{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,11]],"date-time":"2025-11-11T13:05:02Z","timestamp":1762866302496,"version":"3.40.5"},"reference-count":23,"publisher":"Wiley","license":[{"start":{"date-parts":[[2013,1,1]],"date-time":"2013-01-01T00:00:00Z","timestamp":1356998400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2013]]},"abstract":"<jats:p>A modified<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:math>-homotopy analysis method (m<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:math>-HAM) was proposed for solving<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math>th-order nonlinear differential equations. This method improves the convergence of the series solution in the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math>HAM which was proposed in (see Hassan and El-Tawil 2011, 2012). The proposed method provides an approximate solution by rewriting the<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math>th-order nonlinear differential equation in the form of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math>first-order differential equations. The solution of these<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:math>differential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.<\/jats:p>","DOI":"10.1155\/2013\/569674","type":"journal-article","created":{"date-parts":[[2013,11,19]],"date-time":"2013-11-19T00:59:35Z","timestamp":1384822775000},"page":"1-9","source":"Crossref","is-referenced-by-count":8,"title":["Approximate Solutions of Nonlinear Partial Differential Equations by Modified<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>q<\/mml:mi><\/mml:mrow><\/mml:math>-Homotopy Analysis Method"],"prefix":"10.1155","volume":"2013","author":[{"given":"Shaheed N.","family":"Huseen","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Computer Science and Mathematics, Thi-Qar University, Thi-Qar, Iraq"}]},{"given":"Said R.","family":"Grace","sequence":"additional","affiliation":[{"name":"Engineering Mathematics Department, Faculty of Engineering, Cairo University, Giza, Egypt"}]}],"member":"311","reference":[{"key":"1","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2006.07.065"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-009-9351-7"},{"issue":"4","key":"3","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1515\/zna-2010-0404","volume":"65","year":"2010","journal-title":"Zeitschrift fur Naturforschung A"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2008.01.008"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/S0020-7225(03)00207-6"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2007.05.026"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2008.02.004"},{"key":"12","doi-asserted-by":"publisher","DOI":"10.1007\/s00707-004-0085-2"},{"year":"2004","key":"14"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7462(94)00054-E"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.1016\/S0096-3003(02)00790-7"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2004.10.058"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2007.10.014"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1007\/s10409-010-0356-1"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1016\/j.icheatmasstransfer.2009.07.012"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2006.06.006"},{"key":"21","doi-asserted-by":"crossref","first-page":"4062","DOI":"10.1016\/j.physleta.2008.03.026","volume":"372","year":"2008","journal-title":"Physics Letters A"},{"key":"22","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2007.10.007"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2007.09.012"},{"issue":"6","key":"10","doi-asserted-by":"crossref","first-page":"728","DOI":"10.1002\/mma.1400","volume":"34","year":"2011","journal-title":"Mathematical Methods in the Applied Sciences"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2012.06.065"},{"issue":"15","key":"8","first-page":"51","volume":"8","year":"2012","journal-title":"International Journal of Applied Mathematics and Mechanics"},{"issue":"10","key":"9","first-page":"481","volume":"8","year":"2013","journal-title":"International Journal of Contemporary Mathematical Sciences"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/569674.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/569674.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2013\/569674.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,8,8]],"date-time":"2020-08-08T11:09:18Z","timestamp":1596884958000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2013\/569674\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013]]},"references-count":23,"alternative-id":["569674","569674"],"URL":"https:\/\/doi.org\/10.1155\/2013\/569674","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2013]]}}}