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The least\u2010squares residual power series method combines the residual power series method with the least\u2010squares method. These calculations depend on the sense of Caputo. Firstly, using the classic residual power series method, the analytical solution can be solved. Secondly, the concept of fractional Wronskian is introduced, which is applied to validate the linear independence of the functions. Thirdly, a linear combination of the first few terms as an approximate solution is used, which contains unknown coefficients. Finally, the least\u2010squares method is proposed to obtain the unknown coefficients. The approximate solutions are solved by the least\u2010squares residual power series method with the fewer expansion terms than the classic residual power series method. The examples are shown in datum and images.The examples show that the new method has an accelerate convergence than the classic residual power series method.<\/jats:p>","DOI":"10.1155\/2019\/6159024","type":"journal-article","created":{"date-parts":[[2019,10,30]],"date-time":"2019-10-30T23:31:21Z","timestamp":1572478281000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":38,"title":["Least\u2010Squares Residual Power Series Method for the Time\u2010Fractional Differential Equations"],"prefix":"10.1155","volume":"2019","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0424-5067","authenticated-orcid":false,"given":"Jianke","family":"Zhang","sequence":"first","affiliation":[]},{"given":"Zhirou","family":"Wei","sequence":"additional","affiliation":[]},{"given":"Lifeng","family":"Li","sequence":"additional","affiliation":[]},{"given":"Chang","family":"Zhou","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2019,10,30]]},"reference":[{"key":"e_1_2_9_1_2","doi-asserted-by":"publisher","DOI":"10.1109\/tie.2014.2362094"},{"key":"e_1_2_9_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2014.05.029"},{"key":"e_1_2_9_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-016-9489-5"},{"key":"e_1_2_9_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.asej.2014.03.006"},{"key":"e_1_2_9_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2008.07.002"},{"key":"e_1_2_9_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2010.10.004"},{"key":"e_1_2_9_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/s40096-015-0141-1"},{"key":"e_1_2_9_8_2","doi-asserted-by":"publisher","DOI":"10.3390\/e17096519"},{"key":"e_1_2_9_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijleo.2018.02.099"},{"key":"e_1_2_9_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-017-3820-7"},{"key":"e_1_2_9_11_2","first-page":"664","article-title":"Analytical approximate solutions of systems of multi-pantograph delay differential equations using residual power-series method","volume":"8","author":"Komashynska I.","year":"2014","journal-title":"Australian Journal of Basic and Applied Science"},{"key":"e_1_2_9_12_2","doi-asserted-by":"publisher","DOI":"10.5899\/2016\/cna-00235"},{"key":"e_1_2_9_13_2","doi-asserted-by":"publisher","DOI":"10.22436\/jnsa.009.11.10"},{"key":"e_1_2_9_14_2","doi-asserted-by":"crossref","unstructured":"AbuteenE.andFreihetA. 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