{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T21:40:02Z","timestamp":1723066802280},"reference-count":0,"publisher":"Wiley","issue":"51","license":[{"start":{"date-parts":[[2000,1,1]],"date-time":"2000-01-01T00:00:00Z","timestamp":946684800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2004,1]]},"abstract":"<jats:p>Sophus Lie developed a systematic way to solve ODEs. He found that\ntransformations which form a continuous group and leave a\ndifferential equation invariant can be used to simplify the\nequation. Lie\u2032s method uses the infinitesimal generator of these\npoint transformations. These are symmetries of the equation\nmapping solutions into solutions. Lie\u2032s methods did not find\nwidespread use in part because the calculations for the\ninfinitesimals were quite lengthy, needing to calculate the\nprolongations of the infinitesimal generator. Nowadays,\nprolongations are obtained using Maple or Mathematica, and Lie\u2032s\ntheory has come back to the attention of researchers. In general,\nthe computation of the coefficients of the (<jats:italic>n<\/jats:italic>)\u2010prolongation is\ndone using recursion formulas. Others have given methods that do\nnot require recursion but use Fr\u00e9chet derivatives. In this\npaper, we present a combinatorial approach to explicitly write the\ncoefficients of the prolongations. Besides being\nnovel, this approach was found to be useful by the authors for\ndidactical and combinatorial purposes, as we show in the examples.<\/jats:p>","DOI":"10.1155\/s016117120430904x","type":"journal-article","created":{"date-parts":[[2004,10,20]],"date-time":"2004-10-20T09:52:31Z","timestamp":1098265951000},"page":"2741-2753","update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Coefficients of prolongations for symmetries of ODEs"],"prefix":"10.1155","volume":"2004","author":[{"given":"Ricardo","family":"Alfaro","sequence":"first","affiliation":[]},{"given":"Jim","family":"Schaeferle","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2004,10,20]]},"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2004\/754503.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2004\/754503.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/S016117120430904X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,7]],"date-time":"2024-08-07T21:07:06Z","timestamp":1723064826000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/S016117120430904X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,1]]},"references-count":0,"journal-issue":{"issue":"51","published-print":{"date-parts":[[2004,1]]}},"alternative-id":["10.1155\/S016117120430904X"],"URL":"https:\/\/doi.org\/10.1155\/s016117120430904x","archive":["Portico"],"relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"type":"print","value":"0161-1712"},{"type":"electronic","value":"1687-0425"}],"subject":[],"published":{"date-parts":[[2004,1]]},"assertion":[{"value":"2003-09-04","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2004-10-20","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}