-06-12T00:00:00Z","timestamp":1244764800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.emerald.com\/insight\/site-policies"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2009,6,12]]},"abstract":"<jats:sec><jats:title content-type=\"abstract-heading\">Purpose<\/jats:title><jats:p>The purpose of this paper is to find methods for solving linear or non\u2010linear integral equations. The decomposition method may be applied, but a difficulty arises when it is applied to integral equations of the first kind which are not in canonical form. This paper aims to describe some techniques to obtain canonical form of Adomian decomposition method (ADM) for solving integral equations of the first kind.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-heading\">Design\/methodology\/approach<\/jats:title><jats:p>Among various numerical methods for solving integral equations, the ADM has shown a good skill for determining the solutions as polynomial series forms. In this paper, some methods based on regularization, derivation or substitution techniques are presented to permit the application of ADM to integral equations of the first kind.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-heading\">Findings<\/jats:title><jats:p>By using the three proposed methods the decomposition method can then be easily used to solve integral equations of the first kind. The three methods allow to obtain the canonical form and a solution to find using the ADM.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-heading\">Research limitations\/implications<\/jats:title><jats:p>The three methods provide the solution by the application of the decomposition method of George Adomian as extended by other researchers. All are efficient: the derivation and substitution techniques giving interesting results for linear and non\u2010linear equations of Volterra type; the regularization technique is also efficient for the equations of Fredholm type, but difficulties could arise from the calculations of the required series.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-heading\">Practical implications<\/jats:title><jats:p>The research provides three efficient methods for solving integral equations of first kind.<\/jats:p><\/jats:sec><jats:sec><jats:title content-type=\"abstract-heading\">Originality\/value<\/jats:title><jats:p>The three methods illustrate the powerful use of the decomposition techniques pioneered by Adomian and a result of their applications may be applied for solving of functional equations of any kind. This paper shows some methods for applying the Adomian techniques to integral equations of the first kind.<\/jats:p><\/jats:sec>","DOI":"10.1108\/03684920910962632","type":"journal-article","created":{"date-parts":[[2009,7,4]],"date-time":"2009-07-04T07:03:37Z","timestamp":1246691017000},"page":"733-743","source":"Crossref","is-referenced-by-count":4,"title":["Solving integral equations of the first kind by decomposition method"],"prefix":"10.1108","volume":"38","author":[{"given":"Ngarkodje","family":"Ngarasta","sequence":"first","affiliation":[]}],"member":"140","reference":[{"key":"key2022031520284351900_b1","unstructured":"Abbaoui, K. (1995), \u201cLes fondements math\u00e9matiques de la m\u00e9thode d\u00e9compositionnelle d'Adomian et application \u00e0 la r\u00e9solution des \u00e9quations issues de la biologie et de la m\u00e9decine\u201d, th\u00e8se de Doctorat de l'Universit\u00e9 de Paris VI, Universit\u00e9 de Paris VI, Paris, October."},{"key":"key2022031520284351900_b4","unstructured":"Churchhouse, F.R. (1981), \u201cNumerical methods\u201d, Handbook of Applilicable Mathematics, Vol. III, Wiley, New York, NY."},{"key":"key2022031520284351900_b3","unstructured":"Ngarasta, N. (2003), \u201cEtude num\u00e9rque de quelques probl\u00e8mes de diffusion et d'\u00e9quations int\u00e9grales par la m\u00e9thode d\u00e9composionnelle d'Adomian\u201d, th\u00e8se de Doctorat de l'Universit\u00e9 de Ouagadougou, Universit\u00e9 de Ouagadougou, Ouagadougou, 10 January."},{"key":"key2022031520284351900_b2","unstructured":"Ngarasta, N. and Some, B. (2003), \u201cSolving integral equations of first kind by Adomian method\u201d, Far East Journal of Mathematical Sciences (FJMS), Vol. 8 No. 3, pp. 329\u201042."},{"key":"key2022031520284351900_b6","unstructured":"Seng, V. (1997), \u201crecherche de Formes canoniques d'Adomian pour la r\u00e9solution d'\u00e9quations fonctionnelles non lin\u00e9aires par la m\u00e9thode d\u00e9compositionnelle\u201d, th\u00e8se de Doctorat de l'Universit\u00e9 de Paris VI, Universit\u00e9 de Paris, Paris, October."},{"key":"key2022031520284351900_b5","doi-asserted-by":"crossref","unstructured":"Seng, V. and Cherruault, Y. (1997), \u201cThe resolution of non\u2010linear integral equation of first kind using the decomposition method of Adomian\u201d, Kybernetes, Vol. 26 No. 2, pp. 198\u2010206.","DOI":"10.1108\/03684929710163100"}],"container-title":["Kybernetes"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.emeraldinsight.com\/doi\/full-xml\/10.1108\/03684920910962632","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.emerald.com\/insight\/content\/doi\/10.1108\/03684920910962632\/full\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.emerald.com\/insight\/content\/doi\/10.1108\/03684920910962632\/full\/html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,24]],"date-time":"2025-07-24T23:53:38Z","timestamp":1753401218000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.emerald.com\/k\/article\/38\/5\/733-743\/273447"}},"subtitle":[],"editor":[{"given":"Yves","family":"Cherruault","sequence":"first","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2009,6,12]]},"references-count":6,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2009,6,12]]}},"alternative-id":["10.1108\/03684920910962632"],"URL":"https:\/\/doi.org\/10.1108\/03684920910962632","relation":{},"ISSN":["0368-492X"],"issn-type":[{"type":"print","value":"0368-492X"}],"subject":[],"published":{"date-parts":[[2009,6,12]]}}}