{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T07:22:20Z","timestamp":1763018540897,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,26]],"date-time":"2022-12-26T00:00:00Z","timestamp":1672012800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Basque Government","award":["IT1555-22","KK-2022\/00090 MCIN\/AEI 269.10.13039\/ 501100011033","PID2021-1235430B-C21\/C22"],"award-info":[{"award-number":["IT1555-22","KK-2022\/00090 MCIN\/AEI 269.10.13039\/ 501100011033","PID2021-1235430B-C21\/C22"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Numerical methods play an important role in modern mathematical research, especially studying the symmetry analysis and obtaining the numerical solutions of fractional differential equation. In the current work, we use two numerical schemes to deal with fractional differential equations. In the first case, a combination of the group preserving scheme and fictitious time integration method (FTIM) is considered to solve the problem. Firstly, we applied the FTIM role, and then the GPS came to integrate the obtained new system using initial conditions. Figure and tables containing the solutions are provided. The tabulated numerical simulations are compared with the reproducing kernel Hilbert space method (RKHSM) as well as the exact solution. The methodology of RKHSM mainly relies on the right choice of the reproducing kernel functions. The results confirm that the FTIM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed methods.<\/jats:p>","DOI":"10.3390\/sym15010065","type":"journal-article","created":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T07:02:12Z","timestamp":1672124532000},"page":"65","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Analysis of the Fractional Differential Equations Using Two Different Methods"],"prefix":"10.3390","volume":"15","author":[{"given":"Mohammad","family":"Partohaghighi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Clarkson University, Potsdam, NY 13699, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9832-1424","authenticated-orcid":false,"given":"Ali","family":"Akg\u00fcl","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Mathematics, Lebanese American University, Beirut 1102 2801, Lebanon"},{"name":"Department of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, T\u00fcrkiye"},{"name":"Mathematics Research Center, Department of Mathematics, Near East University, Near East Boulevard, Nicosia 99138, T\u00fcrkiye"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Esra Karatas","family":"Akg\u00fcl","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, T\u00fcrkiye"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nourhane","family":"Attia","sequence":"additional","affiliation":[{"name":"\u00c9cole Nationale Sup\u00e9rieure des Sciences de la Mer et de l\u2019Am\u00e9nagement du Littoral (ENSSMAL), Campus Universitaire de Dely Ibrahim, Bois des Cars, B.P. 19, Alger 16320, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9320-9433","authenticated-orcid":false,"given":"Manuel","family":"De la Sen","sequence":"additional","affiliation":[{"name":"Department of Electricity and Electronics, Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, 48940 Leioa, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mustafa","family":"Bayram","sequence":"additional","affiliation":[{"name":"Department of Computer Engineering, Biruni University, Istanbul 34010, T\u00fcrkiye"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,26]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"M\u00e9moire sur quelques questions de g\u00e9om\u00e9trie et de m\u00e9canique, et sur un nouveau genre de calcul pour r\u00e9soudre ces questions","volume":"13","author":"Liouville","year":"1832","journal-title":"JEC Polytech."},{"key":"ref_2","unstructured":"Riemann, G.F.B. 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