{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T08:17:26Z","timestamp":1780733846300,"version":"3.54.1"},"reference-count":25,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,5,21]],"date-time":"2020-05-21T00:00:00Z","timestamp":1590019200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>This paper proposes a new definition of fractional derivative with non-singular kernel in the sense of Caputo which generalizes various forms existing in the literature. Furthermore, the version in the sense of Riemann\u2013Liouville is defined. Moreover, fundamental properties of the new generalized fractional derivatives in the sense of Caputo and Riemann\u2013Liouville are rigorously studied. Finally, an application in epidemiology as well as in virology is presented.<\/jats:p>","DOI":"10.3390\/computation8020049","type":"journal-article","created":{"date-parts":[[2020,5,21]],"date-time":"2020-05-21T11:31:18Z","timestamp":1590060678000},"page":"49","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":143,"title":["A New Generalized Definition of Fractional Derivative with Non-Singular Kernel"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5032-3639","authenticated-orcid":false,"given":"Khalid","family":"Hattaf","sequence":"first","affiliation":[{"name":"Centre R\u00e9gional des M\u00e9tiers de l\u2019Education et de la Formation (CRMEF), Derb Ghalef 20340, Casablanca, Morocco"},{"name":"Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M\u2019sik, Hassan II University of Casablanca, P.O. 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Neurosci."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Boukhouima, A., Hattaf, K., and Yousfi, N. (2019). Modeling the Memory and Adaptive Immunity in Viral Infection. Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, Springer.","DOI":"10.1007\/978-3-030-23433-1_18"},{"key":"ref_5","first-page":"8372140","article-title":"Dynamics of a fractional order HIV infection model with specific functional response and cure rate","volume":"2017","author":"Boukhouima","year":"2017","journal-title":"Int. J. Differ. Equ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1736","DOI":"10.1002\/mma.5999","article-title":"Mathematical and numerical analysis of a three-species predator-prey model with herd behavior and time fractional-order derivative","volume":"43","author":"Ghanbari","year":"2020","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"063103","DOI":"10.1063\/1.5094546","article-title":"Numerical solution of predator-prey model with Beddington-DeAngelis functional response and fractional derivatives with Mittag-Leffler kernel","volume":"29","author":"Ghanbari","year":"2019","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"109811","DOI":"10.1016\/j.chaos.2020.109811","article-title":"Chaotic behaviour of fractional predator-prey dynamical system","volume":"135","author":"Kumar","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"109438","DOI":"10.1016\/j.chaos.2019.109438","article-title":"Fractional economic models based on market equilibrium in the frame of different type kernels","volume":"130","author":"Acay","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1101\/SQB.1933.001.01.014","article-title":"Electric conductance of biological systems","volume":"1","author":"Cole","year":"1933","journal-title":"Cold Spring Harb. Symp. Quant. Biol."},{"key":"ref_11","first-page":"17","article-title":"Introduction to transfer and motion in fractal media: The geometry of kinetics","volume":"9","author":"Crepy","year":"1983","journal-title":"Solid State Ionics"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"109467","DOI":"10.1016\/j.chaos.2019.109467","article-title":"Similarities in a fifth-order evolution equation with and with no singular kernel","volume":"130","author":"Goufo","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"109619","DOI":"10.1016\/j.chaos.2020.109619","article-title":"A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative","volume":"133","author":"Ghanbari","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Kumar, S., Kumar, R., Singh, J., Nisar, K.S., and Kumar, D. (2020). An efficient numerical scheme for fractional model of HIV-1 infection of CD4+ T-cells with the effect of antiviral drug therapy. Alex. Eng. J.","DOI":"10.1016\/j.aej.2019.12.046"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Alshabanat, A., Jleli, M., Kumar, S., and Samet, B. (2020). Generalization of Caputo-Fabrizio Fractional Derivative and Applications to Electrical Circuits. Front. Phys.","DOI":"10.3389\/fphy.2020.00064"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1340","DOI":"10.1016\/j.aml.2011.11.041","article-title":"A new fractional analytical approach via a modified Riemann\u2013Liouville derivative","volume":"25","author":"Khan","year":"2012","journal-title":"Appl. Math. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"123806","DOI":"10.1016\/j.physa.2019.123806","article-title":"Certain new models of the multi space-fractional Gardner equation","volume":"545","author":"Alderremy","year":"2020","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_18","first-page":"73","article-title":"A new definition of fractional derivative without singular kernel","volume":"1","author":"Caputo","year":"2015","journal-title":"Prog. Fract. Differ. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"763","DOI":"10.2298\/TSCI160111018A","article-title":"New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model","volume":"20","author":"Atangana","year":"2016","journal-title":"Therm. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1186\/s13662-019-2471-z","article-title":"On weighted Atangana\u2013Baleanu fractional operators","volume":"2020","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_21","unstructured":"Ross, R. (1911). The Prevention of Malaria, Murray. [2nd ed.]."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1136\/bmj.1.2830.546","article-title":"Some a priori pathometric equations","volume":"1","author":"Ross","year":"1915","journal-title":"Br. Med. J."},{"key":"ref_23","first-page":"700","article-title":"A Contribution to the Mathematical Theory of Epidemics","volume":"115","author":"Kermack","year":"1927","journal-title":"Proc. R. Soc. Lond."},{"key":"ref_24","first-page":"383","article-title":"Mathematical model of the influenza A (H1N1) infection","volume":"1","author":"Hattaf","year":"2009","journal-title":"Adv. Stud. Biol."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14232\/ejqtde.2013.1.3","article-title":"A delayed SIR epidemic model with general incidence rate","volume":"3","author":"Hattaf","year":"2013","journal-title":"Electron. J. Qual. Theory Differ. 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