{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T19:17:13Z","timestamp":1770059833680,"version":"3.49.0"},"reference-count":38,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,21]],"date-time":"2022-02-21T00:00:00Z","timestamp":1645401600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Differential problems with the Riesz derivative in space are widely used to model anomalous diffusion. Although the Riesz\u2013Caputo derivative is more suitable for modeling real phenomena, there are few examples in literature where numerical methods are used to solve such differential problems. In this paper, we propose to approximate the Riesz\u2013Caputo derivative of a given function with a cubic spline. As far as we are aware, this is the first time that cubic splines have been used in the context of the Riesz\u2013Caputo derivative. To show the effectiveness of the proposed numerical method, we present numerical tests in which we compare the analytical solution of several boundary differential problems which have the Riesz\u2013Caputo derivative in space with the numerical solution we obtain by a spline collocation method. The numerical results show that the proposed method is efficient and accurate.<\/jats:p>","DOI":"10.3390\/a15020069","type":"journal-article","created":{"date-parts":[[2022,2,21]],"date-time":"2022-02-21T20:24:21Z","timestamp":1645475061000},"page":"69","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Approximation of the Riesz\u2013Caputo Derivative by Cubic Splines"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7159-0533","authenticated-orcid":false,"given":"Francesca","family":"Pitolli","sequence":"first","affiliation":[{"name":"Department SBAI, Universit\u00e0 di Roma \u201cLa Sapienza\u201d, Via Antonio Scarpa 16, 00161 Rome, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1370-5694","authenticated-orcid":false,"given":"Chiara","family":"Sorgentone","sequence":"additional","affiliation":[{"name":"Department SBAI, Universit\u00e0 di Roma \u201cLa Sapienza\u201d, Via Antonio Scarpa 16, 00161 Rome, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6209-3028","authenticated-orcid":false,"given":"Enza","family":"Pellegrino","sequence":"additional","affiliation":[{"name":"Department DIIIE, University of L\u2019Aquila, E. Pontieri 2, Roio Poggio, 67040 L\u2019Aquila, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hilfer, R. (2000). Applications of Fractional Calculus in Physics, World Scientific.","DOI":"10.1142\/9789812817747"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1016\/S0370-1573(02)00331-9","article-title":"Chaos, fractional kinetics, and anomalous transport","volume":"371","author":"Zaslavsky","year":"2002","journal-title":"Phys. Rep."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s11071-004-3764-6","article-title":"A general formulation and solution scheme for fractional optimal control problems","volume":"38","author":"Agrawal","year":"2004","journal-title":"Nonlinear Dyn."},{"key":"ref_4","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). 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