{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T22:17:56Z","timestamp":1775081876253,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2016,2,5]],"date-time":"2016-02-05T00:00:00Z","timestamp":1454630400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this work; we present a method for solving the second-order linear ordinary differential equation of hypergeometric type. The solutions of this equation are given by the confluent hypergeometric functions (CHFs). Unlike previous studies, we obtain some different new solutions of the equation without using the CHFs. Therefore, we obtain new discrete fractional solutions of the homogeneous and non-homogeneous confluent hypergeometric differential equation (CHE) by using a discrete fractional Nabla calculus operator. Thus, we obtain four different new discrete complex fractional solutions for these equations.<\/jats:p>","DOI":"10.3390\/e18020049","type":"journal-article","created":{"date-parts":[[2016,2,5]],"date-time":"2016-02-05T10:06:16Z","timestamp":1454666776000},"page":"49","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Particular Solutions of the Confluent Hypergeometric Differential Equation by Using the Nabla Fractional Calculus Operator"],"prefix":"10.3390","volume":"18","author":[{"given":"Resat","family":"Yilmazer","sequence":"first","affiliation":[{"name":"Department of Mathematics, Science Faculty, F\u0131rat University, Elaz\u0131\u011f 23119, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4996-8373","authenticated-orcid":false,"given":"Mustafa","family":"Inc","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Science Faculty, F\u0131rat University, Elaz\u0131\u011f 23119, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7855-508X","authenticated-orcid":false,"given":"Fairouz","family":"Tchier","sequence":"additional","affiliation":[{"name":"Department of Mathematics, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, \u00c7ankaya University, Ankara 06530, Turkey"},{"name":"Institute of Space Sciences, P.O. Box MG-23, Magurele-Bucharest RO-76911, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,2,5]]},"reference":[{"key":"ref_1","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equation, Wiley. [1st ed.]."},{"key":"ref_2","unstructured":"Oldham, K.B., and Spanier, J. (1974). The Fractional Calculus, Academic Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Baleanu, D., Guven, Z.B., and Machado, J.A.T. (2010). New Trends in Nanotechnology and Fractional Calculus Applications, Springer.","DOI":"10.1007\/978-90-481-3293-5"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.1063\/1.532809","article-title":"An evolutionary equation in thermoelasticity of dipolar bodies","volume":"40","author":"Marin","year":"1999","journal-title":"J. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/S0020-7225(97)00019-0","article-title":"Thermoelasticity of initially stressed bodies, asymptotic equipartition of energies","volume":"36","author":"Marin","year":"1998","journal-title":"Int. J. Eng. Sci."},{"key":"ref_6","first-page":"165","article-title":"A transform method in discrete fractional calculus","volume":"2","author":"Atici","year":"2007","journal-title":"Int. J. Differ. Equ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1090\/S0002-9939-08-09626-3","article-title":"Initial value problems in discrete fractional calculus","volume":"137","author":"Atici","year":"2009","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14232\/ejqtde.2009.4.0","article-title":"Discrete fractional calculus with the nabla operator","volume":"3","author":"Eloe","year":"2009","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmaa.2010.02.009","article-title":"Modeling with fractional difference equations","volume":"369","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_10","first-page":"41","article-title":"Combined Delta-Nabla Sum Operator in Discrete Fractional Calculus","volume":"1","author":"Bastos","year":"2010","journal-title":"Commun. Frac. Calc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"153","DOI":"10.4067\/S0719-06462011000300009","article-title":"Sum and difference compositions in discrete fractional calculus","volume":"13","author":"Holm","year":"2011","journal-title":"CUBO A Math. J."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"270106","DOI":"10.1155\/2012\/270106","article-title":"On Sumudu transform method in discrete fractional calculus","volume":"2012","author":"Jarad","year":"2012","journal-title":"Abstr. Appl. Anal."},{"key":"ref_13","first-page":"125","article-title":"Solutions of perturbed nonlinear nabla fractional difference equations","volume":"43","author":"Mohan","year":"2013","journal-title":"Novi Sad J. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"7","DOI":"10.26637\/mjm103\/002","article-title":"Solutions of fractional difference equations using S-transforms","volume":"3","author":"Mohan","year":"2013","journal-title":"Malaya J. Math."},{"key":"ref_15","first-page":"228","article-title":"De integralibus quibusdam definitis et seriebus infinitis. J. f\u00fcr die reine und angewandte","volume":"17","author":"Kummer","year":"1837","journal-title":"Mathematik"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/S0377-0427(00)00706-8","article-title":"On some solutions of the extended confluent hypergeometric differential equation","volume":"137","author":"Campos","year":"2001","journal-title":"J. Comput. Appl. Math."},{"key":"ref_17","first-page":"974","article-title":"Generalized entropies and the Langevin and Fokker-Planck equations","volume":"40","author":"Akimoto","year":"2002","journal-title":"J. Korean Phys. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4028","DOI":"10.3390\/e17064028","article-title":"Generalized boundary conditions for the time-fractional advection diffusion equation","volume":"17","author":"Povstenko","year":"2015","journal-title":"Entropy"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2350","DOI":"10.3390\/e16042350","article-title":"Fractional order generalized information","volume":"16","author":"Machado","year":"2014","journal-title":"Entropy"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"109","DOI":"10.5890\/JAND.2012.03.001","article-title":"Entropy analysis of fractional derivatives and their approximation","volume":"1","author":"Machado","year":"2012","journal-title":"J. Appl. Nonlinear Dyn."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.micromeso.2013.02.054","article-title":"Characterization of anomalous diffusion in porous biological tissues using fractional order derivatives and entropy","volume":"178","author":"Magin","year":"2013","journal-title":"Microporous Mesoporous Mater."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5838","DOI":"10.3390\/e16115838","article-title":"New insights into the fractional order diffusion equation using entropy and kurtosis","volume":"16","author":"Ingo","year":"2014","journal-title":"Entropy"},{"key":"ref_23","first-page":"489","article-title":"N-fractional calculus operator N\u00b5 method to a modified hydrogen atom equation","volume":"15","author":"Yilmazer","year":"2010","journal-title":"Math. Commun."},{"key":"ref_24","unstructured":"Kelley, W.G., and Peterson, A.C. (2001). Difference Equations: An Introduction with Applications, Academic Press."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/2\/49\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:18:48Z","timestamp":1760210328000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/2\/49"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,2,5]]},"references-count":24,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2016,2]]}},"alternative-id":["e18020049"],"URL":"https:\/\/doi.org\/10.3390\/e18020049","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,2,5]]}}}