{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:23:14Z","timestamp":1760235794624,"version":"build-2065373602"},"reference-count":55,"publisher":"MDPI AG","issue":"19","license":[{"start":{"date-parts":[[2021,9,30]],"date-time":"2021-09-30T00:00:00Z","timestamp":1632960000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>This paper studies the discretization of fractional operators by means of advanced clustering methods. The Gr\u00fcnwald\u2013Letnikov fractional operator is approximated by series generated by the Euler, Tustin and generalized mean. The series for different fractional orders form the objects to be assessed. For this purpose, the several distances associated with the hierarchical clustering and multidimensional scaling computational techniques are tested. The Arc-cosine distance and the 3-dim multidimensional scaling produce good results. The visualization of the graphical representations allows a better understanding of the properties embedded in each type of approximation of the fractional operators.<\/jats:p>","DOI":"10.3390\/math9192429","type":"journal-article","created":{"date-parts":[[2021,10,11]],"date-time":"2021-10-11T01:57:54Z","timestamp":1633917474000},"page":"2429","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Discretization of Fractional Operators: Analysis by Means of Advanced Computational Techniques"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4274-4879","authenticated-orcid":false,"given":"Jose","family":"Tenreiro Machado","sequence":"first","affiliation":[{"name":"Institute of Engineering, Polytechnic of Porto, Rua Dr. Ant\u00f3nio Bernardino de Almeida, 431, 4249-015 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8262-1369","authenticated-orcid":false,"given":"Alexandra M.","family":"Galhano","sequence":"additional","affiliation":[{"name":"Faculdade de Ci\u00eancias Naturais, Engenharias e Tecnologias, Universidade Lus\u00f3fona do Porto, Rua Augusto Rosa 24, 4000-098 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8577-2779","authenticated-orcid":false,"given":"Carla S.","family":"Cordeiro","sequence":"additional","affiliation":[{"name":"Faculdade de Ci\u00eancias Naturais, Engenharias e Tecnologias, Universidade Lus\u00f3fona do Porto, Rua Augusto Rosa 24, 4000-098 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,30]]},"reference":[{"key":"ref_1","unstructured":"Oldham, K., and Spanier, J. 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