{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T07:55:48Z","timestamp":1648799748236},"reference-count":10,"publisher":"Hindawi Limited","license":[{"start":{"date-parts":[[2016,1,1]],"date-time":"2016-01-01T00:00:00Z","timestamp":1451606400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["41076013"],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2016]]},"abstract":"This study expresses the solution of the Bessel equation in the neighbourhood ofx<\/mml:mi>=<\/mml:mo>\u221e<\/mml:mi><\/mml:math>as the product of a known-form singular divisor and a specific nonsingular function, which satisfies the corresponding derived equation. Considering the failure of the traditional irregular solution constructed with the power series, we adopt the corrected Fourier series with only limited smooth degree to approximate the nonsingular function in the interval[<\/mml:mo>x<\/mml:mi><\/mml:mrow>0<\/mml:mn><\/mml:mrow><\/mml:msub>,<\/mml:mo>\u221e<\/mml:mi>]<\/mml:mo><\/mml:math>. In order to guarantee the series\u2019 uniform convergence and uniform approximation to the derived equation, we introduce constraint and compatibility conditions and hence completely determine all undetermined coefficients of the corrected Fourier series. Thus, what we found is not an asymptotic solution atx<\/mml:mi>\u2192<\/mml:mo>\u221e<\/mml:mi><\/mml:math>(not to mention a so-called formal solution), but a solution in the interval[<\/mml:mo>x<\/mml:mi><\/mml:mrow>0<\/mml:mn><\/mml:mrow><\/mml:msub>,<\/mml:mo>\u221e<\/mml:mi>]<\/mml:mo><\/mml:math>with certain regularities of distribution. During the solution procedure, there is no limitation on the coefficient property of the equation; that is, the coefficients of the equation can be any complex constant, so that the solution method presented here is universal.<\/jats:p>","DOI":"10.1155\/2016\/6826482","type":"journal-article","created":{"date-parts":[[2016,7,31]],"date-time":"2016-07-31T17:03:10Z","timestamp":1469984590000},"page":"1-7","source":"Crossref","is-referenced-by-count":0,"title":["Bessel Equation in the Semiunbounded Intervalx<\/mml:mi>\u2208<\/mml:mo>[<\/mml:mo>x<\/mml:mi><\/mml:mrow>0<\/mml:mn><\/mml:mrow><\/mml:msub>,<\/mml:mo>\u221e<\/mml:mi>]<\/mml:mo><\/mml:math>: Solving in the Neighbourhood of an Irregular Singular Point"],"prefix":"10.1155","volume":"2016","author":[{"given":"Qing-Hua","family":"Zhang","sequence":"first","affiliation":[{"name":"The First Institute of Oceanography, State Oceanic Administration, Qingdao 266061, China"}]},{"ORCID":"http:\/\/orcid.org\/0000-0002-9091-2040","authenticated-orcid":true,"given":"Jian","family":"Ma","sequence":"additional","affiliation":[{"name":"College of Marine Sciences, Shanghai Ocean University, Shanghai 201306, China"}]},{"given":"Yuanyuan","family":"Qu","sequence":"additional","affiliation":[{"name":"The First Institute of Oceanography, State Oceanic Administration, Qingdao 266061, China"}]}],"member":"98","reference":[{"key":"1","year":"1972"},{"key":"2","year":"1991"},{"key":"3","year":"1993"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-05221-7"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1155\/ijmms.2005.33"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2006.08.155"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2016.05.028"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2010.11.001"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2011.07.051"},{"issue":"2","key":"10","first-page":"273","volume":"7","year":"2013","journal-title":"Malaysian Journal of Mathematical Sciences"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2016\/6826482.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2016\/6826482.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2016\/6826482.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2016,7,31]],"date-time":"2016-07-31T17:03:13Z","timestamp":1469984593000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/ijmms\/2016\/6826482\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016]]},"references-count":10,"alternative-id":["6826482","6826482"],"URL":"https:\/\/doi.org\/10.1155\/2016\/6826482","relation":{},"ISSN":["0161-1712","1687-0425"],"issn-type":[{"value":"0161-1712","type":"print"},{"value":"1687-0425","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016]]}}}