{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T16:31:32Z","timestamp":1772555492985,"version":"3.50.1"},"reference-count":34,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T00:00:00Z","timestamp":1655078400000},"content-version":"vor","delay-in-days":163,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2022,1,1]],"date-time":"2022-01-01T00:00:00Z","timestamp":1640995200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2022,1]]},"abstract":"<jats:p>In this paper, we investigate the numerical solution of the coupled fractional massive Thirring equation with the aid of He\u2019s fractional complex transform (FCT). This study plays a significant aspect in the field of quantum physics, weakly nonlinear thrilling waves, and nonlinear optics. The main advantage of FCT is that it converts the fractional differential equation into its traditional parts and is also capable to handle the fractional order, whereas the homotopy perturbation method (HPM) is employed to tackle the nonlinear terms in the coupled fractional massive Thirring equation. An example is illustrated to present the efficiency and validity of the two\u2010scale theory. The solutions are obtained in the form of series with simple and easy computations which confirm that the present approach is good in agreement and is easy to implement for such type of complex systems in science and engineering.<\/jats:p>","DOI":"10.1155\/2022\/4141988","type":"journal-article","created":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T19:20:06Z","timestamp":1655148006000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Numerical Investigation of the Nonlinear Coupled Fractional Massive Thirring Equation Using Two\u2010Scale Approach"],"prefix":"10.1155","volume":"2022","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1071-2231","authenticated-orcid":false,"given":"Jinxing","family":"Liu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9349-4729","authenticated-orcid":false,"given":"Muhammad","family":"Nadeem","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2797-1865","authenticated-orcid":false,"given":"Mustafa","family":"Habib","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2401-5437","authenticated-orcid":false,"given":"Shazia","family":"Karim","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1687-623X","authenticated-orcid":false,"given":"Harun","family":"Or Roshid","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2022,6,13]]},"reference":[{"key":"e_1_2_9_1_2","volume-title":"Fractional Integrals and Derivatives (Theory and Applications)","author":"Kilbas A. A.","year":"1993"},{"key":"e_1_2_9_2_2","volume-title":"An Introduction to the Fractional Calculus and Fractional Differential Equations","author":"Miller K. S.","year":"1993"},{"key":"e_1_2_9_3_2","doi-asserted-by":"publisher","DOI":"10.1155\/s0161171203301486"},{"key":"e_1_2_9_4_2","doi-asserted-by":"publisher","DOI":"10.1080\/09537325.2021.1979205"},{"key":"e_1_2_9_5_2","doi-asserted-by":"publisher","DOI":"10.1177\/09717218211020476"},{"key":"e_1_2_9_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijin.2020.05.001"},{"key":"e_1_2_9_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.techfore.2022.121727"},{"key":"e_1_2_9_8_2","doi-asserted-by":"publisher","DOI":"10.1186\/1687-1847-2013-18"},{"key":"e_1_2_9_9_2","doi-asserted-by":"publisher","DOI":"10.1080\/00207160.2018.1487059"},{"key":"e_1_2_9_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2011.01.111"},{"key":"e_1_2_9_11_2","doi-asserted-by":"publisher","DOI":"10.1155\/2011\/587068"},{"key":"e_1_2_9_12_2","doi-asserted-by":"publisher","DOI":"10.1155\/2013\/160681"},{"key":"e_1_2_9_13_2","doi-asserted-by":"publisher","DOI":"10.1186\/s13662-019-2044-1"},{"key":"e_1_2_9_14_2","doi-asserted-by":"publisher","DOI":"10.1155\/2019\/3456848"},{"key":"e_1_2_9_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2011.09.078"},{"key":"e_1_2_9_16_2","doi-asserted-by":"publisher","DOI":"10.1186\/s13662-021-03247-6"},{"key":"e_1_2_9_17_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0096-3003(01)00312-5"},{"key":"e_1_2_9_18_2","first-page":"28","article-title":"Darboux transformation for soliton solutions of the modified Kadomtsev-Petviashvili-ii equation","volume":"3","author":"Ali M. R.","year":"2018","journal-title":"Communication in Mathematical Modeling and Applications"},{"key":"e_1_2_9_19_2","doi-asserted-by":"publisher","DOI":"10.1515\/nleng-2018-0027"},{"key":"e_1_2_9_20_2","first-page":"1","article-title":"Analytic and numerical solutions of time-fractional linear Schr\u00f6dinger equation","volume":"7","author":"Edeki S.","year":"2016","journal-title":"Communications in Mathematics and Applications"},{"key":"e_1_2_9_21_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0034-4877(16)30002-7"},{"key":"e_1_2_9_22_2","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4916(58)90015-0"},{"key":"e_1_2_9_23_2","doi-asserted-by":"publisher","DOI":"10.3390\/mca15050970"},{"key":"e_1_2_9_24_2","doi-asserted-by":"publisher","DOI":"10.1142\/s1793557119500451"},{"key":"e_1_2_9_25_2","doi-asserted-by":"publisher","DOI":"10.1088\/1402-4896\/abb420"},{"key":"e_1_2_9_26_2","doi-asserted-by":"publisher","DOI":"10.1103\/physrevd.100.094504"},{"key":"e_1_2_9_27_2","doi-asserted-by":"publisher","DOI":"10.3390\/e23121646"},{"key":"e_1_2_9_28_2","doi-asserted-by":"publisher","DOI":"10.2298\/tsci2004507w"},{"key":"e_1_2_9_29_2","doi-asserted-by":"publisher","DOI":"10.1515\/ijnsns.2010.11.s1.335"},{"key":"e_1_2_9_30_2","doi-asserted-by":"publisher","DOI":"10.2298\/tsci1904131h"},{"key":"e_1_2_9_31_2","doi-asserted-by":"publisher","DOI":"10.1108\/hff-11-2020-0703"},{"key":"e_1_2_9_32_2","doi-asserted-by":"publisher","DOI":"10.1142\/s0218348x20501418"},{"key":"e_1_2_9_33_2","doi-asserted-by":"publisher","DOI":"10.1177\/1461348418817869"},{"key":"e_1_2_9_34_2","doi-asserted-by":"publisher","DOI":"10.1177\/1461348418818973"}],"container-title":["Complexity"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2022\/4141988","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1155\/2022\/4141988","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/2022\/4141988","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T20:59:43Z","timestamp":1769115583000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/2022\/4141988"}},"subtitle":[],"editor":[{"given":"Yi","family":"Su","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2022,1]]},"references-count":34,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,1]]}},"alternative-id":["10.1155\/2022\/4141988"],"URL":"https:\/\/doi.org\/10.1155\/2022\/4141988","archive":["Portico"],"relation":{},"ISSN":["1076-2787","1099-0526"],"issn-type":[{"value":"1076-2787","type":"print"},{"value":"1099-0526","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1]]},"assertion":[{"value":"2022-05-04","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2022-05-28","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2022-06-13","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"4141988"}}